Differentiate the following functions with respect to x :

Solution:

$y=\cos ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}$

let $x=\cos \theta$

Now

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

$=\cos ^{-1}\left\{2 \cos \theta \sqrt{\sin ^{2} \theta}\right\}$

Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ and $2 \sin \theta \cos \theta=\sin 2 \theta$

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

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

$y=\cos ^{-1}\left(\cos \left(\frac{\pi}{2}-2 \theta\right)\right)$

Considering the limits,

$\frac{1}{\sqrt{2}}

$\Rightarrow \frac{1}{\sqrt{2}}<\cos \theta<1$

$\Rightarrow 0<\theta<\frac{\pi}{4}$

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

$\Rightarrow 0>-2 \theta>-\frac{\pi}{2}$

$\Rightarrow \frac{\pi}{2}>\frac{\pi}{2}-2 \theta>\frac{\pi}{2}-\frac{\pi}{2}$

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

Therefore,

$y=\cos ^{-1}\left(\cos \left(\frac{\pi}{2}-2 \theta\right)\right)$

$y=\cos ^{-1}\left(\cos \left(\frac{\pi}{2}-2 \theta\right)\right)$

$y=\left(\frac{\pi}{2}-2 \theta\right)$

$y=\frac{\pi}{2}-2 \cos ^{-1} x$

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

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\pi}{2}-2 \cos ^{-1} \mathrm{x}\right)$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=0-2\left(\frac{-1}{\sqrt{1-\mathrm{x}^{2}}}\right)$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2}{\sqrt{1-\mathrm{x}^{2}}}$