Wikipedia:Reference desk/Archives/Mathematics/2017 July 15

= 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)
 * 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)
 * $$\frac{d}{dx}\arctan\left(\frac{\sin(\frac{\pi}{180}x)}{\cos(\frac{\pi}{180}x)+c}\right)=\frac{\pi}{180}\left(\frac{c\cos(\frac{\pi}{180}x)+1}{c^2+2c\cos(\frac{\pi}{180}x)+1}\right)$$ ? יהודה שמחה ולדמן (talk) 21:37, 15 July 2017 (UTC)
 * 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)


 * 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)
 * 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)


 * 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)
 * 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)