Talk:Derivative/Archive 2

Archive 1

Archive 2

Archive 3

GA Review

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS):

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   a (fair representation): b (all significant views):

5. It is stable.

   

6. It contains images, where possible, to illustrate the topic.

   a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):

7. Overall:

   a Pass/Fail:

As 2:b is disputed, I'll declare it as void, and approve this article. _→AzaToth 13:01, 10 October 2006 (UTC)

Additional remarks

This article has great impact as it covers a topic that is not too esoteric and is familiar to most encyclopaedia readers. As another GA reviewer, I would like to add some suggestions on improving this article. Some of these suggestions are simply so that the article conforms to the mathematics Manual of Style and the Wikipedia-generic Manual of Style guidelines. These suggestions should be implemented as soon as possible so that the article is representative of a Good Article.

• The current lead section is broken up into short two-sentence paragraphs with a bulleted list included within it. Following the guidelines of the MoS on the lead section, this section should be rewritten into two or three paragraphs without the bulleted list. It should provide an overview of the main points and should be capable of standing alone. Self-referential statements such as, “The remainder of this article discusses only the simplest case (real-valued functions of real numbers)” should be avoided. The main text covers the simplest case and a final section adds information for generalisations, so such statements are not needed.
• According to the mathematics MoS, there should be some introductory paragraphs, which describes the subject in general terms. This is particularly important for a basic mathematics article. Although this article is supposed to be a break-out article from the main Calculus article, an introduction section would make the article complete in itself which is important if the reader came across the article from another path. It has also been suggested to connect this article with the History of calculus article.
• Also as noted in the mathematics Manual of Style, the use of the first person (“we”) should be avoided.
• In the Physics section, the first clause “Arguably the most important application” is an example of WP:WEASEL words. If it is the most important, then just say so, otherwise please provide a citation where someone states that it is an arguable claim. Another example is “it is usually rather easy to get a rough idea of…”. This should be reworded or dropped.
• As this article covers an uncontroversial topic, it would help to cite to one or two authoritative sources at the start of the article in order to conform to verifiability criteria. The Physics Wikiproject has a proposal on how to deal with this kind of citation. Please consider adding this cite to the sources you have listed under References.

Other comments:

• In Leibniz's notation section, the last sentence is a parenthetical element. Is there a reason why?
• The Physics section has a good examples. Could other examples be provided from other fields? If necessary, the physics example could be shortened to add other examples.

I hope that you find these suggestions helpful in improving your article on the way toward Featured Article status. RelHistBuff 11:51, 11 October 2006 (UTC)

Recent revisions have addressed these helpful suggestions in the following way.

• The new lead section now comprises four short paragraphs and stands alone .
• The idea of differentiation is introduced in the first section before a formal definition is given.
• The History of Differentiation has been expanded using material from the History of calculus and other sources.
• The first person pronoun has (almost?) entirely been eliminated.
• Weasel words have been eliminated.
• The parenthetical element at the end of Leibniz's notation has been replaced by a footnote.

Authoritative sources have not yet been chosen for the start of the article and examples from other fields have not yet been provided. I believe that the latter cannot be done properly until a separate article on applications of derivatives has been written. However, the lead does now at least refer to the fundamental application of differential equations, and mentions applications in economics. Further comments on this article are always welcome. Geometry guy 18:23, 31 March 2007 (UTC)

rate of change - Änderungsrate (de)

A discussion about the lemma Änderungsrate in the german wikipedia lead to the conclusion, that (momentary) "rate of change" (Momentane Änderungsrate) in applied mathematics (e. g. physics) is not exactly the same as analytic derrivative. Therefor we encounter some problems in interlinking our article on Änderungsrate with the english wikipedia (interwiki). Could someone familiar with the english terminology help here? --KleinKlio 17:22, 11 October 2006 (UTC)

English "derivative" = German "Ableitung". However, the German wikipedia does not have a separate article about the mathematical meaning of that term, so I changed the interwiki to point to de:Differentialrechnung which seems the best match. -- Jitse Niesen (talk) 02:57, 12 October 2006 (UTC)

I'm sorry, I misinterpreted your question. You're looking for the proper interwiki link for de:Änderungsrate. Unfortunately, I can't find a good target in the English Wikipedia. (Note for other editors: I think that the central point at the discussion on the German WP is that "(instantaneous) rate of change" can only be used if the function depends on time, while "derivative" is more generally applicable). -- Jitse Niesen (talk) 03:14, 12 October 2006 (UTC)

Thank You for Your help so far, Jitse Niesen. Your decision to interwiki "derivative"- "Differentialrechnung" seems quite right to me. There's a lot discussion about how large (in terms of content, not words) an article should grow, but for the moment there is no "Ableitung". The article de: Änderungsrate does link to "Differentialrechnung" too, when it comes to mathematical details.

The german "Änderungsrate" is like the english "rate of change" used in a metaphorical way synonymous to derivative=Ableitung, but since in experimental physics you can sometimes measure a (formal) derivative in a independent way (for example electric current independently from electric charge), we desided to develop an extra arcticle for this. This was the main justification for doing so: Änderungsrate is (for Germans) not really a special case of derivative. But perhaps are all those german Gedankenexperimente? :-) --KleinKlio 17:01, 12 October 2006 (UTC)

By the way (I'm not a crack in English language, pardon me): Should the disambiguition sentence in the intro not say "For other meanings of this word, see ..."? --KleinKlio 17:43, 12 October 2006 (UTC)

represent

Can anybody explain me what "social science apps often use derivative + or - sign to 'represent' empirical-theory relation" or "derivatives can be used to represent many properties of a function" is supposed to mean? English is not my native language, but I know it pretty well and "represent" seems like the wrong word here. -- Jitse Niesen (talk) 08:33, 25 November 2006 (UTC)

Well, I hope that I can (although you might not be pleased with my answers;):

"derivatives can be used to represent many properties of a function"

I take it that the only term you find puzzling there is 'represent'. But in the article there [was] a [Wiktionary] link to 'represent', which if you mouse-click will take you to definitions of that term. The definition intended is #6. (One way of flagging the intended definition in the article would be include '[#6]' or '[def. 6]' after '(or represent)'.) (This was referenced in the Edit summary, which you might not have seen.)

"social science applications often use a + or - sign of a derivative to 'represent' a hypothesized relation" [slightly expanded and rephrased]

Take my field, economics (please). There it is common and standard to place a +, -, or ? sign over each argument of a function. The signs refer to the (so-called qualitative) relationships implied or hypothesized by the respective partial derivatives (see Samuelson's Foundations of Economic Analysis for a similar use). Equivalently, partial derivatives may be referenced directly, as for example by this shorthand:

δGDP/δM > 0, δGDP/δGovSpending > 0, . . .

Or, in notation for a function of only one independent variable:

dGDP/dM > 0.

These are examples of 'representing'. I appreciated your allowing me to respond, rather than simply reverting (although reverting, preferably with an Edit summary, might be quite appropriate, constructive, or otherwise helpful in many instances). Thomasmeeks 14:05, 25 November 2006 (UTC) Edited by Thomasmeeks 00:11, 26 November 2006 (UTC)

I don't quite see how this is "representing"; I may have to get a copy of Samuelson. The partial derivatives are what they are, and information can be inferred from the signs of them. I see this as a use of the derivative, not that the derivative is doing some "representing". Perhaps this is simply a matter of perspective. In any case, since this is the lead of the article, I think simplicity (without misleading) is paramount, and since "representing" is not, apparently, a simple aspect of derivative, perhaps it could be included elsewhere (say, a section somewhere on applications of derivatives in economics). Or, perhaps it could be split into another sentence so that we have one on the derivative's use to determine properties of a function, and another one on "representing". Note that the first sentence of this 2nd lead paragraph implies that the paragraph is about real-valued, single variable functions, for which partial derivatives are not applicable; so perhaps you could say something about representing in the 3rd paragraph or 4th paragraph? Doctormatt 22:46, 25 November 2006 (UTC)

Thanks for your comment. On the last, you might be right about breaking them out. I'm more focussed on the immediate question right now. I'm afraid that on that, the point I was trying to make above did not come out as plain as it should have. Let me try to respond more directly to your point. Wiktionary def. #6 of 'recognize' at represent' is:

To serve as a sign or symbol of; as, mathematical symbols represent quantities or relations; words represent ideas or things.

That seems to correspond closely to the above usage, closer than 'use' & close enough to everyday use to be helpful (possibly with a link and cite of def. 6). Thomasmeeks 03:12, 26 November 2006 (UTC)

2nd sentence of 2nd paragraph edited

2nd paragraph, 2nd sentence: Earlier edit: Derivatives can be used to determine (or represent) many properties of a function, including whether the function is increasing or decreasing on an interval and whether it has a maximum or minimum at a point.

Current edit: Derivatives can be used to characterize many properties of a function, including

• whether and at what rate the function is increasing or decreasing through a value of the function
• whether and where the function has maximum or minimum values.

Advantages: (1) It is more specific about the uses of derivatives without being complicated. (2) It removes the parenthetical suggestion of the earlier edit that 'representing' is secondary. (3) The wording of "the function is increasing or decreasing on an interval" is avoided. While this is doubtless good shorthand, someone unfamiliar with the usage might well be puzzled (unless he or she studies the increasing or decreasing links). Thomasmeeks 22:28, 27 November 2006 (UTC)

Style of introduction definition

I changed the definition of a derivative in the first sentence from:

In mathematics, a derivative is defined as the instantaneous rate of change of a function. The process of finding the derivative is called differentiation.

to:

In mathematics, a derivative is the instantaneous rate of change of a function, which is to say the rate of change at a particular point in time (or some other variable) and not the average rate of change over a period. Finding a derivative is called differentiation.

I believe that the original definition, while to-the-point, does not make sense to the person who does not already understand calculus. For example, what is an "instantaneous" rate of change? No one talks about the needle of a speedometer pointing to the "instantaneous" speed of a car. It's an obvious jargon term. My revision also removes the unnecessary "The process of" which is redundant.

Doctormatt reverted my edits. I hope we can discuss this ... otherwise I certainly feel that my version, while not perfect, is at least better than what is here.

Great - let's discuss. My feeling is that the very first sentence should be unequivocal, even if it uses a potentially confusing term (instantaneous). Your version is too confusing, with various clauses that, while well intentioned, don't make things any clearer. Why compare instantaneous rate of change to "rate of change over a period"? You claim that original first sentence does not make sense to the person who does not already understand calculus, but who other than someone who has studied calculus has used the term "rate of change over a period", or, for that matter, "rate of change of a function"?

I really think we can do better than to have a "which is to say" phrase in the very first sentence of such a fundamental mathematical concept.

(Truth be told, I don't like this "definition" at all. The derivative is not defined as the instantaneous rate of change of a function at all. This is merely one interpretation of the derivative (another is the slope of a tangent line). But I assume that there is some kind of general agreement (and the history of the page shows much variation) here that this is the way folks want the definition to be stated. So I'm working from that.)

Your second sentence is, I believe, not grammatically correct. I think yours should be "Finding a derivative is differentiating", or perhaps "Finding a derivative is called differentiating". Differentiation is the process of finding the derivative, as stated in the original version. By the way, I don't think there is any reason for this to be the second sentence in the article. I suggest moving it further down the page. Cheers, (and don't forget to sign your comments with four tildes) Doctormatt 04:05, 8 December 2006 (UTC)

Hi Doctormatt, perhaps we could say that "a derivative is the rate of change of a quantity (e.g., function) at a certain time." That seems fairly unequivocal and much more accessible than what is currently there. I'm not so much a proponent of my own version so much as an *opponent* of the use of the meaningless (to the non-mathematical) word "instantaneous."

I don't really mind the differentiation thing being right after the initial sentence, because people might have clicked on the derivative page after going to differentiation (a disambig page). It's certainly grammatical to say "Finding a derivative is called differentiation"; see gerund. However, now that I think about it, maybe we should just say "Differentiation is the process of finding a derivative." (We put the sentence in the active voice.)

Your thoughts?

Best,

128.36.70.147 06:32, 8 December 2006 (UTC)

Well, I think perhaps this is better, but I don't like the "e.g." bit. Again, I think the first sentence should be as definitive as possible. Since you went ahead and changed the article, I'll go ahead and change it, too, so it doesn't have this "for example" kind of thing. Cheers, Doctormatt 05:46, 10 December 2006 (UTC)

Okay, I didn't change it. Frankly, I think the new sentence with the "e.g." just sounds terrible. Perhaps the first sentence could talk about quantities OR functions, but not both? Also, "at a certain time" suggests the question: "what time?" I think "at an instant" or bringing back "instantaneous" would be more clear. Cheers, Doctormatt 06:00, 10 December 2006 (UTC)

Hi Doctormatt, I went ahead and changed it because you hadn't written back. Thanks for your comments. What if we changed the first paragraph to:

"In mathematics, a derivative is the rate of change of a quantity. A derivative is instantaneous, or calculated at a specific instant rather than as an average over time. The process of finding a derivative is called differentiation. The reverse process is integration. The two processes are the central concepts of calculus and are related via the fundamental theorem of calculus."

