If `y=tan^(-1){(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))},"f i n d"(dy)/(dx)`

Explanation

\begin{equation}
\begin{aligned}
&\text { Let } x=\cos 2 \theta=2 \cos ^{2} \theta-1 \\
&y=\tan ^{-1}\left(\frac{\sqrt{1+\cos 2 \theta}-\sqrt{1-\cos 2 \theta}}{\sqrt{1+\cos 2 \theta}+\sqrt{1-\cos 2 \theta}}\right) \\
&y=\tan ^{-1}\left(\frac{\sqrt{2} \cos \theta-\sqrt{2} \sin \theta}{\sqrt{2} \cos \theta-\sqrt{2} \sin \theta}\right) \\
&y=\tan ^{-1}\left(\frac{1-\tan \theta}{1+\tan \theta}\right) \\
&y=\frac{\pi}{4}-\theta=\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x \\
&\frac{d y}{d x}=-\frac{1}{2}-\frac{1}{\sqrt{1-x^{2}}}=\frac{1}{2 \sqrt{1-x^{2}}}
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