Derivative of x^3/(2(x+1)^2)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x^{3}$ and $g{\left(x \right)} = 2 \left(x + 1\right)^{2}$.

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

   1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

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

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

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

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

            The result is: $1$

         The result of the chain rule is:

         $$2 x + 2$$

      So, the result is: $4 x + 4$

   Now plug in to the quotient rule:

   $\frac{- x^{3} \left(4 x + 4\right) + 6 x^{2} \left(x + 1\right)^{2}}{4 \left(x + 1\right)^{4}}$

2. Now simplify:

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

The answer is:

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