Derivative of 1+9/x^2

1. Differentiate $1 + \frac{9}{x^{2}}$ term by term:

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

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

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

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

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

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

         The result of the chain rule is:

         $$- \frac{2}{x^{3}}$$

      So, the result is: $- \frac{18}{x^{3}}$

   The result is: $- \frac{18}{x^{3}}$

The answer is:

$$- \frac{18}{x^{3}}$$