Derivative of log_7⁡(12x+5)

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

   1. Let $u = 12 x + 5$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(12 x + 5\right)$:

      1. Differentiate $12 x + 5$ 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: $12$

         2. The derivative of the constant $5$ is zero.

         The result is: $12$

      The result of the chain rule is:

      $$\frac{12}{12 x + 5}$$

   So, the result is: $\frac{12}{\left(12 x + 5\right) \log{\left(7 \right)}}$

2. Now simplify:

   $$\frac{12}{\left(12 x + 5\right) \log{\left(7 \right)}}$$

The answer is:

$$\frac{12}{\left(12 x + 5\right) \log{\left(7 \right)}}$$