Derivative of (x^3)(4x+5)^4

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 $3 x^{2}$

   $g{\left(x \right)} = \left(4 x + 5\right)^{4}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 4 x + 5$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 5\right)$:

      1. Differentiate $4 x + 5$ term by term:

         1. 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: $4$

         2. The derivative of the constant $5$ is zero.

         The result is: $4$

      The result of the chain rule is:

      $$16 \left(4 x + 5\right)^{3}$$

   The result is: $16 x^{3} \left(4 x + 5\right)^{3} + 3 x^{2} \left(4 x + 5\right)^{4}$

2. Now simplify:

   $$x^{2} \left(4 x + 5\right)^{3} \left(28 x + 15\right)$$

The answer is:

$$x^{2} \left(4 x + 5\right)^{3} \left(28 x + 15\right)$$