Question

If $\frac{\pi }{2}\le x\le \frac{3\pi }{2}\mathrm{and}y={\sin }^{-1}\left(\sin x\right),\mathrm{find}\frac{dy}{dx}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y={\sin }^{-1}\left(\sin x\right) \\ \Rightarrow y=\mathrm{\pi }-x\left[\because {\sin }^{-1}\left(\sin x\right)=\mathrm{\pi }-x,\mathrm{if}x\in \left[\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\right]\right] \end{gathered}$$

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