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

Explanation

1. 01

   Take logs: $\log y = x\log x$, so $\frac{1}{y}\frac{dy}{dx} = \log x + 1$.

2. 02

   $\frac{dy}{dx} = x^{x}(1 + \log x)$.