Question

Find the derivative of the inverse function of the following: $y=x\cos x$

Answer 1

$y=x\cos x$

Differentiating w.r.t. $x,$ we get

$\frac{dy}{dx}=\frac{d}{dx}\left(x\cos x\right)$

$=x\frac{d}{dx}\left(\cos x\right)+\cos x\frac{d}{dx}\left(x\right)$

$=x(-\sin x)+\cos x\times 1$

$=\cos x-x\sin x$

The derivative of inverse function of $y=f\left(x\right)$ is given by

$\frac{dx}{dy}=\frac{1}{\left(\frac{dy}{dx}\right)}$

$=\frac{1}{\cos x-x\sin x}$