My main complaint was the word "instantaneous," which I think the common man has a hard time understanding. When something is instantaneous, it is fast. There other definition, "pertaining to an instant," is rather obscure.

In the revised version, we clean up instantaneous, but also explain what it means in the next sentence. Moreover, we keep the first sentence decisive.

Hope to hear your thoughts soon. 128.36.70.147 07:05, 12 December 2006 (UTC)

Okay, this is better. Can the second sentence be written to avoid the "or"? Would "A derivative is instantaneous: calculated at a specific instant..." be acceptable? The "or" might lead to confusion, since it is not clear that the clause following it is a definition, as opposed to just being an alternative. Cheers, Doctormatt 17:06, 12 December 2006 (UTC)

Done. 128.36.70.147 23:35, 13 December 2006 (UTC)

Thanks to mathematicians, consolation to non-mathematicians, & a plea

The "thanks" is for your interest in Wiki math articles as readers and editors. What a compliment to non-mathematicians (like me) whose fumbling efforts are corrected or improved (or removed where appropriate), by you, often with an Edit summary or Talk Page reference. This is in the best spirit of critical rationalism. The principle of charity is thereby exhibited (where there is any truth worth salvaging, of course).

For bruised non-mathematicians (sometimes including me), take consolation. Corrections or deletions by mathematicians are rarely personal. The mathematicians are just trying to improve the article. Rather, indifference ("Why bother with this mess?") may be the highest form of contempt.

The "plea" referenced above is to mathematicians, but it applies to any professional field, including my own (economics): Your writing with the lay public in mind, especially in the lead, is sooo appreciated. Exposition or usage that is common, even preferred, in the field may come across as "jargon" to interested outsiders. If you Edit for the interested, intelligent layperson, that should be sufficient, provided that the exposition is relatively self-contained (rather than being circular) or that it has transparent links. If on the other hand one term or sentence is explained with another unexplained (or more obscure) term or sentence or with an obscure link, the reader may give up, even though only a little more thought and editing might render the term or sentence clearer. Yes, terseness is a great virtue but not at the cost of clarity, especially in an encyclopedia. Given the beauty, clarity, and logic that mathematician rightly take pride in, this might be "preaching to the choir." Still, y'all know what I'm talking about (Find Lefschetz). Thanks again. Thomasmeeks 15:50, 25 November 2006 (UTC)

Explicative Numerical THEORY

Dosent exactly has to do with derivatives, but EN Theory should revolutionize Diferential Calculus if proved.

If T=Lim of x>1, then. T+1 (c) = Lim >I * dy/dx (T * e) —The preceding unsigned comment was added by Hanek45 (talk • contribs) 19:28, 11 December 2006 (UTC).

cleanup needed

Someone needs to clean up the references. Spivak has written many books, I have no idea which of his books the reference is to. Also, the first reference needs an ISBN. Rick Norwood 14:50, 31 January 2007 (UTC)

Go ahead. :) Oleg Alexandrov (talk) 16:20, 31 January 2007 (UTC)

This article has to do too much work

Reading through the talk page, I see that there have been many issues concerning this article, which seem to have been brushed under the carpet since it acquired GA status. I don't believe these issues have been adequately resolved, mainly because this article does not or should not exist in isolation. However, at present, it is not only the effective main article for the differential calculus category, but also one of the few accessible and reasonably polished articles in the entire category. Because of this, it has to be everyman's article, and has too much work to do. Many of the problems it has (as a consequence) have been raised already, but I think it is worth repeating three of them.

• The article is too long. It covers in detail things like "notations for differentiation" and "rules for finding the derivarive" which really belong in separate articles. Furthermore, it discusses separately three applications, to physics, critical points, and graph sketching, whereas a general article would be much better off with an applications section and links to articles with further details.
• The article is too elementary throughout. A wikipedia article should progress. I agree this article needs to be really really accessible, but at present, it is more elementary than the differentiation section of the calculus article which it cites for a non-technical overview! (The latter article even describes the derivative as an operator! In part, of course, this is a problem with the calculus article. Can we help using material from here?)
• The article restricts attention to the derivative in one variable. I know this has been discussed already, but believe the discussion has been coloured by the amount of work this article has to do. I agree that there is scope for an article entirely on the derivative in one variable, but it should not be called "Derivative", at least, not without disambiguation. At the moment the differential calculus articles (and some of the discussions) convey the mood that one either works in one dimension or in Banach spaces. This is ridiculous. The mainstream of calculus is finite dimensional but not necessarily one dimensional (e.g. what could be a more central and motivating concept in analysis than a partial differential equation?). A fundamental article on the derivative needs to make contact with partial derivatives and the derivative as a linear map or matrix.

I saw one light in the history of this article when Dmharvey pointed out that

The common thread is that the derivative at a point serves as a linear approximation of the function at that point. I think it would be great if something like that could appear in the intro.

It would indeed. It is a pity that this opportunity was missed, as this is an idea that should inform the entire differential calculus category (with exceptions to this idea, such as the Gateaux derivative, duly noted).

This light does not shine at the moment. The multivariable calculus article is almost content-free. Total derivative cannot make up its mind what it is about, and has several vague ideas with no connections between them. The derivative as a linear map is spread all over the place, from bizarre locations such as differential (infinitesimal) to technical and specialized ones concerned with Banach spaces or manifolds.

Despite my concerns about this article, I am mainly posting here because it is one of the best articles in differential calculus, and has many editors. So please comment, whether or not you agree with the above, on how we might hammer the differential calculus category into better shape. Geometry guy 23:42, 23 February 2007 (UTC)

I've now "improved" (I hope) total derivative and differential (infinitesimal), but there is still no satisfactory "homepage" for the derivative in more than one variable. My current thinking is to create a new page at derivative (differential) if I have the energy, unless someone suggests an alternative way forward. Geometry guy 22:57, 28 February 2007 (UTC)

Recent edits

There is an anonymous user replacing italic d with roman d in many articles (see Talk:Integral). I believe these edits are in good faith (although the user's edits in general have been variable and suggest he/she may be a teenager - by which I mean no offense). However I do support them. I think the upright d works particularly well in wikipedia, because of the mixture of wiki-text and math. Geometry guy 00:06, 23 March 2007 (UTC)

Well, I went and reverted them all, sorry. We had a discussion a while ago about that, and people were in favor of italic d (which I think is the standard in US, the upright d seems to be preferred in Europe). Perhaps we can have another discussion at Wikipedia talk:WikiProject Mathematics on the issue. Oleg Alexandrov (talk) 03:55, 23 March 2007 (UTC)

Shame on you Oleg, for referring to prior history to justify your revert without providing a link to back up your assertion! :-) The only discussion I know of is at Wikipedia talk:WikiProject Mathematics/Archive 4#straight or italic d? and I see no consensus there at all. The voice of reason, in my view, is Toby Bartels, who says

I oppose any sort of policy decision for all articles; we should follow the usual rule of tolerance for variation that applies to US/UK spelling differences.

This is more subtle than a US/UK thing: it is a matter of taste with users on both sides of the atlantic, with both opinions held on both sides. There are also mathematical divergences, for instance depending on whether dx (isn't that pretty?) is viewed as a single symbol or the differential of a function. And, as the anonymous editor writes in the edit summary at Integral

Upright d for derivative notation so as not to confuse with d*x notation

For all these reasons, I generally don't like it when people (from the most humble anonymous visitor to the most esteemed administrator) go through articles uniformizing stylistic differences like this. As long as an article is internally consistent, I am okay with variations: I would even support using upright d in wiki-text and italic d in display math. However, in this case the anonymous users edit summary is somewhat better than Oleg's

rv \rm d. Ugly, against Wikipedia conventions.

Hmmm... shades of POV? :-) Anyway, I'll commit the same crime and revert here and at differential form, but leave Oleg's revert intact at Integral. Let's see what happens ;-) Geometry guy 10:41, 23 March 2007 (UTC)

However, the problem is that the italic d notation has been there for a while. It was not me who was going through articles doing mass changes, I just revered back to the notation which has been used in these articles for years. A discussion on this is in order. Oleg Alexandrov (talk) 14:44, 23 March 2007 (UTC)

Yes, you are completely right. I've now found quite a lot of discussions on this issue, and I would guess you are fed up with it cropping up again and again, and constantly having to revert mass changes! To make up for causing trouble, I'll try to put together a list of links to the previous discussions. On the other hand, although, as I said, I generally don't like mass changes to uniformize, it is not so bad if they happen randomly once in a while, as it adds to the diversity and liveliness of wikipedia (with benefits e.g. for NPOV). In this case I also felt a certain sympathy and wish not to bite the newbie.

As for further discussion, from my point of view, the most interesting point concerns the exterior derivative (cf. differential form in the above revert), since I would quite like to make that particular d upright to highlight its status as an operator. For derivatives and infinitesimals, the notation has many interpretations, and we will never reach consensus: I certainly won't revert again if the d's here turn back to italics.

Regarding this article, I think there are much more serious issues to tackle, and I raised them in the previous section. I'm sorry that no one has added further comments to these, so I should probably be bold to rekindle some interest :-) Geometry guy 15:15, 23 March 2007 (UTC)

I think this is a color of the bikeshed type discussion, which we could argue about for eons. Consistency within one article is good consistency across all of the mathematics articles is a grand waste of time. --Salix alba (talk) 15:57, 23 March 2007 (UTC)

I agree striving for consistency across articles is not really worth it.

I reverted the upright d notation per discussion at Wikipedia talk:WikiProject Mathematics where there was good agreement that notation change is likely to cause more trouble than what it's worth.

As far as differential forms are concerned, I understand that some people may prefer using an upright d notation to distinguish the exterior derivative operator from the form it is applied to. I will not try to change notation there one way or another (up to Geometry guy, that is). Oleg Alexandrov (talk) 02:37, 24 March 2007 (UTC)

Very sensible compromises have been made here I think. I'm happy to keep differential form as it is and will not change exterior derivative until I have the time and energy to make substantial improvements. Meanwhile, what about substantial improvements to this article? ... (Comment moved to new section). Geometry guy 21:05, 24 March 2007 (UTC)

Improvements needed? If so, what?

(continued) ... Meanwhile, what about substantial improvements to this article? I am tempted to ship out some of the material to subarticle stubs to focus our attention, but I'm quite busy now and won't do it just yet, so please comment or (better) edit! Geometry guy 21:05, 24 March 2007 (UTC)

Overall I like the derivative article. What is in here that you want to "ship to subarticles"? Oleg Alexandrov (talk) 03:26, 25 March 2007 (UTC)

It's a bit ironic, after all this talk about notation, but I think the "Notation" section in the articles is too long and should be condensed. So one possibility would be notation for derivatives; I can envisage this growing in a proper article with tracking the histories of all these notations (well, that's perhaps a bit optimistic, but hope springs eternal). -- Jitse Niesen (talk) 12:05, 25 March 2007 (UTC)

I agree with this: the notation section occupies 3 pages in my browser and is nearly a third of the body of the article! My full comments are at Talk:Derivative#This article has to do too much work, but I've moved the discussion to a new section rather than there so that my comments don't dominate the section. I think Rules for finding the derivative could be shortened and rewritten and a few standard derivatives could be given, with more prominant (main article?) links to Table of derivatives (which could probably be split into separate articles on rules and standard functions). There is also some reordering, restructuring and rewording that I would like to try. Geometry guy 14:08, 25 March 2007 (UTC)

I agree with shortening the notation section. We can keep the two most used notations, meaning Lagrange and Leibniz, shorten the Leibniz's notation section, and then cut out the entire notation section from this article and move it to a new article as Jitse suggests (and refer to this article at the top of the now shortened notation section). Oleg Alexandrov (talk) 14:43, 25 March 2007 (UTC)

And I would like a source for "Euler's" notation; I was taught that as Heaviside's. Septentrionalis PMAnderson 22:32, 25 March 2007 (UTC)

Progress reports

Okay, I have done most of the tedious work in moving out the notation section and sorting out related articles. I've moved the unsourced tag to the new article Notation for differentiation. I've done very little other rewriting here, but made a start. I think we should mention all the notations briefly. Geometry guy 00:28, 26 March 2007 (UTC)

I've now tidied up the articles on notation a bit, and shortened the section here, as suggested above. The Leibniz notation subsection is still quite long, but I think that is justified as it is probably the most common notation in elementary calculus (even if it somewhat flawed from a more advanced modern point of view). I am inclined to reorder so that it comes first. I have also made what I hope are the least controversial of my other proposed changes, which involve grouping the applications together. Geometry guy 20:26, 26 March 2007 (UTC)

(Reordering done.) After creating a new Differentiation rules article (a fine slogan!), I edited the material here and at Table of derivatives and fixed many links. Today I moved on to some more challenging edits ;-). I spent some time rereading the edit history and the talk history: my feeling is that this article has been a victim of its own history. At some key moments when a good editor (e.g. User:Dmharvey) could have refreshed the article, instead another editor made an incomprehensible mess which had to be reverted and cleaned up. So I decided to rewrite the first section completely, to eliminate the redundant information coming from the "differentiability" point of view and the "Newton's difference quotient" point of view. I tried not to lose any previous editors' insights in this edit, but my main hope is that this is now more accessible and structured for the novice reader. I also incorporated some material from the calculus article. Geometry guy 01:45, 29 March 2007 (UTC)

I think that the article is in a good shape now, thanks to the recent almost single handed efforts from one determined editor. In my opinion, the first section is very well written, and frankly, I am not convinced that the Calculus article is "non-technical introduction" to that. Also, the overall organization of the material seems good to me. I do have a couple of concerns, but they are related to the technovisual aspects.

