If $y=\cos ^{-1}(\cos x)$, then $\frac{d y}{d x}$ at $x=\frac{5 \pi}{4}$, is

-1

We have,

$y=\cos ^{-1}(\cos x)=\left\{\begin{array}{cl} x, & \text { if } 0 \leq x \leq \pi \\ 2 \pi-x, & \text { if } \pi \leq x \leq 2 \pi \end{array}\right.$

∴  $\left(\frac{d y}{d x}\right)_{x=5 \pi / 4}=\left(\frac{d}{d x}(2 \pi-x)\right)_{x=5 \pi / 4}=-1$