Answer
$\frac{{dy}}{{dx}} = \frac{{2{x^{\ln x}}\ln x}}{x}$
Work Step by Step
$$\eqalign{
& y = {x^{\ln x}} \cr
& {\text{using the logarithmic differentiation}} \cr
& \ln y = \ln \left( {{x^{\ln x}}} \right) \cr
& \ln y = \ln x\ln x \cr
& \ln y = {\ln ^2}x \cr
& {\text{Differentiate both sides with respect to }}x \cr
& \frac{d}{{dx}}\left[ {\ln y} \right] = \frac{d}{{dx}}\left[ {{{\ln }^2}x} \right] \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = 2\ln x\left( {\frac{1}{x}} \right) \cr
& {\text{Solve for }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = 2y\ln x\left( {\frac{1}{x}} \right) \cr
& {\text{Substitute }}{x^{\ln x}}{\text{ for }}y \cr
& \frac{{dy}}{{dx}} = 2{x^{\ln x}}\ln x\left( {\frac{1}{x}} \right) \cr
& {\text{Simplify}} \cr
& \frac{{dy}}{{dx}} = \frac{{2{x^{\ln x}}\ln x}}{x} \cr} $$
