Derivative of (1-x^2)^2

1. Let $u = 1 - x^{2}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x^{2}\right)$:

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

      1. The derivative of the constant $1$ 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 of the chain rule is:

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

4. Now simplify:

   $$4 x \left(x^{2} - 1\right)$$

The answer is:

$$4 x \left(x^{2} - 1\right)$$