Use logarithmic differentiation to find the derivative of y with respect the independent
variable x, where
y = x^{ln x thanks to everyone who takes a look }
$\displaystyle \displaystyle \begin{align*} y &= x^{\ln{x}} \\ \ln{y} &= \ln{\left( x^{\ln{x}} \right)} \\ \ln{y} &= \ln{x}\ln{x} \\ \ln{y} &= \left( \ln{x} \right)^2 \\ \frac{d}{dx} \left( \ln{y} \right) &= \frac{d}{dx} \left[ \left( \ln{x} \right)^2 \right] \\ \frac{d}{dy} \left( \ln{y} \right)\,\frac{dy}{dx} &= \frac{2\ln{x}}{x} \\ \frac{1}{y}\,\frac{dy}{dx} &= \frac{2\ln{x}}{x} \\ \frac{dy}{dx} &= \frac{2y\ln{x}}{x} \\ \frac{dy}{dx} &= \frac{2x^{\ln{x}} \ln{x} }{ x } \end{align*}$