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Answer :
Final answer:
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
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