Derivative of (1/9)*x^3*(x+4)

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

   1. Apply the product rule:

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

      $f{\left(x \right)} = x^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

      $g{\left(x \right)} = x + 4$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Differentiate $x + 4$ term by term:

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

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

         The result is: $1$

      The result is: $x^{3} + 3 x^{2} \left(x + 4\right)$

   So, the result is: $\frac{x^{3}}{9} + \frac{x^{2} \left(x + 4\right)}{3}$

2. Now simplify:

   $$\frac{4 x^{2} \left(x + 3\right)}{9}$$

The answer is:

$$\frac{4 x^{2} \left(x + 3\right)}{9}$$