rule\ntake the derivative first\n$y=-x^{3}(3x^{4}-2)$\n$3x^{12}+x^{3}$\n$(12)x^{11}+3(2)x^{2}$

Explanation:

Step1: Apply the product - rule

The product - rule states that if $y = u\cdot v$, where $u=-x^{3}$ and $v = 3x^{4}-2$, then $y^\prime=u^\prime v+uv^\prime$. First, find $u^\prime$ and $v^\prime$. The derivative of $u=-x^{3}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $u^\prime=-3x^{2}$. The derivative of $v = 3x^{4}-2$ is $v^\prime = 12x^{3}$.

Step2: Substitute into the product - rule formula

$y^\prime=u^\prime v+uv^\prime=(-3x^{2})(3x^{4}-2)+(-x^{3})(12x^{3})$.

Step3: Expand the expressions

Expand $(-3x^{2})(3x^{4}-2)=-9x^{6}+6x^{2}$ and $(-x^{3})(12x^{3})=-12x^{6}$.

Step4: Combine like terms

$y^\prime=-9x^{6}+6x^{2}-12x^{6}=-21x^{6}+6x^{2}$.

Answer:

$y^\prime=-21x^{6}+6x^{2}$