Question

Differentiate \[3^{x \log x}\] ?

Answer 1

\[\text{ Let } y = 3^{x \log x} \]

\[\text{Differentiate it with respect to x we get}, \]

\[\frac{d y}{d x} = \frac{d}{dx}\left( 3^{x \log x} \right)\]

\[ = 3^{x \log x} \times \log_e 3\frac{d}{dx}\left( x \log x \right) \left[ \text{Using chain rule} \right]\]

\[ = 3^x \log x \times \log_e 3\left[ x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right) \right] \]

\[ = 3^{x \log x} \times \log_e 3\left[ \frac{x}{x} + \log x \right]\]

\[ = 3^{x \log x} \left( 1 + \log x \right) \times \log_e 3\]

\[So, \frac{d}{dx}\left( 3^{x \log x} \right) = 3^{x \log x} \left( 1 + \log x \right) \log_e 3\]