Answer
$$ y^{\prime}=\left(\frac{\ln (\ln x)+1}{x}\right)(\ln x)^{\ln x}$$
Work Step by Step
Given $$ y=(\ln x)^{\ln x} $$
So, we have
\begin{aligned}
&y=(\ln x)^{\ln x} \\
&\Rightarrow \ln y=(\ln x) \ln (\ln x) \\
&\text{ differentiate with respect to } \ x,\\
&\Rightarrow \frac{y^{\prime}}{y}=\left(\frac{1}{x}\right) \ln (\ln x)+(\ln x)\left(\frac{1}{\ln x}\right) \frac{d}{d x}(\ln x)\\
& \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\ln (\ln x)}{x}+\frac{1}{x}
\\&\Rightarrow y^{\prime}=y \left(\frac{\ln (\ln x)+1}{x}\right) \\&\\&\Rightarrow y^{\prime}=\left(\frac{\ln (\ln x)+1}{x}\right)(\ln x)^{\ln x}
\end{aligned}
