Derivative of x^2(x-2)^2

1. Apply the product rule:

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

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

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

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

   1. Let $u = x - 2$.

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

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

      $$2 x - 4$$

   The result is: $x^{2} \left(2 x - 4\right) + 2 x \left(x - 2\right)^{2}$

2. Now simplify:

   $$4 x \left(x - 2\right) \left(x - 1\right)$$

The answer is:

$$4 x \left(x - 2\right) \left(x - 1\right)$$