Derivative of y=ln(11x)-11x+9

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

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

      1. Let $u = 11 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 11 x$:

         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: $11$

         The result of the chain rule is:

         $$\frac{1}{x}$$

      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: $-11$

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

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

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

The answer is:

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