Derivative of 2/(1+x)^3

1. The derivative of a constant times a function is the constant times the derivative of the function.

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

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

            The result is: $1$

         The result of the chain rule is:

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

      The result of the chain rule is:

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

   So, the result is: $- \frac{6}{\left(x + 1\right)^{4}}$

The answer is:

$$- \frac{6}{\left(x + 1\right)^{4}}$$