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