Derivative of 2x-ln(x+7)^2+4

1. Differentiate $\left(2 x - \log{\left(x + 7 \right)}^{2}\right) + 4$ term by term:

   1. Differentiate $2 x - \log{\left(x + 7 \right)}^{2}$ 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 a constant times a function is the constant times the derivative of the function.

         1. Let $u = \log{\left(x + 7 \right)}$.

         2. Apply the power rule: $u^{2}$ goes to $2 u$

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

            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:

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

            The result of the chain rule is:

            $$\frac{2 \log{\left(x + 7 \right)}}{x + 7}$$

         So, the result is: $- \frac{2 \log{\left(x + 7 \right)}}{x + 7}$

      The result is: $2 - \frac{2 \log{\left(x + 7 \right)}}{x + 7}$

   2. The derivative of the constant $4$ is zero.

   The result is: $2 - \frac{2 \log{\left(x + 7 \right)}}{x + 7}$

2. Now simplify:

   $$\frac{2 \left(x - \log{\left(x + 7 \right)} + 7\right)}{x + 7}$$

The answer is:

$$\frac{2 \left(x - \log{\left(x + 7 \right)} + 7\right)}{x + 7}$$