Final answer:
To find f'(1), differentiate the function f(x) using the chain rule and substitute x = 1. The value of f'(1) is -2/31.
Step-by-step explanation:
To find f'(1), you need to differentiate the function f(x) and evaluate it at x = 1. Let's start by finding the derivative of f(x) = x²[f(x)]⁴. Using the chain rule, the derivative is given by:
f'(x) = 2x[f(x)]⁴ + 4x²[f(x)]³f'(x)
Now, substitute x = 1 into the derivative expression to find f'(1):
f'(1) = 2(1)[f(1)]⁴ + 4(1)²[f(1)]³f'(1)
Given that f(1) = 2, we can substitute this value in:
f'(1) = 2(1)(2)⁴ + 4(1)²(2)³f'(1)
Simplifying the expression gives f'(1) = 2 + 32f'(1). Solving for f'(1), we get:
f'(1) = -2/31