Derivative of (4x^2)(6x+2)

1. Apply the product rule:

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

   $f{\left(x \right)} = 4 x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

      So, the result is: $8 x$

   $g{\left(x \right)} = 6 x + 2$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $6 x + 2$ 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: $6$

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

      The result is: $6$

   The result is: $24 x^{2} + 8 x \left(6 x + 2\right)$

2. Now simplify:

   $$8 x \left(9 x + 2\right)$$

The answer is:

$$8 x \left(9 x + 2\right)$$