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

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

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

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

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

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

   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:

      $$- \frac{x}{\sqrt{1 - x^{2}}}$$

   The result of the chain rule is:

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

The answer is:

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