Derivative of $(\ln x)^{\ln x}$ How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions?
Thank you.

Note that $$a^b = {\left(e^{\ln a}\right)}^b = e^{b\ln a}$$
so $$\left(\ln x\right)^{\ln x} = {\left(e^{\ln\ln x}\right)}^{\ln x}\\
= e^{\ln x\cdot\ln\ln x} = x^{\ln \ln x}$$ either of which which you should be able to do with methods you already know.
We can apply this technique generally to calculate the derivative of $f(x)^{g(x)}$:
$$f(x)^{g(x)} = e^{g\ln f}$$
so $$\begin{align}
\frac d{dx} f^g  = \frac d{dx} e^{g\cdot\ln f} & = \left(\frac d{dx} \left(g\cdot\ln f\right)\right)e^{g\cdot\ln f}\qquad\text{(chain rule)}\\& = \left(\frac d{dx} \left(g\cdot\ln f\right)\right)f^g \\
& = \left(g'\ln f + \frac {gf'}f\right)f^g\qquad\text{(product rule)}
\end{align} $$
where $f' =\frac{df}{dx}$ and $g' = \frac{dg}{dx}$.
