Derivative of 9x-ln(9x+3)

1. Differentiate $9 x - \log{\left(9 x + 3 \right)}$ 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 $u = 9 x + 3$.

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

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

         1. Differentiate $9 x + 3$ 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 the constant $3$ is zero.

            The result is: $9$

         The result of the chain rule is:

         $$\frac{9}{9 x + 3}$$

      So, the result is: $- \frac{9}{9 x + 3}$

   The result is: $9 - \frac{9}{9 x + 3}$

2. Now simplify:

   $$\frac{3 \left(9 x + 2\right)}{3 x + 1}$$

The answer is:

$$\frac{3 \left(9 x + 2\right)}{3 x + 1}$$