• Is it possible to display the table of content in two-column format? Currently, it occupies almost an entire screen, and makes it hard to glimpse at the first section from the introduction.
• I know that this had been discussed before, but I will bring it up again: is there a better solution than posting the list of calculus topics on the right of the lead? Especially in this case, at least on my browser all the lead text appears very ragged as a result. It's not at all a trivial issue, because the blue bands in the topics panel distract too much from the lead.
• Perhaps, the two questions can be addressed together by moving the topics down a bit to make it abut the TOC instead of the lead. If there is anything that should go on the upper right, it'd be some graph like Figures 1-3: sufficiently small not to cause the current problems, pleasant to the eye to increase friendliness (does anyone seriously think putting a crib-sheet or exam-topics-list-style bar makes the article more attractive?), but not too catchy.
• Finally, another question to technosavvy editors: can someone, please, archive the beginning of this talk page, e.g. to around the GA award discussion? It's insanely long at the moment. Arcfrk 04:24, 29 March 2007 (UTC)

Thanks! Unfortunately, I'm not sure what to do about the technical aspects. I moved the calculus banner down a paragraph, which is not much better, but helps a bit. More radically, it could be moved to the "Computing the derivative" section - opinions? As for the table of contents, I don't know a fix. I am partly to blame for expanding the number of subsections, but I feel these add to the readability. Anyway, I reduced the number of subsections by one.

It would be good to have feedback from a (non/less)-mathematical editor about the changes. I believe I have made the material more accessible than it was before, while maintaining (perhaps even enhancing) an encyclopedic tone — but as both a mathematician and the contributing editor, it is hard for me to be objective!

I also expanded the history section, in line with the post-GA recommendations. Geometry guy 19:03, 29 March 2007 (UTC)

I have now completed the last main task to address the issues I raised on 23 February by adding a section which is an introductory survey on derivatives in higher dimensions. This material is, of course, intrinsically more advanced than the one dimensional derivative, but I have done my best to make it as accessible as possible by relating the various notions to the one dimensional derivative and to each other, and also by providing clear links to articles that elaborate these ideas. There is still plenty of scope for improvement, though, I am sure! As I mentioned before, as long as this article is the main article on derivatives, I think it is essential that it should address the central concept of multivariable and higher dimensional differentiation in as elementary way as possible.

I have also rewritten the "generalizations" section in the light of this addition. I would also like to tidy up the "applications" section (which is not entirely encyclopedic right now) and update the introduction. I will have a go at this shortly. In the long run, it would be nice to have an Applications of differentiation article. And of course, it is necessary to make sure this article is well-sourced... Geometry guy 16:56, 31 March 2007 (UTC)

Progress?

With a redraft of the introduction, the rewriting I wanted to do is done. The article is no shorter than before, but I believe it is richer in content and branches out to several improved supporting articles. My fondest wish is that those who liked the article before still do, and those who didn't, do now. Of course (WP-)life is not that simple, so long may the comments, discussions, criticisms and edits continue! I would really like to see this article reach A-class: it is such a fundamental topic in mathematics that it is a great pity that it isn't A-class yet. Geometry guy 18:38, 31 March 2007 (UTC)

Recent edits to intro and first section

141.211.120.199 (aka 141.211.63.35) has made some useful edits to the introduction and the first section. Some of them seem to me to be definite improvements, but I'm not completely convinced by all of them. I've only made fairly minor corrections to them so far. However, I do have a couple of requests for 141.211.120.199.

• Please do not use the word "copyedit" in the edit summary to describe an edit which adds to or changes the mathematical content.
• Please break your edits into stages so that others can see more easily what changes you have made using the diff's in the edit history.

This is just a matter of courtesy to other editors on wikipedia. Thanks! Geometry guy 13:01, 18 April 2007 (UTC)

Hmm. Most of the time, I'm not sure what else to say besides "copyedit". I'll try to be more specific, and I'll try to commit things in smaller chunks—but I should warn you that I'll be writing it the same way as I have, in big chunks, so it may happen that some of those individual edits will look strange in isolation. We'll see how it goes.

Also, I think I should give you an explanation, because you haven't seen me before. (Before this, I've mostly stuck to advanced articles like spectral sequence and sheaf (mathematics)—I'm rather proud of those two—and my only serious foray into more elementary topics, Riemann integral, didn't elicit any response.) First, I take the commandment be bold very seriously, and if I want to change something then I change it. If you don't like it, you should be bold and change it yourself. With an article as good as this one I'm changing mainly the exposition, not the content, and that's something that can be tinkered with endlessly. Second, I like my anonymity, and I want to remain anonymous even among Wikipedians. I'm well-aware that you can see my IP address and that an account will bring privileges I don't have. I'm still not getting one. (I seem to be an Exopedian.) I hope that clears things up a little. 141.211.62.20 15:14, 18 April 2007 (UTC)

Many thanks for the clarifications. I'll copy the second part of your message over to User Talk:Geometry guy and reply further there. I agree with you about being bold: if I don't like something about an article, I try to be bold and change it too. The only comment I am making is that if there are two or three different things I don't like about an article, I tend to change them one at a time. The more elementary and evolved an article is, the more likely I am to do this. I find that the easier I make it for other editors to see where I am going, the easier it is to improve the article, with less work for everyone (including me!). Anyway, this is just a suggestion, and you already do this to some extent anyway. I cannot and do not wish to change the way you work! Geometry guy 16:28, 18 April 2007 (UTC)

PS. (Over edit conflict with Salix alba.) I'll also move Salix Alba's reply to my talk page. Please revert if you have any objections to me doing this.

Question

Hi, isn't the line in the first picture function technically also a secant line; for it passes the peak and the trough simultaneously. Just wanted to clarify this VinceyB 19:28, 23 April 2007 (UTC)

Yes, it's also a secant line. This is a very common phenomenon to which the most important exception is quadratic polynomials. If you have a polynomial which is not a quadratic, there are usually lots of tangent lines which are also secant lines. If you consider intersections over the complex numbers, then every tangent line (to a non-quadratic polynomial) is a secant line. This is because of Bézout's theorem. 141.211.62.20 23:05, 25 April 2007 (UTC)

Oops, that's not true. For example, the tangent line at an inflection point of a cubic polynomial is not a secant line. (E.g., f(x) = x³ at x = 0.) However, it holds for all but finitely many points on the graph of the polynomial, the exceptions being the points where the tangent line meets the graph with multiplicity equal to the degree of the polynomial. That is, if you take the Taylor series of f(x) around a, and if that starts out f(a) + f'(a)(x - a) and has no more terms until the term of degree n, then the tangent line at a is not a secant line. 141.211.63.35 18:08, 27 April 2007 (UTC)

This article needs reorganization

I'm pretty happy with the text of the article at this point. We could all tinker with it ad infinitum, of course, but I think that, when looked at on the paragraph-by-paragraph level, the article is okay as it stands. (Though I've decided I don't like the way footnotes are used. I know I'm guilty of adding some. I repent, I won't do it anymore.)

What I'm not happy with is its organization. The article is scattered and only somewhat coherent. It starts off with the single variable case, then switches to notation, then switches back to the single variable case for computations. Then it generalizes to the higher variable case, switches to history and applications, then switches back to generalizations at the end. So while the article is find on the paragraph-by-paragraph level, I don't think it works on the section-by-section level, because the sections don't fit well together.

Here's a proposal. Tell me what you think:

• Break off most of the single-variable stuff into a new article, say derivative (single variable). This would absorb most of the first section. It could also absorb some, maybe even all, of the computation section.
• Change the focus of the explanations to the derivative as the best linear approximation. This is, of course, what the derivative "is", in the philosophical sense. There would be a section titled "The derivative as the best linear approximation", and its plan would look something like:
  • The derivative in single variable case. Leave out the stuff about the derivative as a difference quotient, but emphasize the derivative as the slope of the tangent line and as the best linear approximation. This has a geometric feel to it and can be easily explained through pictures (which, fortunately, we already have).
  • The derivative in the multivariable case. The "best linear approximation" stuff is built in to the definition of the total derivative, of course.
  • Other generalizations. The current generalizations section could be used as-is, though if we're going to focus on the linear approximation property it might want some touching-up.
• And everything else would go in other sections: Notation, History, Applications, Bibliography.

I think this would make a more consistent article than what we've got now. Thoughts? 141.211.120.130 23:15, 29 May 2007 (UTC)

Sounds good to me, but since it is a massive change I think that editors will want to see it first. Feel free to use my sand box to implement your idea User:Cronholm144/Derivative.Good luck!--Cronholm144 01:38, 30 May 2007 (UTC)

My how things have changed since February when this article only treated the one variable case! As can be seen from the comments I made then, I am personally a big fan of the "best linear approximation" point of view. However, as long as this is the lead article for Category:Differential calculus, I think it needs to be really accessible, and I am concerned that the proposed changes would reduce its accessibility. Also, while "best linear approximation" may be a good unifying theme for the category, overemphasising it in a given article at the expense of other important viewpoints (rates of change, difference quotients) may not be encyclopedic.

Apart from the location of the history and applications sections, the sections are currently in order of increasing depth, but I can see the case for a reordering such as

Definition, Computing, Higher dimensions, Generalizations, History, Notation, Applications

similar to that proposed by 141.211. An alternative (which is quite often recommended at WP) would be to lead with the history, although I am not a fan of this approach. It is also conventional to end with generalizations, but there is no reason to be bound by convention.

The thing I am most unsure about is moving the one variable stuff out. I'm not against the idea in principle, but it needs to be done for the right reasons. For instance, concepts such as higher derivatives, and rules of differentiation (especially the product rule) are fundamental to the idea of differentiation. If these are moved out so that the best linear approximation viewpoint flows together more seamlessly, then I think something may be lost.

Anyway, I'm not set against some restructuring, and Cronholm's suggestion to sandbox it first is a great idea. Then we can evaluate the pros and cons more carefully. Geometry guy 11:11, 30 May 2007 (UTC)

I will very strongly disagree with breaking off the single variable case. Let us keep in mind that Wikipedia articles should be accessible, and 95% of the people who will want to read the derivative article will want to see the single-variable case. Oleg Alexandrov (talk) 15:23, 30 May 2007 (UTC)

Rewrite available for feedback

OK, I think I've finished my complete rewrite. It's available at User:Cronholm144/Derivative. Many thanks to him for giving me a sandbox. This is such a big rewrite that I don't know if I could have done it without that. Here are some notes:

• I decided to write "pushforward" where a differential geometer might have said "differential", and I adopted the pushforward-like notation f_* rather than the differential-like notation df. I did this because df looks like the differential form df and like something coming from the Leibnitz notation. The pushforward isn't the same thing, so I thought that using df would be confusing.
• In light of an article referenced on the Calculus talk page, (Conceptual Knowledge in Introductory Calculus, Paul White; Michael Mitchelmore Journal for Research in Mathematics Education, Vol. 27, No. 1. (Jan., 1996), pp. 79-95 [1]) I decided to avoid using variables for constants. Saying "Fix a constant a" might make us happy, but apparently it does not make students happy. (Strangely enough, I remember the moment when I realized that some variables were fixed and some were variable. It must have made quite an impression on me.) Consequently you see actual numbers everywhere. This can be changed if we agree it's not a good idea.
• The proposed article Derivative (single-variable) would contain most of the material that was deleted: More detail on the difference quotient, a worked example, and comments about continuity. There's additional stuff which can go there; for instance, apparently there are some interesting restrictions on what functions can possibly be derivatives of some other function.
• Finally, as with any draft, there are probably solecisms lurking. There are places where the formatting can be improved. Please excuse my errors.

Moving forward, there are four paths I can see from here:

1. Adopt the rewrite mostly as it is.
2. Adopt parts of the rewrite, adapting them to fit the current article.
3. Significantly modify the rewrite, but keep the same general structure and outlook.
4. Throw out the rewrite.

As you might expect, I'm partial to the first option, but this is a team effort. Opinions? 141.211.120.86 00:24, 10 June 2007 (UTC)

Wow, nice work. I don't know if the pushforward stuff belongs in the main article though, although I agree that from a mathematical point of view, it's much better to think of the derivative as a linear map. By the way, the differential is the same as the pushforward in the sense that if f : M → N is a mapping of manifolds, then f_* = df : TM → TN is a linear map of vector bundles covering f. When N = R, you recover the usual differential form df you're thinking of. As for what to do with your rewrite, I'm not sure. May I make another possible suggestion:

• Incorporate the new changes with one or both of the existing articles linearization or total derivative (the relevant redirect from differential (calculus)).

