Answer
$$y = 1$$
Work Step by Step
$$\eqalign{ & y = {x^{\ln x}} \cr & {\text{Use }}{b^x} = {e^{x\ln b}},{\text{ then}} \cr & y = {e^{\ln x\ln x}} \cr & y = {e^{{{\ln }^2}x}} \cr & {\text{Differentiating}} \cr & \frac{{dy}}{{dx}} = {e^{{{\ln }^2}x}}\frac{d}{{dx}}\left[ {{{\ln }^2}x} \right] \cr & \frac{{dy}}{{dx}} = {e^{{{\ln }^2}x}}\left( {2\ln x} \right)\left( {\frac{1}{x}} \right) \cr & \frac{{dy}}{{dx}} = \frac{{2{e^{{{\ln }^2}x}}\ln x}}{x} \cr & \frac{{dy}}{{dx}} = \frac{{2\left( {{x^{\ln x}}} \right)\ln x}}{x} \cr & \frac{{dy}}{{dx}} = 2{x^{\ln x - 1}}\ln x \cr & \cr & {\text{Set the derivative equal to 0}} \cr & 2{x^{\ln x - 1}}\ln x = 0 \cr & 2{x^{\ln x - 1}} = 0{\text{ or }}\ln x = 0 \cr & 2{x^{\ln x - 1}}{\text{ is always positive with }}x > 0,{\text{ then}} \cr & \ln x = 0 \cr & x = 1 \cr & \cr & {\text{Then the function has a horizontal line at the point}} \cr & y\left( 1 \right) = {1^{\ln 1}} \cr & y\left( 1 \right) = 1 \cr & {\text{The horizontal line is }} \cr & y = 1 \cr} $$
