Let \( y = 4x^4(x^5 - 8) \). Calculate the derivative using the product rule.

1. \( f(x) = \)
2. \( g(x) = \)

This implies:
- \( f'(x) = \)
- \( g'(x) = \)

Thus, \( y' = \)

(It doesn't matter which part of the sum you enter first)

Options:
a) \( (x^9, 4x^4), (9x^8, 4x^4 - 32x^3), (4x^9 - 32x^8) \)
b) \( (4x^4, x^9), (4x^4, 9x^8), (4x^9 - 32x^8) \)
c) \( (4x^4, x^9), (4x^9, 9x^8), (4x^9 + 32x^8) \)
d) \( (x^9, 4x^4), (9x^8, 4x^4 - 32x^3), (4x^9 + 32x^8) \)

The derivative of y = 4x⁴(x⁵ - 8) using the product rule is y' = 36x⁸ - 128x³, calculated by first determining f(x) = 4x⁴, g(x) = x⁵ - 8, and then their derivatives f'(x) = 16x³, g'(x) = 5x⁴.

Explanation:

When calculating the derivative of the function y = 4x⁴(x⁵ - 8) using the product rule, we consider one function as f(x) and the other as g(x).

To apply the product rule, which states that the derivative of a product of two functions f(x)g(x) is f'(x)g(x) + f(x)g'(x), we first need to determine f(x) and g(x).

Here, we set f(x) = 4x⁴ and g(x) = x⁵ - 8. The derivatives of these functions are f'(x) = 4(4x³) and g'(x) = 5x⁴ respectively.

Then, applying the product rule:

f'(x)g(x) = 16x³(x⁵ - 8)

f(x)g'(x) = 4x⁴(5x⁴)

Combining these results, the derivative y' is:

y' = 16x³(x⁵ - 8) + 20x⁸ = 16x⁸ - 128x³ + 20x⁸ = 36x⁸ - 128x³