[tex]y=x^{\ln{x}}
\\ \ln{y}=\ln{x^{\ln{x}}}
\\ \ln{y}=\ln{x}\ln{x}
\\ \frac{1}{y} dy= (\frac{1}{x} \ln{x}+ \frac{1}{x} \ln{x})dx
\\
\\\frac{1}{y} y'= \frac{2}{x} \ln{x}
\\
\\y'=y\frac{2}{x} \ln{x}
\\
\\y'=x^{\ln{x}}\frac{2}{x} \ln{x}
\\
\\y'= \frac{x^{\ln{x}}2\ln{x}}{x}
\\
\\y'=2x^{\ln{x}-1}\ln{x}[/tex]
