Derivative of y=8ln(x+7)-8x+3

1. Differentiate $\left(- 8 x + 8 \log{\left(x + 7 \right)}\right) + 3$ term by term:

   1. Differentiate $- 8 x + 8 \log{\left(x + 7 \right)}$ term by term:

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

         1. Let $u = x + 7$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 7\right)$:

            1. Differentiate $x + 7$ term by term:

               1. Apply the power rule: $x$ goes to $1$

               2. The derivative of the constant $7$ is zero.

               The result is: $1$

            The result of the chain rule is:

            $$\frac{1}{x + 7}$$

         So, the result is: $\frac{8}{x + 7}$

      2. 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: $-8$

      The result is: $-8 + \frac{8}{x + 7}$

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

   The result is: $-8 + \frac{8}{x + 7}$

2. Now simplify:

   $$- \frac{8 x + 48}{x + 7}$$

The answer is:

$$- \frac{8 x + 48}{x + 7}$$