Derivative of 9x-ln(x+11)+7

1. Differentiate $\left(9 x - \log{\left(x + 11 \right)}\right) + 7$ term by term:

   1. Differentiate $9 x - \log{\left(x + 11 \right)}$ 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: $9$

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

         1. Let $u = x + 11$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 11\right)$:

            1. Differentiate $x + 11$ term by term:

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

               2. The derivative of the constant $11$ is zero.

               The result is: $1$

            The result of the chain rule is:

            $$\frac{1}{x + 11}$$

         So, the result is: $- \frac{1}{x + 11}$

      The result is: $9 - \frac{1}{x + 11}$

   2. The derivative of the constant $7$ is zero.

   The result is: $9 - \frac{1}{x + 11}$

2. Now simplify:

   $$\frac{9 x + 98}{x + 11}$$

The answer is:

$$\frac{9 x + 98}{x + 11}$$