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Question
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
Solution
\[\text{ We have, y } = \sin^{- 1} \left( \sin x \right)\]
\[ \Rightarrow y = x \left[ \because \sin^{- 1} \left( \sin x \right) = x , \text {if }x \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( x \right)\]
\[ \Rightarrow \frac{dy}{dx} = 1\]
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