Final answer:
To find f'(1), we take the derivative of the function f(x) = ln(x⁸) using the chain rule. The derivative is 8x⁶, and plugging in x = 1 gives f'(1) = 8.
Step-by-step explanation:
To find the derivative of the function f(x) = ln(x⁸), we can use the chain rule. The chain rule states that if we have a function g(x) inside another function f(x), then the derivative of f(g(x)) is given by f'(g(x)) * g'(x). In this case, g(x) = x⁸ and f(x) = ln(x), so f(g(x)) = ln(x⁸). The derivative of f(x) = ln(x) is 1/x, so the derivative of f(g(x)) = ln(x⁸) is 1/x * 8x⁷ = 8x⁶.
Now we need to find f'(1). Plugging in x = 1 into the derivative function, we get f'(1) = 8(1)⁶ = 8.