Derivative of y=ln(x+11)^12-12x

1. Differentiate $- 12 x + \log{\left(x + 11 \right)}^{12}$ term by term:

   1. Let $u = \log{\left(x + 11 \right)}$.

   2. Apply the power rule: $u^{12}$ goes to $12 u^{11}$

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

      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:

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

      The result of the chain rule is:

      $$\frac{12 \log{\left(x + 11 \right)}^{11}}{x + 11}$$

   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: $-12$

   The result is: $-12 + \frac{12 \log{\left(x + 11 \right)}^{11}}{x + 11}$

2. Now simplify:

   $$\frac{12 \left(- x + \log{\left(x + 11 \right)}^{11} - 11\right)}{x + 11}$$

The answer is:

$$\frac{12 \left(- x + \log{\left(x + 11 \right)}^{11} - 11\right)}{x + 11}$$