Derivative of y=-x³(3x⁴-2)

1. Apply the product rule:

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

   $f{\left(x \right)} = - x^{3}$; 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: $x^{3}$ goes to $3 x^{2}$

      So, the result is: $- 3 x^{2}$

   $g{\left(x \right)} = 3 x^{4} - 2$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $3 x^{4} - 2$ 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: $x^{4}$ goes to $4 x^{3}$

         So, the result is: $12 x^{3}$

      2. The derivative of the constant $\left(-1\right) 2$ is zero.

      The result is: $12 x^{3}$

   The result is: $- 12 x^{6} - 3 x^{2} \cdot \left(3 x^{4} - 2\right)$

2. Now simplify:

   $$x^{2} \cdot \left(6 - 21 x^{4}\right)$$

The answer is:

$$x^{2} \cdot \left(6 - 21 x^{4}\right)$$