Derivative of x^2(16-x^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 $2 x$

   $g{\left(x \right)} = 16 - x^{2}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $16 - x^{2}$ term by term:

      1. The derivative of the constant $16$ 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: $x^{2}$ goes to $2 x$

         So, the result is: $- 2 x$

      The result is: $- 2 x$

   The result is: $- 2 x^{3} + 2 x \left(16 - x^{2}\right)$

2. Now simplify:

   $$4 x \left(8 - x^{2}\right)$$

The answer is:

$$4 x \left(8 - x^{2}\right)$$