Question

If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?

Answer 1

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

\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( \pi - x \right)\]

\[ \Rightarrow \frac{dy}{dx} = 0 - 1\]

\[ \Rightarrow \frac{dy}{dx} = - 1\]