Question

Find $\frac{dy}{dx}$ if $y=x^x$

Answer 1

Given$y=x^x$

Applying $\log$ to both sides,

$\log y=\log x^x=x\log x$ since $\log a^m=m\log a$

Differentiating both sides w.r.t $x$

$\frac{d\left(\log y\right)}{dx}=\frac{d\left(x\log x\right)}{dx}$

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

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

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

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

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

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

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