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Question
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Solution
\[\text{ Let y} = \left( \sin x \right)^{\cos x }. . . \left( i \right)\]
Taking log on both sides
\[\log y = \log \left( \sin x \right)^{\cos x }\]
\[ \Rightarrow \log y = \cos x \log \sin x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \cos x\frac{d}{dx}\left( \log \sin x \right) + \log \sin x\frac{d}{dx}\left( \cos x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos x\frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right) + \log \sin x\left( - \sin x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos x}{\sin x}\left( \cos x \right) - \sin x \log \sin x\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \cos x \cot x - \sin x \log \sin x \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left( \sin x \right)^{\cos x} \left[ \cos x \cot x - \sin x \log \sin x \right] \left[ \text{using equation }\left( i \right) \right]\]
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