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

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

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

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

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

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

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

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

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

         2. The derivative of the constant $1$ is zero.

         The result is: $2 x$

      The result of the chain rule is:

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

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: