Derivative of -x-(9/(x+2))

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

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

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

      So, the result is: $-1$

   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 $\frac{d}{d x} \left(x + 2\right)$:

         1. Differentiate $x + 2$ term by term:

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

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

            The result is: $1$

         The result of the chain rule is:

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

      So, the result is: $\frac{9}{\left(x + 2\right)^{2}}$

   The result is: $-1 + \frac{9}{\left(x + 2\right)^{2}}$

2. Now simplify:

   $$-1 + \frac{9}{\left(x + 2\right)^{2}}$$

The answer is:

$$-1 + \frac{9}{\left(x + 2\right)^{2}}$$