Derivative of x^2*(2*log(x)-3)

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 $2 x$

   $g{\left(x \right)} = 2 \log{\left(x \right)} - 3$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $2 \log{\left(x \right)} - 3$ term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

         So, the result is: $\frac{2}{x}$

      2. The derivative of the constant $-3$ is zero.

      The result is: $\frac{2}{x}$

   The result is: $2 x \left(2 \log{\left(x \right)} - 3\right) + 2 x$

2. Now simplify:

   $$4 x \left(\log{\left(x \right)} - 1\right)$$

The answer is: