Answer
\[{y^,} = \frac{{2{x^{\ln x}}\ln x}}{x}\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = {x^{\ln x}} \hfill \\
\hfill \\
f\,\left( x \right) = y\,\,\,then\,\,y = {x^{\ln x}} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\ln y = \ln {x^{\ln x}} \hfill \\
\hfill \\
use\,\,\log \,\,properties \hfill \\
\hfill \\
\ln y = \left( {\ln x} \right)\,\left( {\ln x} \right) \hfill \\
\hfill \\
\ln y = {\ln ^2}x \hfill \\
\hfill \\
Differentiate \hfill \\
\hfill \\
\frac{{{y^,}}}{y} = 2\ln x\,\left( {\frac{1}{x}} \right) \hfill \\
\hfill \\
Solve\,\,for\,\,{y^,} \hfill \\
\hfill \\
{y^,} = 2y\ln x\,\left( {\frac{1}{x}} \right) \hfill \\
\hfill \\
then \hfill \\
\hfill \\
{y^,} = \frac{{2{x^{\ln x}}\ln x}}{x} \hfill \\
\end{gathered} \]
