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