Derivative of 4ln(2x+5)-4x+17

1. Differentiate $- 4 x + 4 \log{\left(2 x + 5 \right)} + 17$ term by term:

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

      1. Let $u = 2 x + 5$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 5\right)$:

         1. Differentiate $2 x + 5$ 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: $2$

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

            The result is: $2$

         The result of the chain rule is:

         $$\frac{2}{2 x + 5}$$

      So, the result is: $\frac{8}{2 x + 5}$

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

      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: $4$

      So, the result is: $-4$

   3. The derivative of the constant $17$ is zero.

   The result is: $-4 + \frac{8}{2 x + 5}$

2. Now simplify:

   $$\frac{4 \left(- 2 x - 3\right)}{2 x + 5}$$

The answer is:

$$\frac{4 \left(- 2 x - 3\right)}{2 x + 5}$$