To me this seems like a more reasonable solution than adopting a comparatively advanced viewpoint in the derivative article. Thoughts from other editors would also be appreciated. Silly rabbit 08:54, 11 June 2007 (UTC)

Thank you. It's great to hear that at least one person thinks the rewrite is nice!

You're right, pushforwards and differential forms are the same. (I usually think of differential forms as sections of T*M.) But the Leibnitz notation is a subtle thing, and using both df and dy/dx suggests that you can divide differential forms; I don't know how to explain that that's no good in elementary terms. You may have thought more about that than I have, and if you can do it, it might make a good addition to this article or to some other article. (On the other hand, the distinction between a differential form and a derivative is important, and if we consistently use df then that sets the reader up to integrate differential forms. Maybe that outweighs the possible confusion? The notation would get prettier: df_{(5, -2)} versus f_*(5, -2).)

I don't think the viewpoint is as advanced as it appears. I don't wait until the total derivative to talk about linear transformations, so it does look a little more advanced. But most of the rewritten material doesn't really use linear transformations or vector spaces, the big exception being "The single-variable derivative as a linear transformation", and even the parts that do use mostly one-dimensional vector spaces. The main difference from the present article is the emphasis on the derivative as the best linear approximation, which, I think, comes closer to capturing the soul of the derivative than the present article's technique. That was what I really wanted to capture with this rewrite, but I found that I couldn't avoid talking about linear transformations without doing the material a serious injustice; if we left linear transformations out, how could we sleep at night? 141.211.120.108 15:54, 11 June 2007 (UTC)

If there is a way to introduce the linear approximation (or linearization) approach to differentiation here in an unobtrusive way, then I think the article would benefit. The main advantage is that this gives a uniform way to think about derivatives as geometrical objects. The disadvantage is that, in an article intended for the masses such as this one, organizing the article around this notion is going to lose most of the prospective readers. (The phrase pearls before swine comes to mind.) The existing articles on linearization and total differentiation are in a sufficiently poor state that I would suggest incorporating some of this material there. The linear-approximation approach to the derivative could then be summarized here (see WP:Summary style), with a {{main|Linearization}} or somesuch link template. Silly rabbit 21:35, 12 June 2007 (UTC) PS: I note the deafening silence of other editors. Does anyone else have any thoughts?

I think it is good content but agree with SR that its probably not appropriate full scale in this article. Maybe there is a new article where it could go, modern approach to calculus or some such title.

One concern is that it is really discussing tangent spaces without explicitly mentioning them, so in some respects its not technical enough.

I don't know if this is a sill question but how does del fit in this framework? --Salix alba (talk) 22:18, 12 June 2007 (UTC)

Re: User:Silly rabbit. Introducing the linear approximation approach was exactly my intent. User:Geometry guy said above that he is a "big fan" of that viewpoint. User:Dmharvey also seemed to be in favor of this approach (see Talk:Derivative/archive1). You say you think the article would benefit. And, as I said above, I think that "best linear approximation" is what the derivative really is—hence my desire to make the article reflect that.

"Slope of the tangent line" seems to go over better in class than "instantaneous rate of change". One semester I had a class really get hooked on the tangent line approach; it was hard to get them to compute algebraically, because they liked drawing tangent lines so much! Rather than being "pearls before swine" I think that our readers will appreciate the fine pearl of linear approximation.

Re: User:Salix alba. I really do intend this as an introduction, and not as a modern approach—that would also be good, but it's a different article. I agree that it would be nice if our readers understood tangent spaces, but because we're working on open subsets of Rⁿ we can, and should, avoid them. Almost all of the proposed article doesn't require that the reader know anything about vectors, and the parts that do only require a very modest knowledge. We could even segment off "The single-variable derivative as a linear transformation" into a subsubsubsection which began with a warning that this part, and the other mentions of linear transformations, is necessary only for understanding the total derivative.

Regarding del, I believe that del is another way of writing the exterior derivative. This means that it is a connection. The one-variable analog is to take the differential form df associated with f; since df = (∂f/∂x)dx this is another way of approaching the derivative. However I should defer to Geometry guy, who is actually a differential geometer and understands these things better. 141.211.120.77 23:30, 12 June 2007 (UTC)

Apologies for the silence. I wanted to step back a bit and let others comment, because the current structure has a lot to do with my handiwork, and so was concerned to be too close. However, the discussion has died down, so it is time to add my view.

I think 141.211 has produced a lot of good material here, but I am not sure what should be done with it, and I stand by some of the concerns I raised (about this being the main article for Category:Differential calculus and about the need for encyclopedic content) before the draft was written.

First, though, some replies to the above.

• del in the sense of nabla is indeed a standard notation for a connection on a vector bundle. This is different from the exterior derivative although the exterior derivative on functions is a connection on a trivial bundle. I guess nabla is more notable to most readers as a major tool in vector calculus to express div, grad, curl and all that. Its use for the gradient is of course closely related to the exterior derivative on functions, except that the gradient of a function is viewed as a vector field (using e.g. a Riemannian metric) whereas the exterior derivative is a 1-form.
• del in the sense of ∂ is rarely used for the exterior derivative, although it is sometimes used for the "complex linear part" of the derivative, as a counter-part to the del-bar or Cauchy-Riemann operator (which is the complex antilinear part). It is of course, also used for boundary operators in topology and homological algebra.
• In one variable you can take ratios of differential forms because the vector space of differential forms at each point is one dimensional. This is consistent with Leibniz notation, although explaining this to a general reader is, admittedly, somewhat challenging!

Second, a couple of minor things. I agree with comments above that using f_∗ for the derivative as a linear map is not a good idea at this level. I'm not a fan of the overarrow notation for tangent vectors (and bold font is used in a couple of placed). I also don't like all the explicit numbers: even if it is good pedagogy, I find it unencyclopedic (and a bit cluttered).

Which brings me to the general issues I mentioned before. It is interesting to note that the context for the aforementioned remarks of Dmharvey very much concerns the challenge of making encyclopedic material approachable. It is like teaching with your hands tied behind your back (no handwaving!). It is a real pain because, for instance, standard good teaching practice such as "Don't introduce a second point of view until students have mastered the first" often conflicts with encyclopedic principles.

The problem for me is that Derivative is (still) the main article in Category:Differential calculus, and so it has to be a scholarly reference piece as well as being accessible. 141.211 done a fantastic work to explain the derivative as a linear map in an elementary way, and there could be a great article on Introduction to differentiation as linear approximation (or something like that) in the making here, but at the moment it seems difficult to incorporate this into Derivative while maintaining a good balance of content, encyclopedic rather that textbook style, accessibility, NOR, NPOV and all that jazz.

I am a big fan of the linear approximation point of view, but this appears in three ways initially: first, there is the tangent line, then there is the derivative as the slope of the tangent line (which is subtly different, because it does not locate the line), and finally there is the derivative as a 1x1 matrix or linear map. I agree that the tangent line and linear approximation is a great way to introduce and explain the derivative, but would question whether one can throw all three of these variations at the reader so quickly, although 141.211 has given it a pretty good shot!

I suspect the issue again is partly that Derivative still has to do too much work. When I did some rewriting a while back, I was surprised, having read the talk archive, that the rather sweeping changes I made were virtually uncontested. It may have partly been because I developed a couple of subarticles first, so that when I made changes here they were clearly seen as steps in the right direction. I'm not sure what the right direction is at the moment.

This might become clearer if some related articles such as Linearization, Derivative (generalizations) and Total derivative are improved. Maybe it is time to attempt an overview at Differential calculus (which currently redirects here). Maybe we should try to write Derivative (single variable): some of 141.211s material would certainly be useful for that! I favour implementing the idea of linear approximation as a unifying idea for the entire category, before deciding how best to add it to the main article.

So my view is that it would be good incorporate some of 141.211s ideas here, but not all of it, at least not yet. Some of it may also be useful for related articles, and I would favour revisiting the issue at a later date. However, this is not www.geometryguyknowsbest.com, so I leave these rather long thoughts for other editors to consider. Geometry guy 16:59, 23 June 2007 (UTC)

Let me try to summarize everyone's remarks before I go on. If I misinterpret or err, please correct me:

• The article Derivative is currently both an introduction to both differential calculus and all of its trappings (optimization, differential equations, etc.) as well as an introduction to the derivative operator.
• The article I produced is heavily weighted towards the derivative as an operator and does not provide a balanced introduction to differential calculus.
• The article I produced is too advanced to serve as the main article for Category:Differential calculus.
• Related articles such as Linearization, Total derivative, and Derivative (generalizations) need to be developed.

Assuming that's a correct summary, I think I know what needs to be done.

• Develop some of the articles in Category:Differential calculus.
• Write Differential calculus, a new main article for Category:Differential calculus which explains the derivative in vague terms and puts a lot of emphasis on its applications.
• Remove from Derivative anything not immediately relevant to the derivative operator.

My vision is that Differential calculus will contain things like history and applications (to differential equations, physics, optimization, etc.), while Derivative will contain things like the definition, rules for computation, and so on. The really lacking thing, and the whole reason why Derivative has been doing too much work, is the absence of Differential calculus. This should include at least:

• A non-technical overview of the derivative.
• History of differential calculus.
• Applications:
  • Physics
  • Chemistry (e.g., reaction rates)
  • Optimization problems
    • Calculus of variations
    • Operations research
    • Game theory
    • Pattern matching (e.g., minimizing energy functionals in computer vision)
  • Related rate problems
  • Taylor series
  • ODEs and PDEs
  • Implicit function theorem

I'm sure there are lots of other applications out there—I only spent a few minutes looking around to get all of that. Assuming that we had an article like that, it should be easy to adapt either the present derivative article or my draft to be the new derivative article—cut the history and applications and write a new lead. (Either one would be better if they meshed with expanded articles on linearization and so on, but they're both good first approximations. (Linearizations! :-) )) What would everyone think of that? 141.211.120.87 17:44, 23 July 2007 (UTC)

It has become very quiet around here! The vision you suggest sounds very good to me. The only thing I would add is that Differential calculus should contain the overarching idea, that differentiation is about linear approximation, explained in an elementary way. Geometry guy 09:51, 27 July 2007 (UTC)

Okay, I've made a first start. It's rough and incomplete, but it's there. 141.211.62.20 03:28, 28 July 2007 (UTC)

Picture

I made a pic illustrating a function and its derivative (I was thinking of x^2/3 when I drew it) anyway I noticed that Kingbee had a problem with the current pic so here is. BTW things sure have gotten quiet around here since this article made GA...--Cronholm¹⁴⁴ 15:03, 25 June 2007 (UTC)

I think the problem kingbee had was with the caption, which gave the misleading impression that continuity implied differentiability. (In fairness to its author, it did not say this explicitly, but rather left out an important qualification.) I have added an image of the absolute value function to the Continuity and differentiability section, since the text deals with the absolute value as well. Silly rabbit 15:53, 25 June 2007 (UTC)

Thanks Silly rabbit! ... I feel strange typing that... Oh well that was a good addition, it is important to "differentiate" between necessary and sufficient conditions...I obviously need to go to bed if I am make puns that bad...and if I am talking to myself...sigh...ending post. --Cronholm¹⁴⁴ 16:02, 25 June 2007 (UTC)

GA on hold

This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed, particularly references. Many equations are not referenced, and some of the simple equations should be relatively easy to find (in a book or on website). I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. (oldid reference #:153229464) OhanaUnited^{Talk page} 02:40, 24 August 2007 (UTC)

The article is now on GA/R. OhanaUnited^{Talk page} 21:24, 8 September 2007 (UTC)

I understand your concerns about referencing, however in this article there is a reason why the formula are not referenced, they are all very common appearing in every text book. More to the point the references given at the end are more than adequate for the reader to verify the formula. The reader is better served by providing a a range of references. I can't see how it wily help to add lots of links to the same book. --Salix alba (talk) 22:40, 8 September 2007 (UTC)

I concur. Inline references for every mathematical statement in every mathematical article on Wikipedia would be excessive and unnecessary (and would not reflect the style of works in the field). Points that can be verified in standard texts need only be referenced by a reference at the end. Inline references would be suited for claims of particular notability or authorship, or that are non-standard in some way. --Cheeser1 12:52, 14 September 2007 (UTC)

Paring external links

There are currently 4 derivative calculators in the external links. I think it would be best if we could just pick one or two and remove the others. Any suggestions as to which to pick? —Cronholm¹⁴⁴ 08:01, 14 September 2007 (UTC)

Well, I figured a test was in order. I input the function e^(2x+sin(x)) into each and got:

e^(2x+sin(x))* (((2x+sin(x))*0)/e+ (cos(x)*1)*log(e)) from Solve My Math

f'x = (2+cos(x))*e^(2*x+sin(x)) from Online Derivatives Calculator

a correct graphical representation like this* from WIMS Function Calculator

Syntax Error from ADIFF (I honestly can't get it to work - the syntax here is notably over-complicated)

In light of this, I'd say the Online Derivatives Calculator and the WIMS Function Calculator should stay. I've gone ahead and removed the other two to reflect this. *Warning - unreliable server (sorry!). --Cheeser1 12:50, 14 September 2007 (UTC)

