Final answer:
To find g'(13), we must first compute f'(x), which is the derivative of f(x) = x³ - 7x². Then we solve for x in g(x) = 13. Finally, substitute that x-value into g'(x) = 5f'(x) to get g'(13).
Step-by-step explanation:
The given function is f(x) = x³ - 7x². The function g(x) = 5f(x), so to find g'(x), we first need to find f'(x). The derivative of f(x) with respect to x is f'(x) = 3x² - 14x. Substituting x = 1 into f'(x) doesn't give us g'(13), because we're looking for g'(x) evaluated at an x-value that makes g(x) equal to 13. However, the information about f(1) = 13 is a typo and irrelevant to solving for g'(13). Instead, we need to evaluate f'(x) at the x-value where g(x) = 13, which requires solving g(x) = 13 for x, then using that x-value in g'(x) = 5f'(x).