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.

We define $f(x):= x^x \Rightarrow (f\circ\ln)(x)=\ln(x)^{\ln(x)}$
And by the chain rule $f'(x)=(x^x)'=(e^{x\ln(x)})'= (x\ln(x))'e^{x\ln(x)}=e^{x\ln(x)}(\ln(x)+1)= (\ln(x)+1)x^x$  
Now also by the chain rule $(f\circ\ln)'(x) = f'(\ln(x))\cdot(\ln(x))'=((\ln(\ln(x))+1)\ln(x)^{\ln(x)})\cdot \frac{1}{x} = \frac{(\ln(\ln(x))+1)\ln(x)^{\ln(x)}}{x}$
