Final answer:
To find the derivative f'(1) we differentiate the given f(x) = x² obtaining f'(x) = 2x, and thus f'(1) = 2, without needing to solve for f(x) explicitly.
Step-by-step explanation:
A student has requested help with finding the derivative f'(1) for a function given that f(x) = x² and f(x)⁴ = 18 when f(1) = 2. To solve this, it's necessary to differentiate f(x) with respect to x, then evaluate the derivative at x = 1.
We know that f(x)⁴ = 18, which can be rewritten as f(x)´ = (18)¹⁄₄ or f(x) = (18)¹⁄₈. Taking the eighth root of both sides won't help much, but we can use the given value f(1) = 2 and remember that f(x) = x². With these values, we can differentiate f(x) with respect to x to find f'(x) = 2x. Evaluating at x = 1, we have f'(1) = 2(1) = 2.
The mistake to avoid here is to try to solve for f(x) when in fact, it's not necessary. Since we only need the derivative, we can use the rules of differentiation directly.