What is $\frac{d}{dx}\left(x^{\log x}\right)$ for $x > 0$?

Explanation

1. 01

   $\log y = \log x \cdot \log x = (\log x)^{2}$, so $\frac{1}{y}\frac{dy}{dx} = \frac{2\log x}{x}$.

2. 02

   Multiply back by $y$: $\frac{dy}{dx} = x^{\log x} \cdot \frac{2\log x}{x}$.