What is the derivative of x^(lnx)?

Aug 9, 2015

$\frac{d}{\mathrm{dx}} {x}^{\ln x} = 2 {x}^{\ln x - 1} \ln x$

Explanation:

Let $y = {x}^{\ln x}$
Taking the natural logarithm:
$\ln y = {\left(\ln x\right)}^{2}$

Taking the derivative with respect to x:
$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 \ln x}{x}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 {x}^{\ln x - 1} \ln x$