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Question
Differentiate \[x^{\tan^{- 1} x }\] ?
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Solution
\[\text{ Let y } = x^{\tan^{- 1} x} . . . \left( i \right)\]
\[\text{ Taking log on both sides }, \]
\[\log y = \log x^{\tan^{- 1} x } \]
\[ \Rightarrow \log y = \tan^{- 1} x \log x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \tan^{- 1} x\frac{d}{dx}\log x + \log x\frac{d}{dx}\left( \tan^{- 1} x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \tan^{- 1} x\left( \frac{1}{x} \right) + \log x\left( \frac{1}{1 + x^2} \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\tan^{- 1} x}{x} + \frac{\log x}{1 + x^2} \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^{\tan^{- 1} x} \left[ \frac{\tan^{- 1} x}{x} + \frac{\log x}{1 + x^2} \right] \left[\text{ using equation } \left( i \right) \right]\]
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