Derivative of (x-6)x^3

1. Apply the product rule:

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

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

   1. Differentiate $x - 6$ term by term:

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

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

      The result is: $1$

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

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

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

2. Now simplify:

   $$x^{2} \left(4 x - 18\right)$$

The answer is:

$$x^{2} \left(4 x - 18\right)$$