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