Derivative of x^3/(2*x+4)

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 $g{\left(x \right)} = 2 x + 4$.

   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. 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: $2$

   Now plug in to the quotient rule:

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

2. Now simplify:

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

The answer is:

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