Derivative of ln(x/(x+2))+1

1. Differentiate $\log{\left(\frac{x}{x + 2} \right)} + 1$ term by term:

   1. Let $u = \frac{x}{x + 2}$.

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

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

      1. Apply the quotient rule, which is:

         $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

         $f{\left(x \right)} = x$ and $g{\left(x \right)} = x + 2$.

         To find $\frac{d}{d x} f{\left(x \right)}$:

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

         To find $\frac{d}{d x} g{\left(x \right)}$:

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

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

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

            The result is: $1$

         Now plug in to the quotient rule:

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

      The result of the chain rule is:

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

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

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

2. Now simplify:

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

The answer is:

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