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

Question

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

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Answer

Let $y=\cos ^{-1}\left(\sqrt{\frac{1+x}{2}}\right)$
Differentiate w.r.t.x.
$
\begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left[\cos ^{-1}\left(\sqrt{\frac{1+x}{2}}\right)\right] \\
& =-\frac{1}{\sqrt{1-\left(\sqrt{\frac{1+x}{2}}\right)^2}} \cdot \frac{d}{d x}\left(\sqrt{\frac{1+x}{2}}\right) \\
& =-\frac{1}{\sqrt{1-\frac{1+x}{2}}} \times \frac{1}{2 \sqrt{\frac{1+x}{2}}} \times \frac{d}{d x}\left(\frac{1+x}{2}\right) \\
& =-\frac{\sqrt{2}}{\sqrt{1-x}} \times \frac{1}{\sqrt{2} \sqrt{1+x}} \times \frac{1}{2} \\
\therefore \frac{d y}{d x} & =-\frac{1}{2 \sqrt{1-x^2}}
\end{aligned}
$

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