#### Answer

\[{y^,} = \frac{1}{{x\ln x}}\]

#### Work Step by Step

\[\begin{gathered}
y = \ln \left| {\ln x} \right| \hfill \\
Differentiate \hfill \\
{y^,} = \,\,{\left[ {\ln \left| {\ln x} \right|} \right]^,} \hfill \\
Use\,\,the\,\,\,formula \hfill \\
\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\
Here\,\,g\,\left( x \right) = \ln x \hfill \\
{y^,} = \frac{{\,\,{{\left[ {\ln x} \right]}^,}}}{{\ln x}} \hfill \\
Then \hfill \\
{y^,} = \frac{{1/x}}{{\ln x}} \hfill \\
{y^,} = \frac{1}{{x\ln x}} \hfill \\
\end{gathered} \]