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October 2

Is there any short-cut method for finding the derivative or integral of a function?

Finding limit of a function may be lengthy in some cases, but those limits can simply be solved using L'Hôpital's rule. In fact, almost every question related to limits can be solved easily using L'Hôpital's rule. I am searching for some short-cut methods (like L'Hôpital's rule) for finding the derivative or integral of a function. You know finding integral or derivative of a function in some cases is really very lengthy. So, I need a method (or rule) by which I could find derivative or integral of a given function in shortest time possible. Thank you for you response! Concepts of Physics (talk) 15:34, 2 October 2013 (UTC)[reply]

Graphical methods might work if you only need approximate numeric values within a certain range. That is, graph it, then find the slope, area under the curve, etc. StuRat (talk) 15:38, 2 October 2013 (UTC)[reply]

See our articles on automatic differentiation and symbolic integration. In general, integration is more difficult than differentiation. Gandalf61 (talk) 15:44, 2 October 2013 (UTC)[reply]

This may not be what you are looking for, but Wolfram Alpha and programs like Maple (software) and Mathematica do them. Bubba73 ^{You talkin' to me?} 15:49, 2 October 2013 (UTC)[reply]

In fact, almost every question related to limits can be solved easily using L'Hôpital's rule. ^{[citation needed]}, I think. In any case, application of L'Hôpital's rule involves finding the derivatives of two functions, so it's hardly a 'shortcut' compared to differentiation. AndrewWTaylor (talk) 16:15, 2 October 2013 (UTC)[reply]

I think the OP meant that just as L'Hopital's Rule is a shortcut method for finding limits, so also method "?????" is a shortcut for differentiating. Duoduoduo (talk) 17:43, 2 October 2013 (UTC)[reply]

You think that "every question related to limits can be solved easily using L'Hôpital's rule" because in your course you have been given exercises that are to be solved using L'Hôpital's rule. This doesn't apply generally.

If you want a symbolic expression for a derivative, the "shortcut" is to use Differentiation rules rather than the limit definition. I don't believe there is a shorter way.

For integration, while there are some systematic methods, in practice it is more of an art, with many integration methods to try. -- Meni Rosenfeld (talk) 08:00, 3 October 2013 (UTC)[reply]

If you need to evaluate a high order derivative in a point, then the fastest method is to use the Taylor expansion around that point. To evaluate a Taylor expansion to order N, the required number of computations is of order of some power of Log(N), so to find the trillionth derivative of a complicated function in some point only requires some dozens of computations. Count Iblis (talk) 14:06, 3 October 2013 (UTC)[reply]

I'm curious — how does one derive a Taylor series that efficiently? --Tardis (talk) 21:12, 6 October 2013 (UTC)[reply]

Definite integrals can sometimes be immediately evaluated from the series expansion of the integrand e.g. using Ramanujan's master theorem or some suitable generalization of this based on Umbral calculus. These methods sometimes allow you to immediately write down the result of an integral that Wolfram Alpha or Mathematica cannot evaluate. Count Iblis (talk) 14:40, 3 October 2013 (UTC)[reply]