Someone put the ADIFF calculator link back in, so I removed it again. It is very clunky and inelegant, and we already have other derivative calculators on the page that are nicer. By default ADIFF doesn't know what to do with sin x, for instance. As a result, this doesn't seem useful for the general reader (see the guidelines for external links at WP:EL). Doctormatt 17:08, 3 October 2007 (UTC)

Adiff is a general purpose derivative solver. People have been using it. To enable Trig functions, it is only necessary to refer to Example 1 on the page. I provide a bulletin board, and a help page for people to ask for further assistance ufnoise. —Preceding signed but undated comment was added at 20:07, 3 October 2007 (UTC)

I've also run the provided example in this forum, and I get the correct answers inputing the following.

declare(cos(x))

define(sin(x),cos(x))

define(cos(x),-sin(x))

diff(exp(2*x+sin(x)),x) —Preceding unsigned comment added by Ufnoise (talk • contribs) 20:12, 3 October 2007 (UTC)

But the fact is, we don't need an extensive list of all existing online derivative calculators. We have a couple, they work well, and there's no reason to include more. Especially when they are exceedingly difficult to use - I'm sure there are how-tos and help pages to help people learn how to (essentially) write in this specific programming language, but that's not going to be helpful to the reader. --Cheeser1 03:15, 7 October 2007 (UTC)

People have been using adiff quite successfully. I don't see what the harm having an additional link at the bottom of the list. This tool has features that the other tools don't. —Preceding unsigned comment added by 70.123.153.7 (talk) 03:34, 13 October 2007 (UTC)

Wikipedia is not a collection of links, nor a collection of "valuable service[s]." Only a few links are necessary. There is a prevailing opinion that this link is not helpful, because it requires extensive programming to take simple derivatives, and it often does not have mathematically-readable input or output. Furthermore, I would ask you to explain why you are so interested in including the link. You have made no other edits to Wikipedia, and you traceroute straight back to the same place as adiff.com. I find this troubling. Please note that conflicts of interest are considered problematic here. Please also note that commenting on the talk page is not, as you seem to think, reason enough for you to continue to re-add the link. consensus to re-add the link is required. --Cheeser1 04:01, 13 October 2007 (UTC)

My link has been here for over a year. And I'd appreciate it being left in. There is no conflict of interest here. I'm not making any money, I'm just trying to provide a useful service and hopefully get some feedback and make improvements to the software. Writing an automatic differentiator is a very interesting hobby for me. Based on the feedback of this section, I went ahead and implemented all of the trig, hyperbolic trig, and inverse trig functions. The example given earlier in this section can now be run, by just doing:

diff(exp(2*x+sin(x)),x)

and it yields the correct result:

((2 + cos(x)) * exp(((2 * x) + sin(x)))) —Preceding unsigned comment added by 70.123.153.7 (talk) 23:04, 24 October 2007 (UTC)

Wikipedia is not a place to beta test or advertise your software. --Cheeser1 20:59, 28 October 2007 (UTC)

One-sided derivatives

I can't find anything in the article about left- and right- derivatives and the equivalent definition stated in terms of these. Rolle's theorem mentions these and I tried to link them but couldn't. Perhaps this article should mention them somewhere ? MP ^{(talk•contribs)} 18:39, 20 October 2007 (UTC)

I've found something on Semi-differentiability. MP ^{(talk•contribs)} 18:53, 20 October 2007 (UTC)

Differentials

(Context: Bo wants to introduce the non-standard derivatives notation $d(x^{3})=3x^{2}dx$ that is not supported by references in this article. Oleg Alexandrov (talk) 16:00, 4 February 2008 (UTC))

Hello Oleg Alexandrov. I take you advice to continue the discussion in (talk) here. Thanks for pointing my attention to separation of variables where $dx$ is freely used. Is't that sufficient reference? Leibniz did not consider functions, as they were not yet invented, but he used variables which depend on one another by means of equations. So $df/dx$ , where $f$ is a function, is not really Leibniz notation. And of course the term 'Leibniz notation' for $dy/dx$ does not imply that Leibniz did not use the differentials $dy$ and $dx$ separately, which he did. I agree that the differential is not an infinitesimal. It is a formal variable, a useful algebraic tool. People once restricted the use of the imaginary unit, $i$, to expressions that eventually evaluate to reals, because "$i$ is not a number, and should not be misused". You restrict differentials to expressions that eventually evaluate to limits, such as derivatives and integrals, because "dx is not an infinitesimal, and should not be misused". Even if you have never seen it during your eight years of teaching mathematics at good U.S. universities, you have indeed seen it now, because I have just shown it to you, and the U.S. university teaching may improve just a little bit. By relaxing you opposition against my WP contributions, the university teaching may improve even more. You are welcome to improve, but you are not welcome to revert, according to the WP rules. Bo Jacoby (talk) 10:36, 4 February 2008 (UTC).

Nonstandard abuses of notation should be reverted. It's not up to you to tell people not to revert your changes to the article. We've already jumped the infinitesimal hurdle by way of nonstandard analysis, but this is not an article on nonstandard analysis but rather, an article on traditional calculus. Notation should follow the accepted practices, and should not be changed because you personally intend to make some change in how notation is used, or because you personally prefer it, or because you personally have some compelling argument to change it. --Cheeser1 (talk) 14:07, 4 February 2008 (UTC)

Thanks Cheeser1. I am using the standard notation that for centuries have been used by mathematicians since Leibniz. I refer to WP rules regarding reverting. I described traditional calculus and not nonstandard analysis, and I repeat that I am not referring to infinitesimals but to differential algebra. I did not change notation, but supplemented the Lagrange notation with the well-established Leibniz notation for differentials. There is nothing personal in that. Having delt with these misunderstandings, the question remains: Why shouldn't the readers use the differential "dy = a dx", but only the derivative "a = dy/dx" and the integral "y = ∫a dx"? No reason has so far been given. Bo Jacoby (talk) 14:46, 4 February 2008 (UTC).

Reason: that is the standard, and no consensus exists that your proposed changes are standard or otherwise merit inclusion. You can insist that you're working within "WP rules" but consensus is the #1 rule. Respect it. I have nothing more to say on this. --Cheeser1 (talk) 14:50, 4 February 2008 (UTC)

Of course I respect consensus. That's why I appeal to argumentation and avoid an edit war. But there is no consensus. There are arguments both ways regarding the use of differentials: User:JRSpriggs, myself, and the article on separation of variables use differentials. User:Cheeser1 feels personally that it is not standard, but there exists actually no standard for mathematical notation. The rest of Cheeser1's argumentation consists of misunderstandings. User:Oleg Alexandrov requests references to established calculus textbooks using that notation. Like JRSpriggs I have used differential notation for decades and so far have not found the textbook, but I find that separation of variables is sufficient reference for not banning the notation or reverting my contribution. Those opposing differential notation must rewrite the article on separation of variables. Bo Jacoby (talk) 15:21, 4 February 2008 (UTC).

Separation of variables uses dx as a variable for good reason, it makes the formalism easier. That one is also widely referenced in the literature.

There is no need to use $d(x^{3})=3x^{2}dx$ in calculating derivatives, and there are no referenfes for that. Nothing more to say. Oleg Alexandrov (talk) 15:59, 4 February 2008 (UTC)

Bo might find Differential form appealing, plenty of raw dx's about there. I concur with others that we should stick to standard notations here. Indeed the differential forms article explains why that notation should be avoided here, dx makes sense when you consider if as a map between tangent spaces, and only provides a real number when it is composed with a tangent vector. These concepts are to advanced for this article. --Salix alba (talk) 19:01, 4 February 2008 (UTC)

Even if mathematicians do not seem to know it, and certainly do not want to explain it, the differential notation is commonly used in physics. See for example Gibbs-Duhem equation, and Hamiltonian mechanics#Deriving Hamilton's equations. Bo Jacoby (talk) 00:19, 5 February 2008 (UTC).

Once again, if mathematicians don't use it and it isn't standard, then we shouldn't be using it here. I'm glad you like it, and think it's helpful, but you are not the source of information for this article. No calculus or introductory analysis text (and I doubt many advanced analysis texts) use this notation. Using it would be inappropriate. Why are you wasting your time and energy advocating for this minor change to notation notation? You seem to have made a habit of spending your time doing so. --Cheeser1 (talk) 00:41, 5 February 2008 (UTC)

If you follow the links of the first you find that they refer to Total differential which in turn refers to Differential form. Also note in the second it talks of differential not derivative. There may be scope to discuss differential forms later in the article, but it is already pretty long. --Salix alba (talk) 00:45, 5 February 2008 (UTC)

to Cheeser1: WP should explain it because physicists use it. I repeat: the argument of nonstandard is void as no standard for mathematical notation exists. I repeat: I am not advocating for a change in notation, but for supplementing the Lagrange notation with Leibniz differential notation, which is widely used. Did you compare the two versions of derivative before and after Oleg' reverting of my edit? It seems to me as if you do not know what I was trying to do. I am using time and energy to improve wikipedia because I find it important. Why are you using (perhaps even wasting) time and energy preventing me doing that?

to Salix alba: Thank you for the link to Total differential#The total derivative via differentials which explains the multivariate case: "Differentials provide a simple way to understand the total derivative". When there is only one independent variable it says "$dy=(dy/dx)dx$". Differentials provide a simple way to understand the derivative.

to Oleg. In addition to the two references above I also found Mathematics of general relativity#The metric tensor. How many references do you need? "$d(x^{3})=3x^{2}dx$" uses dx as a variable for good reason, it makes the formalism easier. Bo Jacoby (talk) 01:36, 5 February 2008 (UTC).

Find me references using dx for derivatives, not differential equations, differential forms, or tensors. Oleg Alexandrov (talk) 03:47, 5 February 2008 (UTC)

Bo, there is a standard - regular calculus notation, used in all calculus books. Yours is not it. I read both versions. What you're adding isn't particularly different from anything else in the article, except your nonstandard notation (standard notations do exist in mathematics, if nothing else, there are certainly notations that are not widely used - yours, for example). You might assume it's helpful for physicists, or that they use it, but this is an article about the derivative. This notation is not used in the calculus or in analysis. It is not appropriate for the article. --Cheeser1 (talk) 04:01, 5 February 2008 (UTC)

To Oleg. The differential "$dx$" is not the derivative, but the differential quotient "$dy/dx$" is the derivative. In WP, differentiation is "the act of finding the derivative in mathematics", but literally, differentiation is the act of finding the differential. However, having found the differential "$dy=adx$" it is trivial to find the derivative "$dy/dx=a$". Derivation (abstract algebra) redirects to differential algebra which says: "A natural example of a differential field is the field of rational functions over the complex numbers in one variable, C(t), where the derivation is differentiation with respect to t" (my italics). The derivative is fundamental to "differential equations, differential forms, or tensors" where differential notation is used, so this article on derivative is where the differential notation should be explained in elementary terms. Why don't you do that yourself? Obviously you don't trust me to do it. Bo Jacoby (talk) 10:07, 5 February 2008 (UTC).

To Cheeser1. Standard (disambiguation)#Mathematics does not include calculus. You may consider the notation you know to be a de facto standard, but no relevant law or standard exists, so do not use the words 'standard' or 'non-standard' in this context. These articles use differential notation:

1. Total differential#The total derivative via differentials: "The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if $p_{1}^{2}=p_{2}p_{3}$ then $2p_{1}\operatorname {d} p_{1}=p_{3}\operatorname {d} p_{2}+p_{2}\operatorname {d} p_{3}$"
2. Gibbs-Duhem equation: $\sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p\,$
3. Hamiltonian mechanics#Deriving Hamilton's equations: $\mathrm {d} {\mathcal {H}}=\sum _{i}\left[\left(F_{i}-{\dot {p}}_{i}\right)\mathrm {d} q_{i}+{{\dot {q}}_{i}}\mathrm {d} p_{i}\right]-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t$
4. Special_relativity#The_geometry_of_space-time: $ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}$

So it is simply not true that the "notation is not used in the calculus or in analysis". Your using bold doesn't make it true. And don't say that you know "all calculus books" if you want to be taken seriously. Bo Jacoby (talk) 10:07, 5 February 2008 (UTC).

Bo, don't take this the wrong way, but I think you should honestly take a look at yourself and consider if you are in any position to tell others when and where to worry about being taken seriously. --Cheeser1 (talk) 10:22, 5 February 2008 (UTC)

Perhaps Bo Jacoby will agree to drop this matter if he sees that Oleg and Cheeser1 are not standing alone. So let me see say unequivocally, that his addition was not an improvement, it goes against the notation of text books that explain what a derivative is, considering dy/dx as a quotient of two formally defined differentials goes against their definitions, and it is thus confusing. I'm glad it was reverted. I'm not going to discuss it, I'll just revert. - Jitse Niesen (talk) 10:35, 5 February 2008 (UTC)

"I'll just revert" — oh Jitse bad guy. :) Oleg Alexandrov (talk) 16:20, 5 February 2008 (UTC)

Thank you, everyone, for joining the discussion. I agree with Cheeser1 that I too am not always taken seriously, but that is for other reasons - I have been accused for contributing original research. I do not argue ad hominem, nor do I revert contributions made in good faith. Which definitions of formally defined differentials are Jitse referring to? I encourage Jitse and Oleg to acknowledge the problems and improve the articles with the readers, rather than the editors, in mind. Bo Jacoby (talk) 12:11, 5 February 2008 (UTC).

