To find the derivative of the function $f(x) = (-x^5 + 6x^2 - 3)(5x^4 - 4x)$, we will use the product rule, which states that:

$(fg)' = f'g + fg'$

Let: $u(x) = -x^5 + 6x^2 - 3$ $v(x) = 5x^4 - 4x$

Now, compute the derivatives of $u(x)$ and $v(x)$:

1. Derivative of $u(x)$: $u'(x) = \frac{d}{dx}(-x^5 + 6x^2 - 3) = -5x^4 + 12x$

2. Derivative of $v(x)$: $v'(x) = \frac{d}{dx}(5x^4 - 4x) = 20x^3 - 4$

Now, apply the product rule:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Substitute $u(x), u'(x), v(x),$ and $v'(x)$:

$f'(x) = (-5x^4 + 12x)(5x^4 - 4x) + (-x^5 + 6x^2 - 3)(20x^3 - 4)$

Now, expand each term:

1. Expand $(-5x^4 + 12x)(5x^4 - 4x)$: $= -5x^4(5x^4 - 4x) + 12x(5x^4 - 4x)$ $= -25x^8 + 20x^5 + 60x^5 - 48x^2$ $= -25x^8 + 80x^5 - 48x^2$

2. Expand $(-x^5 + 6x^2 - 3)(20x^3 - 4)$: $= -x^5(20x^3 - 4) + 6x^2(20x^3 - 4) - 3(20x^3 - 4)$ $= -20x^8 + 4x^5 + 120x^5 - 24x^2 - 60x^3 + 12$ $= -20x^8 + 124x^5 - 24x^2 - 60x^3 + 12$

Now, combine the two expanded expressions:

$f'(x) = (-25x^8 + 80x^5 - 48x^2) + (-20x^8 + 124x^5 - 24x^2 - 60x^3 + 12)$

Simplify:

$f'(x) = (-25x^8 - 20x^8) + (80x^5 + 124x^5) + (-48x^2 - 24x^2) - 60x^3 + 12$ $f'(x) = -45x^8 + 204x^5 - 72x^2 - 60x^3 + 12$

Thus, the derivative is:

$f'(x) = -45x^8 + 204x^5 - 60x^3 - 72x^2 + 12$

Would you like more details on any step?

Here are 5 related questions:

1. How is the product rule applied in derivative calculations?
2. Can the chain rule be used in this derivative calculation?
3. How do you combine like terms after expanding products?
4. What other methods exist to differentiate complex functions?
5. How do you compute the second derivative of the given function?

Tip: Simplifying expressions step by step ensures fewer errors in complex calculations.