From Förberedande kurs i matematik 2

We use the product rule with \displaystyle x^{2} and \displaystyle \ln x as factors:

\displaystyle \begin{align} & \left( x^{2}\ln x \right)^{\prime }=\left( x^{2} \right)^{\prime }\ln x+x^{2}\left( \ln x \right)^{\prime } \\ & \\ & =2x\centerdot \ln x+x^{2}\centerdot \frac{1}{x}=2x\ln x+x \\ \end{align}