Derivative of (x²-3x)(2x-3)²

1. Apply the product rule:

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

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

   1. Differentiate $x^{2} - 3 x$ term by term:

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

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

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

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

            So, the result is: $3$

         So, the result is: $-3$

      The result is: $2 x - 3$

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

   1. Let $u = 2 x - 3$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)$:

      1. Differentiate $2 x - 3$ term by term:

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

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

            So, the result is: $2$

         2. The derivative of the constant $\left(-1\right) 3$ is zero.

         The result is: $2$

      The result of the chain rule is:

      $$8 x - 12$$

   The result is: $\left(2 x - 3\right) \left(2 x - 3\right)^{2} + \left(8 x - 12\right) \left(x^{2} - 3 x\right)$

2. Now simplify:

   $$16 x^{3} - 72 x^{2} + 90 x - 27$$

The answer is:

$$16 x^{3} - 72 x^{2} + 90 x - 27$$