I will just point out the fact that no one has made personal attacks, and and the fact that reverting good-faith contributions is reasonable when those contributions are original research or otherwise vacuous is completely appropriate. I'm not interested in responding to Bo, but would like to make clear the facts of the matter. --Cheeser1 (talk) 13:01, 5 February 2008 (UTC)

The two facts of the matter: 1. Cheeser1, you made no personal attack, but by referring to my position you made an argumentum ad hominem, which is inappropriate. 2. Oleg, you made an immediate revert, thus inhibiting further progress of the article, and thus violating Wikipedia:Revert#Do_not : "If what one is attempting is a positive contribution to Wikipedia, a revert of those contributions is inappropriate unless, and only unless, you as an editor possess firm, substantive, and objective proof to the contrary". Bo Jacoby (talk) 14:52, 5 February 2008 (UTC).

• rolls eyes* Okay Bo, except that I was pointing out that you are not being taken seriously because you constantly nit-pick at meaningless notational issues because you prefer a non-standard (if not entirely unused) notation. Call that ad hominem, it's still clearly relevant. Your contributions are vacuous and contribute little (or nothing) to this article, and they clearly merit reversion (the grounds for which have been explained repeatedly), and then you go on and on in these arguments, wasting your time and ours. --Cheeser1 (talk) 17:09, 5 February 2008 (UTC)

Difference Quotient Continuous at h=0?

"In practice, the continuity of the difference quotient Q(h) at h = 0 is shown by modifying the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many short cuts are commonly used to simplify the process."

But the difference quotient is undefined at h=0, much less is it continuous; of course, you can only cancel h for h\ne 0. Anyone? [Further edit: the example uses this same approach, but I disagree: you cannot talk about derivatives without limits, the difference quotient is simply not enough.]

[Final edit: Okay, I wrote too soon (sorry!). The last bit explains that the limit needs to be taken, but the caveat that the manipulation of the difference equation only holds for h\ne 0 (at which it is never defined anyway) should probably come earlier.

74.70.124.27 (talk) 04:58, 3 May 2008 (UTC)

As you say, it isn't really a big deal, since the text mentions that it is necessary to form a continuous extension. I have modified the text slightly to emphasize this. I am uncertain whether this addresses your concerns. Please take a look. silly rabbit (talk) 11:38, 3 May 2008 (UTC)

Bad Heading ???

The term Derivative is used to head this section on Calculus, but it then points to http://en.wikipedia.org/wiki/Derivative_%28disambiguation%29 where we find the generic definition of Derivative which reads: In English, derivative primarily refers to anything derived from a source — not primitive or original.

What should happen:

Because in this mathematical section, the text solely refers to Derivative (Calculus), it should be titled accordingly, and not headed as the generic Derivative. Then the Disambiguation page should inherit the generic heading, so that editors of other Wikipedia pages, seeking to make a reference to the generic concept of derivative rather than the mathematical, may do so following the proper Wikipedia protocol.

I place this as a discussion item, rather than execute a change, because there seems to be considerable "ownership" of the page, where it may be seen as arrogant if I were to make the change myself.

This is therefore submitted as a proposal with a recommendation for change with a request that a senior guardian of this page make the changes.

ClassicalScholar (talk) 04:16, 16 June 2008 (UTC)

A quick check on "What links here" shows that about 600 pages link to the present page, whereas about 300 link to Derivative (finance), the next most popular. I propose that we keep everything as is. Ozob (talk) 16:44, 16 June 2008 (UTC)

In other words, you favour expediency over accuracy. I question is this is appropriate for an entity that calls itself an encyclopaedia.

ClassicalScholar (talk) 03:39, 17 June 2008 (UTC)

No, I believe that if a user types "derivative" into the search box, he most likely wants to come here. I justify this on the basis of the number of links, and I therefore conclude that the present arrangement is most accurate (in the sense of being most likely to reflect actual usage of the term). See also Wikipedia:Disambiguation#Primary topic. Ozob (talk) 15:50, 17 June 2008 (UTC)

 Dear Friends,
   I'm just touch the subject on drrivatives, so i'm seeking to upgrate my knowledge on this subject. it is very greatful 2 you if can u provide study text or other study materials.

   My mail adress. samith_withanachchi@yahoo.com

Thank. —Preceding unsigned comment added by 220.247.235.118 (talk) 07:57, 3 July 2008 (UTC)

Minor notational query - italics

I'm curious as to non-italicization of the letter "d" in derivatives and differentials. Most authors and wiki articles have it italicised, but some don't. That is, some write dy / dx and others write dy / dx. Is this a difference between physicists and mathematicians? Or is the roman d used more in multivariate calculus? Or is it mostly used with differential notation like dy = a dx?

By the way, I prefer dy/dx, as I usually think of this as a single thing, and unitalicising bits of it seems psychologically unhelpful.

Anyway, are there widely agreed criteria for when to italisize the d or not, or is it just down to author preference? —Preceding unsigned comment added by 62.77.181.1 (talk) 08:16, 25 July 2008 (UTC)

I believe it's an American vs. UK thing, with Americans preferring italics and the British preferring roman. The French and German books I'm familiar with use italics. Myself, I think italics looks nicer. However, like all regional differences in spelling, pronunciation, etc., we don't go around changing articles just so that they're consistent with other articles (though making them self-consistent is good). Ozob (talk) 22:25, 25 July 2008 (UTC)

Retrofit topic year headers/subpages

27-Nov-2008: I have added subheaders above as "Topics from 2006" (etc.) to emphasize the dates of topics in the talk-page. Older topics might still apply, but using the year headers helps to focus on more current issues as well. The topic-year boundaries were located by searching from bottom for the prior year#.
Then I added "Talk-page subpages" beside the TOC. -Wikid77 (talk) 19:31, 27 November 2008 (UTC)

Fixing misleading information

27-Nov-2008: This article begins with some misleading information (in the revision for 28-Nov-08), such as showing a 2-hump graph (cubic or higher equation) with a tangent line (linear equation) as the derivative. A linear equation is the derivative of a quadratic equation (one-hump graph), while a qradratic is the derivative of a cubic equation (of x^3, x-cubed). Much of the wording seems confused, and I added "rate of change" to begin to sort the wording into some standard notions of derivative. I will reword more phrases later. -Wikid77 (talk) 19:31, 27 November 2008 (UTC)

Hi Wikid77. As you noted the picture shows the graph of a function and a tangent line at a point on the graph. It doesn't assert that the given tangent line is the derivative of the function, of course this is rarely the case. I think that once you're happy with the difference between the two concepts, you will agree that the picture is fine. I've also added a wikilink to the image caption in case others come along and aren't sure either. Finally, I wanted to note that there exist many more functions than polynomials, so we shouldn't restrict our attention to them as if they were, in the lead (or even assume the given image represents a polynomial). Cheers, Ben (talk) 19:43, 27 November 2008 (UTC)

Several misleading issues

28-Nov-2008: This sub-topic of the above "Fixing misleading information" is a further clarification. There are several issues to address:

• Say rate of change: I think the intro is misleading (in not mentioning "rate of change"), so that's why I had added that phrase.
• Geometric tangent lines are an analogy: The use of the geometric approach, of the tangent lines can be very confusing, because the derivative function is the set of their slopes in points (at each x), so that f '(x)=m_x, where the slope m depends on each value of x. I guess it's like saying:

The derivative of apples is like orange juice, without the rest of the line of citric acid, seeds & orange color, just the sugar/water portion: that derivative is "apple juice".

The function f(x) is the apple, and the orange is the tangent line, and they happen to have one thing in common: when they are squeezed, they yield the same amount of a sugary water (slope). The point-derivative f '(a) yields the same quantity m as the point-derivative of that tangent line, the slope m (for y=mx+b). However, those point-derivatives, as a whole, define a totally different function, which is not illustrated in the graph.

• Whole function versus points: The intro mentions a derivative for the whole function but then focuses on points. The derivative of a function is another function, which describes the rate of change of the first function. The use of each tangent line also focuses on the derivative at just a point, f '(a), rather than the derivative as a whole function, f '(x), as the general case for any x. Start the intro by just stating the full concept directly:

"The derivative of a function is another function, which
 describes the rate of change of the first function."

Also, don't worry (at the top) about functions that don't have derivatives, because the phrase "derivative of a function" implies when a derivative exists, like saying the "speed of the car" implies that the car can move.

• Car analogy then omits talk of points for integral: It can be very confusing, when focusing on the car's speed "at some point in time" to then mention the integral without a specific point. For that reason, I had added "time" to the phrase as: "(conversely the integral of the velocity, at a point in time, is the car's cumulative position, up to that time)". Again, the blurring of distinctions between whole functions and individual points has been confusing in the intro. The whole "integral" is quite different from the "integral at a point".

I mention all the above issues because, if a mathematician finds the terminology confusing, then imagine what general readers think about the wording:

• not even realizing the subject shifts between a whole function-derivative versus a point-derivative;
• getting tangent-line examples which mix apples & oranges because they both produce "juice" (slope) when squeezed;
• omitting the classic phrase "rate of change" in the intro, even though Google matches 388,000 hits for "derivative" plus "rate of change".

Those issues are things to ponder about the focus of the article. I really feel that an intro example would be much better by illustrating the derivative of 4 line segments (linear equations) showing 4 changed car speeds, where the derivatives (slopes) of the 4 segments are the 4 different speeds of that car: such as, a car travels on an empty street, halts at a stopsign (f '=0), resumes speed, then zips down to a fast highway. Later in the article, present the tangent lines, when people are more able to relate "apples to oranges" as the common "juice" (slope). Otherwise, the intro is just well-formatted, tangent-line confusion for general readers: make the intro something the readers can easily understand, so the readers could remember, "Okay, a derivative is like the set of speeds that a car traveled" compared to the total distance traveled at each minute. Again, those issues are long-term concerns about the article. -Wikid77 (talk) 14:01, 28 November 2008 (UTC)

I'll respond point by point.

• While everyone is familiar with the phrase "rate of change", on its own it is vague and imprecise. It runs into the sort of trouble that you'd have if you tried to define the derivative without talking about limits. This is, IIRC, why it was taken out of the lead. I have no objection to including it later in the article, but I think putting it in the lead is actually confusing: "Rate of change" is more like a slogan than a description.
• Geometric tangent lines are not an analogy. They are the graph of the differential of the function, that is, the graph of the derivative considered as a linear transformation between tangent spaces. You might find it useful to think about the multivariable case, where the total derivative must be defined as a linear transformation.
• The fact that the derivative of a function can be made into a function is an accident of the shape of R. On almost all other spaces this is false. IIRC, the only spaces where it is true are one-dimensional spaces with trivial tangent bundle, so for real derivatives that's R and its open subsets, and for complex derivatives that's C, D, elliptic curves, and their open subsets.
• I agree that the one sentence description of integration is misleading, so I've changed it. Hopefully this is clearer?

By the way, you might find the study of differential geometry interesting. Ozob (talk) 18:54, 28 November 2008 (UTC)

Following up to myself: I remembered wrongly in point 3 above. A correct statement works like this: Let M be a smooth n-manifold and f be a smooth real-valued function. f determines a differential form df. If the cotangent bundle Ω_M is a rank one trivial bundle, then df is again a smooth function. This is what happens over R, for example, but it's very special. For Ω_M to be rank one, M must be one-dimensional over its ground field R or C. The smooth 1-manifolds over R are (disjoint unions of) circles and open line segments, and both of these have trivial cotangent bundle. The smooth one dimensional complex manifolds are Riemann surfaces. Compact Riemann surfaces are classified by their genus g and we know that their cotangent bundle is trivial iff g = 1, i.e., iff M is an elliptic curve. If M is simply connected, then by the uniformization theorem it's CP¹, C, or D, and only the latter two have trivial cotangent bundle. In the general case, where M is non-compact non-simply connected, I don't see an obvious way to determine whether M has trivial cotangent bundle or not: M could be D minus a point, which does have trivial cotangent bundle, or it could be a genus 2 surface minus a point, which does not.

