Differentiate the following functions with respect to x :

$y=\sin ^{-1}\left\{\frac{1}{\sqrt{1+x^{2}}}\right\}$

Let $x=\cot \theta$

Now

$y=\sin ^{-1}\left\{\frac{1}{\sqrt{1+\cot ^{2} \theta}}\right\}$

Using, $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$

Now

$y=\sin ^{-1}\left\{\frac{1}{\sqrt{\operatorname{cosec}^{2} \theta}}\right\}$

$y=\sin ^{-1}\left\{\frac{1}{\operatorname{cosec} \theta}\right\}$

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

$y=\theta$

$y=\cot ^{-1} x$

Differentiating w.r.t $x$ we get

$\frac{d y}{d x}=\frac{d}{d x}\left(\cot ^{-1} x\right)$

$\frac{d y}{d x}=-\frac{1}{1+x^{2}}$