#### Answer

\[ = \frac{1}{{x\ln x}}\]

#### Work Step by Step

\[\begin{gathered}
\frac{d}{{dx}}\,\,\left[ {\ln \,\left( {\ln x} \right)} \right] \hfill \\
\hfill \\
Use\,\,\frac{d}{{dx}}\,\,\left[ {\ln u} \right] = \frac{{{u^,}}}{u} \hfill \\
\hfill \\
Set\,\,u = \ln x \hfill \\
\,\,\,\,\,\,\,\,\,\,\,{u^,} = \frac{1}{x} \hfill \\
\hfill \\
Therefore \hfill \\
\hfill \\
= \frac{{\frac{1}{x}}}{{\ln x}} \hfill \\
\hfill \\
Simplify \hfill \\
\hfill \\
= \frac{1}{{x\ln x}} \hfill \\
\end{gathered} \]