Anyway, my point is really that the derivative of a function is not naturally a function except in special cases. This is true even in R² or CP¹. Ozob (talk) 02:18, 29 November 2008 (UTC)

Proof-reading

I'm not a mathematician so I haven't changed the article, but to me the phrase "Let y=ƒ(x) be a function of x" seems misleading. Would "Let y be a function of x (ie y=f(x)) be a better way of phrasing this? Empyema (talk) 21:03, 27 November 2008 (UTC)

I've changed this to: Let ƒ be a real valued function. Cheers, Ben (talk) 23:25, 27 November 2008 (UTC)

Consistency of varibale name: a vs. x

In the Definition via difference quotients section, the variable a is used to talk about a real number and is used as the input to the function f. In the figures to the right of this section, and also throughout the rest of the article, the variable x is used. Is there some reason for using a when x would seem a more suitable choice? Is this just an inconsistency that could be corrected? Edam (talk) 06:08, 11 May 2009 (UTC)

External Link Addition

I would like to ask you about a new derivative calculator to be introduced in the external links section. Beside to the differentiation result, it also plots the graphs of the function and it's derivative toghether, for a better visualization of the theory concepts explained throughout the article (e.g. the points where the derivative is zero are local extremum points for the function). The link is here: derivative calculator with graphs. I would like to have a discussion about this, and if you have nothing against it please add it to the external links section of this article. —Preceding unsigned comment added by Livius3 (talk • contribs) 12:27, 20 October 2009 (UTC)

Vandalism?

Can someone explain what the hell is going on with the Kim Hyun Bin nonsense? Only dead fish go with the flow. 17:52, 16 May 2010 (UTC)

Actually, you know what, can someone do something about whatever 211.117.11.123 has been doing? Someone with rollback? Only dead fish go with the flow. 18:18, 16 May 2010 (UTC)

Picture of function and derivative

It would be nice I think for the picture showing the tangent at the begining to show the graph of the derivative function as well.

An idea for linking the two might be to show a triangle base length one on the x axis with the same direction hypotenuse as the tangent line the top point would move along the graph of the derivative. Dmcq (talk) 12:07, 18 June 2010 (UTC)

derivative in physics

If x is time t, y is speed v and if c is acceleration a, then line y=c*x=v. So x=t, y=v, c=a. And y'=(c*x)'=c*1=c. So speed v derivative v'=a. And v=t*a, v'=a. Or more precise can be written v'(t)=(t*a)'=a. Or ${dv \over dt}={d(t\cdot a) \over dt}=a$. Seems that no more else derivative in physics is used related with acceleration, speed, time or distance.

HOW CAN I KNOW CALCULUS? —Preceding unsigned comment added by 41.204.170.59 (talk) 12:54, 19 October 2010 (UTC)

If I interpret you correctly, you seem to be saying that the page on derivatives should point out the connection to physics, namely derivative=speed and second derivative=acceleration. This is a very good point. The relation to physics should certainly be discussed, as it is of great benefit to all, and particularly to beginners. Tkuvho (talk) 13:12, 19 October 2010 (UTC)

A more general applications section may be more useful. It could include, for example, the reaction rate and rate equation of chemical kinetics, in addition to the the example listed above. Mindmatrix 15:58, 19 October 2010 (UTC)

Personally I think this is a major oversight indeed, but perhaps this is treated at a different page? Should we bring this up at WPM? Tkuvho (talk) 16:08, 19 October 2010 (UTC)

Sure. Perhaps a more generic discussion about the inclusion of applications in articles about various mathematics concepts is warranted. The project may have already had such a discussion many years ago (perhaps 2005?), but I don't recall with certainty. One of the probelms with including applications is the eventual laundry list of every application being added, overwhelming the article. One possibility is to create a List of applications of derivatives et al. Mindmatrix 17:14, 19 October 2010 (UTC)

Bah - I completely forgot about Differential calculus#Applications of derivatives. Sigh. Mindmatrix 17:16, 19 October 2010 (UTC)

Wrong Derivative in animated Gif

The derivative that appears in this picture is wrong. It should be:

$\scriptstyle f'(x)=\sin x^{2}+2x*\cos x^{2}$

But I don't know how to modify this animated gif! —Preceding unsigned comment added by 68.173.54.237 (talk) 12:06, 14 December 2010 (UTC)

You are incorrect. --Izno (talk) 14:35, 14 December 2010 (UTC)

Section "Partial derivatives": incorrect formula?

The section "Partial derivatives" has as the second-last formula

   ${{\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{1},\ldots ,a_{n})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})}$

This seems to me to be incorrect. Looking at the notation in the equation preceding it in the section, I think it should rather be

   ${{\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})}$

Any comments? I will leave this edit to those more familiar with the process, unless asked to do it. Quondum (talk) 09:16, 27 December 2010 (UTC)

I've fixed it. Thanks! Ozob (talk) 11:55, 27 December 2010 (UTC)

Symmetric derivatives

An anon has recently been trying to add the symmetric derivative here as an alternate definition of the derivative. For everyone's benefit, I would like to prove that the symmetric derivative is not equivalent to the usual one. I'll give three counterexamples.

First: Let f(0) = 1 and f(x) = 0 for x not equal to zero. That is, f is zero everywhere except at 0, where it jumps up to the value one. This is not differentiable at x = 0 because it's not continuous. But it does have a symmetric derivative:

$\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {0-0}{2h}}=0.$

This is because when we take the limit, the value of f when h is zero is irrelevant. Because of this, the symmetric derivative never notices the discontinuity at x = 0.

Second: Let f(x) = |x| be the absolute value function. f is continuous at zero, unlike before, but it is not differentiable there because its one-sided derivatives are not equal:

$\lim _{h\to 0^{+}}{\frac {f(0+h)-f(0)}{h}}=\lim _{h\to 0^{+}}{\frac {|0+h|-0}{h}}=\lim _{h\to 0^{+}}{\frac {h}{h}}=1,$

$\lim _{h\to 0^{-}}{\frac {f(0+h)-f(0)}{h}}=\lim _{h\to 0^{-}}{\frac {|0+h|-0}{h}}=\lim _{h\to 0^{-}}{\frac {-h}{h}}=-1.$

Its symmetric derivative isn't equal to either of these:

$\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {|h|-|\mathop {-} h|}{h}}=\lim _{h\to 0}{\frac {0}{h}}=0.$

That is, the symmetric derivative does not notice that the one-sided derivatives disagree. Whenever both one-sided derivatives exist, then the symmetric derivative is their average.

Third: Let f(0) = 0 and f(x) = xsin(1/x) for x not zero. f is continuous at zero because |f(x)| is bounded by |x|. Neither one-sided derivative of f exists at zero:

$\lim _{h\to 0^{+}}{\frac {f(0+h)-f(0)}{h}}=\lim _{h\to 0^{+}}{\frac {h\sin(1/h)-0}{h}}=\lim _{h\to 0^{+}}\sin(1/h),$

which does not exist; similarly for the other side. But f does have a symmetric derivative at 0:

$\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {h\sin(1/h)-(-h)\sin(1/(-h))}{2h}}=\lim _{h\to 0}{\frac {h\sin(1/h)-h\sin(1/h)}{2h}}=0.$

So the symmetric derivative does not notice that the one-sided derivatives do not exist.

I hope this clears up any confusion there might be. Ozob (talk) 01:46, 4 April 2011 (UTC)

I have just a minor issue with your math, and it is the third line:

$\lim _{h\to 0^{-}}{\frac {f(0+h)-f(0)}{h}}=\lim _{h\to 0^{-}}{\frac {|0+h|-0}{h}}=\lim _{h\to 0^{-}}{\frac {-h}{h}}=-1.$

How is abs(h) equal to -h?... --Izno (talk) 15:29, 10 May 2011 (UTC)

In that limit, h is approaching zero from below. That means that h is negative, so |h| = −h. Ozob (talk) 01:21, 11 May 2011 (UTC)

Petition to reorganize a sentence in the first paragraph.

The sentence in question: "Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends." is abrupt and has no direct connection to derivatives.

Suggestions received happily. —Preceding unsigned comment added by 60.250.204.234 (talk) 20:00, 19 May 2011 (UTC)

I've never liked that sentence, and for the same reason (that it has no direct connection). I think we ought to delete it. Ozob (talk) 12:20, 20 May 2011 (UTC)

Nonstandard Calculus

Should we also introduce the nonstandard approach here? Dalcde (talk) 09:14, 1 August 2011 (UTC)

I'd prefer not to discuss it in detail. The definition by difference quotients is, nowadays, the standard one, and giving a full exposition of other approaches would give those approaches undue weight. But right now the article mentions other approaches only in a single footnote, and that might not be enough. I think it's worth having a sentence or two on other ways of defining the derivative. Ozob (talk) 11:22, 1 August 2011 (UTC)

Apostol's use of "in the direction" vs "directional derivative"

I don't have a copy of Apostol's book, but based on controversies I've seen on the web, can you clarify whether he actually uses the term "directional derivative" when defining a derivative that is taken with respect to a general vector v? Or does he use the terminology of a "derivative in the direction v"? There are some sources (for example the Wolfram site) that reserve the term "directional derivative" to refer to the case where v is a unit vector. Does Apostol's definition contradict this use of the term "directional derivative" or does he avoid using the term "directional derivative"?

Tashiro (talk) 15:03, 5 September 2011 (UTC)

section needed for the formal definition of a derivative

The information here looks terribly disorganised and could do with tidying up. I was thinking of a section that can give a formal definition using modern maths. The rest of the article is ok, it just seems to be more suited for high-school children. — Preceding unsigned comment added by Jmc2000 (talk • contribs) 01:34, 5 August 2012 (UTC)

Merge Higher derivative here

A recently forked article higher derivative has sprung up. I don't feel that there is enough new content to warrant a separate article, nor do I feel that there will ever be enough content to warrant such an article apart from what can be covered here and in third derivative, fourth derivative, etc. I thus believe that the new article should be redirected back to the section Derivative#Higher derivatives. Opinions? Sławomir Biały (talk) 22:49, 4 October 2012 (UTC)

Higher total derivatives section ending

The section about higher total derivatives ends with a somewhat vague paragraph saying that higher differentials are not linear transformations and references the page on jets. However, it the second example on that page assumes knowledge of the concept of higher-order differential, which is not explained there, as far as I can tell, on that page. I think the generalization that is intended in both cases is the Fréchet derivative (in the special case Rⁿ → R^m), but I'm not sure as the Fréchet derivative is a linear transformation (from the n-fold product space of the domain, which is how it appears to be used in the jet article). I think it should be made clearer just what that last paragraph refers to, because it doesn't seem like it's referring to jets or any other concept I can find. Cyrapas (talk) 20:33, 14 December 2012 (UTC)

Longest Derivative Solution By Sandeep D. Sontakkey

Hell, There is new developing ideas in solving the Problems in Derivative............................ thank u..............

See u soon............ — Preceding unsigned comment added by 117.228.129.213 (talk) 14:01, 2 January 2013 (UTC)

Continuity: differenciation & integration, step function graph.

The function does have a derivative at the marked point, you need to change the axis of observation to obtain that derivative. Pertinent, that a straight line results in a function delta(x)/delta(y), where delta(y)=0, which is indeterminant solely due the form of observation. The other aspect, where delta(x)=0 (or a constant), is known to lead to a result of zero, delta(x=0)/delta(y). Mathematically, NEITHER division nor multiplication by zero are allowable operations, both leading to one to many, or many to one solutions, which are solely solvable using gaussian functions. That particular function, is defacto continuous, by rotating the axis of observation slightly so that there are no 0/y nor x/0 aspects.

Leads to an interesting observation. Is a constant continuous, being it so that a constant has a derivative=0? The answer is, no, a constant is NOT continuous, it is bound on both sides by a sudden jump (a constant is a dirac delta), and is therefore not differenciable by definition. Because a constant is definitely differenciable (differencial of zero), all such sudden jumps are differenciable, and the platform itself, on the other side of that junction, collapses onto the axis. What does that imply? It implies that you are required to resolve the instance of discontinuity using a double axis, which is similar to using two graphs, one superimposed onto the other, whose resultant is that graph (math isn´t physics).

Someone needs to change the incongruities, errors, in relation to that function. — Preceding unsigned comment added by 186.94.187.76 (talk) 12:08, 26 February 2013 (UTC)

This statement both has nothing to do with the article and is wrong. — Arthur Rubin (talk) 20:59, 26 February 2013 (UTC)

Proposed addition to section on higher derivatives

I would like to tentatively propose an addition to this article in the section on higher derivatives. My reluctance to simply edit the article is due in part to a conflict of interest: I am the author of the publication from which the proposed addition comes. Also, I am not a mathematician by training, so I seek feedback from professional mathematicians to assure that this addition is sufficiently significant and appropriate for an encyclopedia entry.

The addition I propose is a generalized equation for the computation of the n^th derivative. It is the expression for f⁽ⁿ⁾(x) on page 387 of this reference. Subsequent to publishing, I learned that the difference functions, DIF(n,j), which are the subject of this article, are actually based on the Thue-Morse sequence. So the notation in the equation would have to be altered to conform to the accepted notation for the Thue-Morse sequence.

An argument for including this equation in the article on derivatives is that this compact general equation might prove useful. An argument against including it might be that this equation is not included in textbooks or in widespread use. A response to that argument might be that this equation may be undiscovered because it is not published in a math journal and omits reference to the Thue-Morse sequence, so Wikipedia would perform a service by identifying it. A response to that argument might be that Wikipedia is not the the proper forum to bring an unknown equation to the attention of mathematicians (then what is?), or that it is flawed (please state how), or that it is not significant (please state why). I would be grateful for any responses.

