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

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

   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 $g{\left(x \right)} = x + 1$.

      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 $x + 1$ term by term:

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

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

         The result is: $1$

      Now plug in to the quotient rule:

      $\frac{1}{\left(x + 1\right)^{2}}$

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

      1. Let $u = x + 1$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$:

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

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

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

            The result is: $1$

         The result of the chain rule is:

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

      So, the result is: $- \frac{2}{x + 1}$

   The result is: $- \frac{2}{x + 1} + \frac{1}{\left(x + 1\right)^{2}}$

2. Now simplify:

   $$\frac{x - 2 \left(x + 1\right)^{2} + 1}{\left(x + 1\right)^{3}}$$

The answer is:

$$\frac{x - 2 \left(x + 1\right)^{2} + 1}{\left(x + 1\right)^{3}}$$