Derivative of (x+1)^1/3

1. Let $u = x + 1$.

2. Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$

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:

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

4. Now simplify:

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

The answer is:

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