Question

Differentiate w.r.t $x$:

$(\log x{)}^x$

Answer 1

$y=(\log x{)}^x$

Taking log on both sides

$\log y=x\log \left(\log x\right)$

Differentiating w.r.t. x, we get,

$\frac{1}{y}\frac{dy}{dx}=\frac{x}{x\log x}+\log \left(\log x\right)$

$\frac{dy}{dx}=\left(\log x{)}^x\right[\frac{1}{\log x}+\log \left(\log x\right)]$