Derivative of ln(4+2x)

1. Let $u = 2 x + 4$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 4\right)$:

   1. Differentiate $2 x + 4$ term by term:

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

      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: $2$

      The result is: $2$

   The result of the chain rule is:

   $$\frac{2}{2 x + 4}$$

4. Now simplify:

   $$\frac{1}{x + 2}$$

The answer is:

$$\frac{1}{x + 2}$$