Question

If $y=x^x$, find $\frac{dy}{dx}$.

Answer 1

$y=x^x$

$\log y=\log x^x$

$\log y=x\log x$

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=x\times \frac{1}{x}+\log x$

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=1+\log x$

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

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