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