Derivative of (1+1/x)^3

1. Let $u = 1 + \frac{1}{x}$.

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

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

   1. Differentiate $1 + \frac{1}{x}$ term by term:

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

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

      The result is: $- \frac{1}{x^{2}}$

   The result of the chain rule is:

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

4. Now simplify:

   $$- \frac{3 \left(x + 1\right)^{2}}{x^{4}}$$

The answer is:

$$- \frac{3 \left(x + 1\right)^{2}}{x^{4}}$$