# How do you find the derivative of x^(2 lnx)?

Apr 15, 2018

$y ' = 4 {x}^{2 \ln x} \ln \frac{x}{x}$

#### Explanation:

$y = {x}^{2 \ln x}$

by taking the natural logarithm to both sides

$\ln y = \ln {x}^{2 \ln x}$

using the properties of the logarithmic functions
$\textcolor{g r e e n}{\ln {u}^{v} = v \ln u}$

$\ln y = 2 \ln x \cdot \ln x$

$\ln y$ =$2 {\left(\ln x\right)}^{2}$

Differentiate

$\frac{y '}{y} = 4 \ln \frac{x}{x}$

$y ' = 4 y \ln \frac{x}{x}$

Substitute for $y$

$y ' = 4 {x}^{2 \ln x} \ln \frac{x}{x}$