Derivative of log x raised to the power x at e

Differentiate $f(x)=(\ln x)^x$ and evaluate the derivative at $x=e$.

$1$. Taking logs, $\ln f=x\ln(\ln x)$, so $f'=f\left[\ln(\ln x)+\tfrac{1}{\ln x}\right]$. At $x=e$, $f=1$, $\ln(\ln e)=0$ and $1/\ln e=1$, so $f'(e)=1$.