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Answer :
Final answer:
The derivative of the function f(x) = (2x³ - 7x² - 9)⁵ is found using the chain rule. After differentiating the inner function and multiplying by the derivative of the outer function, we get that the correct answer is d) 10x(2x³ - 7x² - 9)⁴.
Explanation:
We are asked to find the derivative of the function f(x) = (2x³ - 7x² - 9)⁵.
To find the derivative, we can apply the chain rule. The outer function is raised to the fifth power, while the inner function is the cubic polynomial. When we apply the chain rule, we take the derivative of the outer function (considering the inner function as a single variable) and multiply it by the derivative of the inner function. Therefore, the derivative of f(x) = u⁵ (where u = 2x³ - 7x² - 9) will be f'(x) = 5u⁴(u'), where u' is the derivative of u with respect to x.
The derivative of u = 2x³ - 7x² - 9 is u' = 6x² - 14x. Now, we substitute u' back into the derivative f'(x) = 5u⁴ × (6x² - 14x), and we get:
f'(x) = 5(2x³ - 7x² - 9)⁴ × (6x² - 14x)
Factoring out the common x, the result simplifies to:
f'(x) = 10x(2x³ - 7x² - 9)⁴
Therefore, the correct answer is d) 10x(2x³ - 7x² - 9)⁴.
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