Differentiate the following functions with respect to x :

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

Now

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

$y=\cos ^{-1}\left\{\cos x \cos \left(\frac{\pi}{4}\right)+\sin x \sin \left(\frac{\pi}{4}\right)\right\}$

Using $\cos (A-B)=\cos A \cos B+\sin A \sin B$

$y=\cos ^{-1}\left\{\cos \left(x-\frac{\pi}{4}\right)\right\}$

Considering the limits,

$-\frac{\pi}{4}

$-\frac{\pi}{2}

Now,

$y=-x+\frac{\pi}{4}$

Differentiating it w.r.t $x$,

$\frac{\mathrm{dy}}{\mathrm{dx}}=-1$