Derivative of y=(x^3+6)(x-2)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{3} + 6$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x^{3} + 6$ term by term:

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

      2. The derivative of the constant $6$ is zero.

      The result is: $3 x^{2}$

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

   1. Differentiate $x - 2$ term by term:

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

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

      The result is: $1$

   The result is: $x^{3} + 3 x^{2} \left(x - 2\right) + 6$

2. Now simplify:

   $$4 x^{3} - 6 x^{2} + 6$$

The answer is:

$$4 x^{3} - 6 x^{2} + 6$$