Derivative of x^3(2-x)

1. Apply the product rule:

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

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

   1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

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

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

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

      2. 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: $-1$

      The result is: $-1$

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

2. Now simplify:

   $$x^{2} \cdot \left(6 - 4 x\right)$$

The answer is:

$$x^{2} \cdot \left(6 - 4 x\right)$$