Derivative of x^2lnx

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{2}$ goes to $2 x$

   $g{\left(x \right)} = \log{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

   The result is: $2 x \log{\left(x \right)} + x$

2. Now simplify:

   $$x \left(2 \log{\left(x \right)} + 1\right)$$

The answer is:

$$x \left(2 \log{\left(x \right)} + 1\right)$$