Robert Richman — Preceding unsigned comment added by Rmrichman (talk • contribs) 19:39, 27 February 2013 (UTC)

Suitable for Wikibooks, perhaps. If not published, it's not really appropriate here, per WP:OR which shows why the first two arguments against are appropriate. And I'd use the central formulation, instead; rather than

$f^{(n)}(x)\approx {\frac {\sum _{i=0}^{2^{n}-1}(-1)^{n-\sum _{k}i_{k}}f(x+ih)}{2^{\frac {n(n-1)}{2}}h^{n}}},$ use

$f^{(n)}(x)\approx {\frac {\sum _{i=-(2^{n}-1),\mathrm {odd} }^{2^{n}-1}(-1)^{\sum _{k}(2^{n}-1-i)_{k}}f(x+ih)}{2^{\frac {n(n+1)}{2}}h^{n}}}.$

But, even so, the even more compact

$f^{(n)}(x)\approx {\frac {\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f(x+ih)}{h^{n}}}.$

could be used. Potentially, the first equation could be written:

$f^{(n)}(x)\approx {\frac {\sum _{i=0}^{2^{n}-1}\operatorname {TM} (2^{n}-1-i)f(x+ih)}{2^{\frac {n(n-1)}{2}}h^{n}}},$

but I don't really see the benefit. — Arthur Rubin (talk) 20:42, 27 February 2013 (UTC)

Problem for Q being the empty set

Let f: {a} \to R for some a \in R. f is then a real-valued function from a subset of the reals to the real numbers, in compliance with the definition in the section. According to the Weierstrass definition of continuity (as well as the sequential limit definition), f is also a continuous function.

Then Q is the empty set, Q: \emptyset \to R because for h \neq 0 f(a+h) doesn't exist and for h=0 there's division by zero. Any choice for Q(0) will make Q continuous at 0, by the same definitions as before.

So the question is, if Q: {0} \to R, does limit_{h \to 0} Q exist? Not according to the definitions of limit of a function on wikipedia (nor in my Analysis textbook), since 0 isn't an accumulation point. But suppose we were to say that the limit of Q at 0 equals Q(0). This means that f'(a) = Q(0), so not only does the derivative of a "one-point function" exist, but its value is the entire set of real numbers. That can't be right either. — Preceding unsigned comment added by 109.131.139.84 (talk) 11:01, 15 August 2013 (UTC)

I thought that the derivative of a function on X at a was only defined if a is a limit point of X. Does that solve the problem, or are we having trouble with the definition of "limit". — Arthur Rubin (talk) 20:30, 15 August 2013 (UTC)

I should have referred to the last few edits on this article's page, which was why i added this section on the talk page here. As it stands now we have an inconsistency between the articles on continuity, limit of a function, and derivative (this one). Let f be as above, then let

P = "f is continuous at a"

Q = "the limit x->a of f exists and equals f(a)"

The article on continuity states P=true. The article on limit of a function states Q=false. This article on derivative states P <=> Q.

I had edited this article to remove the P <=> Q statement, but another user disagreed, so instead of back-and-forth editing i made this section on the talk page. I think the P <=> Q statement is incorrect, continuity is a weaker condition than the existence of the limit (it doesn't have the 0 < |x-a| condition, which makes all the difference here), it's the implication P <= Q rather than an equivalence.109.131.139.84 (talk) 23:15, 15 August 2013 (UTC)

OK, I think I see the problem, in terms of a not being a limit point of the domain of f. In this case:

1. If f is defined at a, then f is continuous at a.
2. $lim_{x\rightarrow a}f(x)$ exists and is equal to all numbers.
3. If f is defined at a, then the derivative of f at a exists and is equal to all numbers.

There's something wrong, here. I would say that none of those are defined, but we need to adjust the definition of "limit" (and the corresponding definition of "derivative") and of "continuous". — Arthur Rubin (talk) 01:31, 16 August 2013 (UTC)

(I'm the same user as 109.131.139.84 above, i've created an account in the meantime) I checked my Analysis textbook on this (it's a Dutch textbook "Analyse 1" by prof. Johan Quaegebeur at KU Leuven). It defines continuity in terms of the Weierstrass definition (noting that the sequential limit definition could also be used), so f is continuous at a, consistent with the wikipedia article on continuity. It defines the limit of a function in the same terms as the topological definition on wikipedia's page on limit of a function (ie requiring a to be a limit point), so $lim_{x\to a}f$ does not exist, which is also consistent with wikipedia's article. It has a theorem stating the equivalence of those two only for limit points, which is inconsistent with the statement on this page on derivative.

I think the culprit here is the unqualified use of the equivalence of continuity and existence of the limit, whereas this equivalence is only correct for limit points. I believe my original edit resolves the problem, at least in accordance with my textbook, but this may be one of those borderline cases due to different authors using slightly different definitions. When viewed according to the definitions used in my textbook the article on "continuity" is strictly speaking correct, since it only uses "if" and never "if and only if" in its definition in terms of a limit, but it is very misleading due to the placement in the article that one could interpret it as "if and only if".

My proposal would be to remove the statement in this article using the equivalence of continuity and existence of the limit as per my original edit, and change the article on continuity to use the Weierstrass definition as main definition and delegate the "limit of a function" definition to a theorem applicable to limit points.B01010100 (talk) 03:02, 16 August 2013 (UTC)

The page does not say that Q is meant to be defined on an arbitrary subset of the reals. The only time the word "subset" appears is much later, in the section on Jacobians, and there the article says "open subset". I think it would be perfectly acceptable for the article to assume that the function is always defined in a open neighborhood of the point in which we're differentiating. That would avoid this problem, and it's the intersection of the definitions available in the literature. Ozob (talk) 03:23, 16 August 2013 (UTC)

The section starts with "Let f be a real-valued function." So maybe if we change this to "Let f be a real-valued function defined on an open neighbourhood of a.", that would solve the problem?

I do still think we should change either the continuity article to point out that it's "if" and not "if and only if", or change the limit of a function article to remove the condition that it must be a limit point. Either on its own is used in the literature, but together they are inconsistent - it's picking one definition to use in one article and another to use in another, even though both definitions may be used they can't be used together if one is to be consistent.B01010100 (talk) 03:41, 16 August 2013 (UTC)

I changed the article; while I thought the context was clear, I guess it does no harm to be explicit.

I don't have a strong opinion on what to do about the limit of a function and continuity articles. Since there are several alternatives available in the literature, it's probably best to mention them all (with citations) and point out their relative merits. The fact that the literature is inconsistent makes it very hard for us to be consistent – as an encyclopedia, not a primary or secondary source, we're really not supposed to be picking and choosing definitions, we're just supposed to be reporting on what definitions other people use. Ozob (talk) 15:24, 17 August 2013 (UTC)

The literature as a whole is inconsistent, but each textbook on its own isn't (at least any well-written textbook is supposed to be free of contradictions). It's true that an encyclopedia isn't a primary or secondary source, but as a reference work I think we should still uphold the same standard of being free of contradictions. I believe this can be done while primarily reporting on what definitions other people use, rather than having to pick and choose. For example, the statements in the continuity article that rely on a different definition of "limit" than the one used in the limit article could be grouped under a section "Alternative definitions", where it can be explained that those definitions/statements rely on a slightly different definition of "limit" (one that also allows limits at non-limit points). That way everything is still being reported on, but we don't have the article containing straight contradictions (through the hyperlink to the "limit" article). Or of course the other way around with the limit article having an "Alternative definitions" section instead.B01010100 (talk) 01:20, 20 August 2013 (UTC)

Newton's notation

I recall Newton using ${\dot {r}}$ rather than $f'$ or $f^{\prime }$. Comments? — Arthur Rubin (talk) 23:26, 14 October 2013 (UTC)

This notation is yet used in physics for the derivative with respect to the time. D.Lazard (talk) 01:37, 15 October 2013 (UTC)

Structure of the article

The main question of the previous thread has been discussed by only two editors, Ozob and myself. This question is about the place in the article of an accurate definition of the derivative. Such a definition does not appear clearly in the present state of the article. I have written a new section including the definition and also the most common jargon (also lacking) here. I have placed this new section at the beginning, because it is always better to define things before talking about them. I have been reverted, and I have not reverted the revert for avoiding an edit war.

Thinking about a solution that may satisfies everybody, it appears to me that the elementary part of this article should have three parts:

• An accurate definition of the notion and of the jargon: "Definition and terminology". This part should also introduce the main notation (Leibniz's and Lagrange's notation), with links for the less common notation and for notation for higher order derivative.
• An informal explanation of the notion and of the motivations for the definition. This is essentially covered by section "Differentiation and the derivative", although my opinion is that section must be strongly edited.
• "Computing the derivative". Presently two things are lacking in this section: First the generalized chain rule: the derivative of f(u, v) where u and v are functions of x is ${\frac {df}{dx}}={\frac {\partial f}{\partial u}}\,{\frac {du}{dx}}+{\frac {\partial f}{\partial v}}\,{\frac {dv}{dx}}$); this allows, for example, to compute the derivative of x^x = e^{x log x}. Secondly the fact that these formulas make an algorithm to compute the derivative, which is implemented in computer algebra system.

Surprisingly, these three parts are almost independent: the formal definition is based on a formal definition of "limit", while the explanation uses (and explains) the intuitive notion of limit (it must be said somewhere and clearly that the formal definition is exactly the formalization of the intuitive notion). The derivation algorithm uses the formal definition only for establishing, once for all, the basic formulas, and does not imply to understand the notion (computer algebra systems do not understand anything :-) Despite this relative independence, my opinion is that the formal definition should appear first, because it is not mathematically correct to develop reasoning about a topic which is not defined. However the formal definition must be preceded by a caveat saying that this formal definition is rarely used in practice, that the next sections do not use it explicitly, and that skipping this section is not a problem.

For these reasons, I'll reinsert the reverted section, with a caveat added, and then (but probably not the same day) edit the remainder of the article to make its structure clearer. D.Lazard (talk) 14:01, 9 December 2013 (UTC)

I haven't followed all the details of the debate about the definition, but at any rate I find your comment about x^x puzzling. Why does one need functions of two variables to differentiate this? Apparently chain rule is enough. Tkuvho (talk) 14:23, 9 December 2013 (UTC)

There is a part which the article needs but does not have, and that is history. An encyclopedic article about the derivative is incomplete if it does not have any history. I am not qualified to write such a section. However, I think that if the article had a well-written history section, it would remove the need for an informal section on motivation.

I also think the single variable case should be treated totally separately from the multivariate case. I have never encountered a situation where I would use the multivariate chain rule to compute a single variable derivative, but I have no experience implementing computer algebra systems. I think I understand the x^x example as follows: Computer algebra systems do not understand how to take the derivative of u^v; but they can be given the rule u^v = e^{vlog u}, and they can be taught how take the derivative of e^{vlog u} using the single variable chain rule (for functions of the form e^{f(u, v)}) and the multivariate chain rule. Is that accurate? Ozob (talk) 14:57, 9 December 2013 (UTC)

(Edit conflict) Ooops. You are right, the generalized chain rule is needed only if there is, among the basic functions, a bivariate function, which cannot be reduced to univariate functions, but whose partial derivatives are known. This may occur with special functions), but this is out of the scope of this article. Thus I agree with the second part of Ozob's post. D.Lazard (talk) 15:20, 9 December 2013 (UTC)

About the history, I am also not qualified to write it, and I agree that it is needed. However, explanation section is, more of less, the description on how Leibniz and Newton came to the notion of derivative. Moreover it seems that there are only two important dates in this history: invention by Leibniz and Newton, and formalization by Weierstrass. It therefore possible that historical comments in the text could be more convenient that a separate section. D.Lazard (talk) 15:20, 9 December 2013 (UTC)

I think the history of multivariable derivatives might be more complicated. For example, I don't know who discovered the total derivative. Maybe it was Newton or Leibniz, but maybe not. It might have been Lagrange or Euler, or it might have been much later. Ozob (talk) 02:29, 10 December 2013 (UTC)

I am also confused about the basis for the discussion of the first bullet point. Does a rigorous definition of the derivative (along with a motivation) not already appear in the Rigorous definition section? Is the terminology not already covered in the Notation for differentiation section? What additions are exactly being proposed? Sławomir Biały (talk) 14:52, 10 December 2013 (UTC)

After cleaning out some of the more glaring issues, I think I can now see more clearly what the issues Professor Lazard raises are. My opinion is that some Introduction section should be split out of the current Differentiation and the derivative section, probably with some editing. This section should include a discussion of the slope of a line, the secant line to a function, culminating in the limit definition (though just an informal treatment of it). Then, the next section (say Definition and notation) can include a brief formal definition of the derivative, a summary of the most popular notations—probably just the Leibniz and Lagrange notations, with a link to the main Notation for differentiation article (and the existing two notations section of this article should be removed). This would look something along the lines of what Lazard originally added to the article. I'm less clear on what is to be done with The derivative as a function, Higher derivatives, and Inflection points, but I would suggest for now grouping these under a top level The derivative as a function heading. Sławomir Biały (talk) 15:42, 10 December 2013 (UTC)