Derivative of y=x-2/(x+1)^2

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

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

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

      1. Let $u = \left(x + 1\right)^{2}$.

      2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)^{2}$:

         1. Let $u = x + 1$.

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

         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:

            $$2 x + 2$$

         The result of the chain rule is:

         $$- \frac{2 x + 2}{\left(x + 1\right)^{4}}$$

      So, the result is: $\frac{2 \left(2 x + 2\right)}{\left(x + 1\right)^{4}}$

   The result is: $1 + \frac{2 \left(2 x + 2\right)}{\left(x + 1\right)^{4}}$

2. Now simplify:

   $$\frac{4 x + \left(x + 1\right)^{4} + 4}{\left(x + 1\right)^{4}}$$

The answer is:

$$\frac{4 x + \left(x + 1\right)^{4} + 4}{\left(x + 1\right)^{4}}$$