Question

Find the derivation of $f\left(x\right)=x^x(x>0)$ by first principle.

Answer 1

$f\left(x\right)=x^x\log f\left(x\right)=x\log x\frac{1}{f\left(x\right)}{f}^{{}^{\prime }}\left(x\right)=\log x+1f^{{}^{\prime }}\left(x\right)=f\left(x\right)(\log x+1)f^{{}^{\prime }}\left(x\right)=x^x(\log x+1)$