Question

Prove that : $\frac{d\left(x^x\right)}{dx}=x^x(1+\log x)$

Answer 1

L.H.S

$\frac{d}{dx}{x}^x$

$\Rightarrow x^x\frac{d}{dx}\left(\log x^x\right)$

$\Rightarrow x^x\frac{d}{dx}\left(x\log x\right)$

$\Rightarrow x^x(x\times \frac{1}{x}+\log x)$

$\Rightarrow x^x(1+\log x)$

Hence, this is the answer.