Derivative of 1/(x+1)^3

1. Let $u = \left(x + 1\right)^{3}$.

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

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

   1. Let $u = x + 1$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

      1. Differentiate $x + 1$ term by term:

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

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

         The result is: $1$

      The result of the chain rule is:

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

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is:

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