What is $\frac{dy}{dx}$ for $y = \tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$, $-\frac{\pi}{4} < x < \frac{\pi}{4}$?

Explanation

1. 01

   Divide the top and the bottom by $\cos x$: $\frac{1 - \tan x}{1 + \tan x} = \tan\left(\frac{\pi}{4} - x\right)$.

2. 02

   For $-\frac{\pi}{4} < x < \frac{\pi}{4}$, $\frac{\pi}{4} - x$ lies in $\left(0, \frac{\pi}{2}\right)$, so $y = \frac{\pi}{4} - x$ and $\frac{dy}{dx} = -1$.