Question

Diffrentiate w.r.t x :

$y={\cos }^{-1}\left[\frac{1-x}{1+x}\right]$

Answer 1

$y={\cos }^{-1}\left[\frac{1-x}{1+x}\right]\Rightarrow \cos y=\frac{1-x}{1+x}\Rightarrow -\sin y\frac{dy}{dx}=\frac{(1+x)\frac{d}{dx}(1-x)-(1-x)\frac{d}{dx}(1+x)}{{(1+x)}^2}\Rightarrow \frac{dy}{dx}=\frac{2}{\sin y{(1+x)}^2}\Rightarrow \frac{dy}{dx}=\frac{2}{\sin \left({\cos }^{-1}\left[\frac{1-x}{1+x}\right]\right){(1+x)}^2}$