Derivative of log(x+5)-2*x+9

1. Differentiate $\left(- 2 x + \log{\left(x + 5 \right)}\right) + 9$ term by term:

   1. Differentiate $- 2 x + \log{\left(x + 5 \right)}$ term by term:

      1. Let $u = x + 5$.

      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 5\right)$:

         1. Differentiate $x + 5$ term by term:

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

            2. The derivative of the constant $5$ is zero.

            The result is: $1$

         The result of the chain rule is:

         $$\frac{1}{x + 5}$$

      4. 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: $-2$

      The result is: $-2 + \frac{1}{x + 5}$

   2. The derivative of the constant $9$ is zero.

   The result is: $-2 + \frac{1}{x + 5}$

2. Now simplify:

   $$- \frac{2 x + 9}{x + 5}$$

The answer is:

$$- \frac{2 x + 9}{x + 5}$$