Let $f(x)=x^x$.
What is the derivative of $f$?
This function can't be treated by chain rule or product rule or (e^x)'=e^x
Let $f(x)=x^x$.
What is the derivative of $f$?
This function can't be treated by chain rule or product rule or (e^x)'=e^x
$\begin{align*}\frac{d}{dx}(x^x) &= \frac{d}{dx}(e^{x \ln x}) & \textrm{(Using the fact that $x^x = e^{x \ln x})$}\\ & = e^{x \ln x} \frac{d}{dx}(x \ln x) & \textrm{(Using the chain rule)} \\ &=x^x (\ln x + 1) & \textrm{(Using the product rule)}\end{align*}$
let $y=x^x$ $\implies$ $\log y=x\log x$ Differentiate on both sides $\dfrac{dy}{dx}\left(\dfrac{1}{y}\right)=\log x+1$ $\implies $ $\dfrac{dy}{dx}=x^x(\log x+1)$
$f(x)=x^x=e^{x\ln x}$ from here you use the chain rule for $(e^g)'=e^g g'$.