Question

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

Answer 1

\[\text { We have, y } = \cos^{- 1} \left( \cos x \right) \]

\[ \Rightarrow y = 2\pi - x ........\left[ \because \cos^{- 1} \left( \cos x \right) = 2\pi - x , \text{ if }x \in \left[ \pi, 2\pi \right] \right] \]

\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( 2\pi - x \right)\]
\[ \Rightarrow \frac{dy}{dx} = 0 - 1\]
\[ \Rightarrow \frac{dy}{dx} = - 1\]