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

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = x$ and $g{\left(x \right)} = \sqrt{1 - x^{2}}$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x$ goes to $1$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   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:

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

   Now plug in to the quotient rule:

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

2. Now simplify:

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

The answer is:

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