Question

If $y={\sin }^{-1}\left(\sin x\right),-\frac{\pi }{2}\le x\le \frac{\pi }{2}$. Then, write the value of $\frac{dy}{dx}\mathrm{for}x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$

Answer 1

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

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