How to differentiate $(\ln n) ^{\ln n}$?

Question

$$f(n) = (\ln n)^{\ln n}$$

Can someone explain me how to differentiate the above function?

I am trying the following solution

$$f'(n) = \frac{\ln n\left((\ln n) ^ {\ln n} - 1\right)}{(\frac1n \ln 2)}$$

Answer 1

Let $f(x)=(\log x)^{\log x}$; then $$ \log f(x)=\log x\cdot \log\log x $$ and so, differentiating both sides, $$ \frac{f'(x)}{f(x)}= \frac{1}{x}\cdot\log\log x+\log x\cdot\frac{1}{\log x}\cdot \frac{1}{x}= \frac{1+\log\log x}{x} $$

If $f(x)=(\log_a x)^{\log_ax}$, then we have again $$ \log f(x)=\log_ax\cdot \log\log_ax $$ (no subscript means natural logarithm). So $$ \frac{f'(x)}{f(x)}= \frac{1}{x\log a}\cdot\log\log_ax+\log_ax\cdot\frac{1}{\log_ax}\cdot\frac{1}{x\log a}= \frac{1+\log\log_ax}{x\log a} $$ remembering that if $g(x)=\log_ax$, then $g'(x)=1/(x\log a)$.