Differentiate the following functions with respect to x :
Question:

Differentiate the following functions with respect to $x$ :

$\cos ^{-1}\left(\frac{1-x^{2 n}}{1+x^{2 n}}\right), 0<x<\infty$

Solution:

$y=\cos ^{-1}\left\{\frac{1-x^{2 n}}{1+x^{2 n}}\right\}$

Let $x^{n}=\tan \theta$

Now

$y=\cos ^{-1}\left\{\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right\}$

Using $\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}=\cos 2 \theta$

$y=\cos ^{-1}\{\cos 2 \theta\}$

Considering the limits,

$0<x<\infty$

$0<x^{n}<\infty$

$0<\theta<\frac{\pi}{2}$

Now,

$y=\cos ^{-1}(\cos 2 \theta)$

$y=2 \theta$

$y=\tan ^{-1}\left(x^{n}\right)$

Differentiating w.r.t $\mathrm{x}$, we get

$\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1}\left(x^{n}\right)\right)$

$\frac{d y}{d x}=\frac{2 n x^{n-1}}{1+\left(x^{n}\right)^{2}}$

$\frac{d y}{d x}=\frac{2 n x^{n-1}}{1+x^{2 n}}$