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July 15

Trig function derivative (degrees)

I think I asked this question a few years ago:

What is the derivative of $\arctan \left({\frac {\sin(x)}{\cos(x)+c}}\right)$ while $x$ is in degrees? יהודה שמחה ולדמן (talk) 20:51, 15 July 2017 (UTC)[reply]

It is the derivative of the function with the argument expressed in radians multiplied by

\pi /180

. Ruslik_Zero 20:56, 15 July 2017 (UTC)[reply]

{\frac {d}{dx}}\arctan \left({\frac {\sin({\frac {\pi }{180}}x)}{\cos({\frac {\pi }{180}}x)+c}}\right)={\frac {\pi }{180}}\left({\frac {c\cos({\frac {\pi }{180}}x)+1}{c^{2}+2c\cos({\frac {\pi }{180}}x)+1}}\right)

? יהודה שמחה ולדמן (talk) 21:37, 15 July 2017 (UTC)[reply]

This is an easy consequence of the chain rule. Let

u={\frac {\pi }{180}}x

and calculate

{\frac {dy}{dx}}={\frac {dy}{du}}{\frac {du}{dx}}={\frac {dy}{du}}{\frac {\pi }{180}}

.--Jasper Deng (talk) 04:09, 16 July 2017 (UTC)[reply]

But if you want the result to also be in degrees, don't you have to multiply this result by ${\frac {180}{\pi }}$ ? Then the result is just the unadjusted formula. Loraof (talk) 15:25, 16 July 2017 (UTC)[reply]

The function he has is dimensionless, so the units of the derivative should be inverse degrees, which agrees with the expression given above.--Jasper Deng (talk) 17:41, 16 July 2017 (UTC)[reply]

Actually the OP's function is the arctan function ("the angle whose tangent is ..."), whose dimensions are degrees or radians. Loraof (talk) 19:17, 16 July 2017 (UTC)[reply]

Right, I got the order of functions reversed. The conversion factor you mentioned should be applied at the beginning; then it drops out as you mentioned.--Jasper Deng (talk) 19:52, 16 July 2017 (UTC)[reply]