Derivative of 1/(1-x^2)^2

1. Let $u = \left(1 - x^{2}\right)^{2}$.

2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

   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:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$- \frac{4 x}{\left(x^{2} - 1\right)^{3}}$$

The answer is:

$$- \frac{4 x}{\left(x^{2} - 1\right)^{3}}$$