Derivative of 10ln(x-2)-10x+11

1. Differentiate $\left(- 10 x + 10 \log{\left(x - 2 \right)}\right) + 11$ term by term:

   1. Differentiate $- 10 x + 10 \log{\left(x - 2 \right)}$ term by term:

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

         1. Let $u = x - 2$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$:

            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:

            $$\frac{1}{x - 2}$$

         So, the result is: $\frac{10}{x - 2}$

      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: $-10$

      The result is: $-10 + \frac{10}{x - 2}$

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

   The result is: $-10 + \frac{10}{x - 2}$

2. Now simplify:

   $$\frac{10 \left(3 - x\right)}{x - 2}$$

The answer is:

$$\frac{10 \left(3 - x\right)}{x - 2}$$