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