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

Question

Diffrentiate the following w. r. t. x.

$\cos ^{-1}\left(\sqrt{\frac{1+\cos x}{2}}\right)$

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Answer

Let$\begin{aligned} y & =\cos ^{-1}\left(\sqrt{\frac{1+\cos x}{2}}\right) \\ & =\cos ^{-1}\left(\sqrt{\frac{2 \cos ^2\left(\frac{x}{2}\right)}{2}}\right) \\ & =\cos ^{-1}\left[\cos \left(\frac{x}{2}\right)\right] \\ & =\frac{x}{2} \end{aligned} $

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

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left(\frac{x}{2}\right)=\frac{1}{2} \frac{d}{d x}(x) \\ & =\frac{1}{2} \times 1=\frac{1}{2} .\end{aligned}$

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