Diffrentiate the following w. r. t. x. ^ -1 ( 1- (x^2 ) 2 ) - Vidyadip

Question

Diffrentiate the following w. r. t. x.$\cos ^{-1}\left(\sqrt{\frac{1-\cos \left(x^2\right)}{2}}\right)$

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Answer

Let $y=\cos ^{-1}\left(\sqrt{\frac{1-\cos \left(x^2\right)}{2}}\right)$
$=\cos ^{-1}\left(\sqrt{\frac{2 \sin ^2\left(\frac{x^2}{2}\right)}{2}}\right)$
$=\cos ^{-1}\left[\sin \left(\frac{x^2}{2}\right)\right]$
$=\cos ^{-1}\left[\cos \left(\frac{\pi}{2}-\frac{x^2}{2}\right)\right]$
$=\frac{\pi}{2}-\frac{x^2}{2}$
Differentiating w.r.t. $x$, we get
$\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\pi}{2}-\frac{x^2}{2}\right)$
$=\frac{d}{d x}\left(\frac{\pi}{2}\right)-\frac{1}{2} \frac{d}{d x}\left(x^2\right)$
$=0-\frac{1}{2} \times 2 x=-x$

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