The problem asks us to find the derivative of the function $y = -x^3(3x^4 - 2)$.

1. Problem Description

The problem asks us to find the derivative of the function $y = -x^3(3x^4 - 2)$.

2. Solution Steps

First, we distribute the $-x^3$ term:

$y = -3x^7 + 2x^3$

Now, we apply the power rule for differentiation, which states that if $y = ax^n$, then $\frac{dy}{dx} = nax^{n-1}$.

Applying the power rule to the first term, $-3x^7$, we get:

$\frac{d}{dx}(-3x^7) = -3(7)x^{7-1} = -21x^6$

Applying the power rule to the second term, $2x^3$, we get:

$\frac{d}{dx}(2x^3) = 2(3)x^{3-1} = 6x^2$

Adding the derivatives of the two terms gives the derivative of $y$:

$\frac{dy}{dx} = -21x^6 + 6x^2$

3. Final Answer

$\frac{dy}{dx} = -21x^6 + 6x^2$