Differentiate the following functions with respect to x :

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

let $x=\cos 2 \theta$

Now

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

$y=\cos ^{-1}\left\{\sqrt{\frac{2 \cos ^{2} \theta}{2}}\right\}$

Using $\cos 2 \theta=2 \cos ^{2} \theta-1$

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

Considering the limits,

$-1

$-1<\cos 2 \theta<1$

$0<2 \theta<\pi$

$0<\theta<\frac{\pi}{2}$

Now, $y=\cos ^{-1}(\cos \theta)$

$y=\theta$

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

Differentiating w.r.t $\mathrm{x}$, we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}\left(-\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$