Question

If $\pi \le x\le 2\pi \mathrm{and}y={\cos }^{-1}\left(\cos x\right),\mathrm{find}\frac{dy}{dx}$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y={\cos }^{-1}\left(\cos x\right) \\ \Rightarrow y=2\mathrm{\pi }-x\left[\because {\cos }^{-1}\left(\cos x\right)=2\mathrm{\pi }-x,\mathrm{if}x\in \left[\mathrm{\pi },2\mathrm{\pi }\right]\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{d}{dx}\left(2\mathrm{\pi }-\mathrm{x}\right) \\ \Rightarrow \frac{dy}{dx}=0-1 \\ \Rightarrow \frac{dy}{dx}=-1 \end{gathered}$$