Derivative of x(lnx)^2

1. Apply the product rule:

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

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

   1. Apply the power rule: $x$ goes to $1$

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

   1. Let $u = \log{\left(x \right)}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$:

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

      The result of the chain rule is:

      $$\frac{2 \log{\left(x \right)}}{x}$$

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

2. Now simplify:

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

The answer is:

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