Derivative of ln(9-5x)+(x+11)/6

1. Differentiate $\frac{x + 11}{6} + \log{\left(9 - 5 x \right)}$ term by term:

   1. Let $u = 9 - 5 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 - 5 x\right)$:

      1. Differentiate $9 - 5 x$ term by term:

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

         2. 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: $-5$

         The result is: $-5$

      The result of the chain rule is:

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

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

      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: $1$

      So, the result is: $\frac{1}{6}$

   The result is: $\frac{1}{6} - \frac{5}{9 - 5 x}$

2. Now simplify:

   $$\frac{5 x + 21}{6 \left(5 x - 9\right)}$$

The answer is:

$$\frac{5 x + 21}{6 \left(5 x - 9\right)}$$