Final answer:
The derivative of f(x) = (4x - 1)^3 using the chain rule is f'(x) = 12(4x - 1)^2.
Step-by-step explanation:
To find the derivative of the function f(x) = (4x - 1)^3 using the chain rule, we first recognize that it's a composition of two functions, where the outer function is g(u) = u^3 and the inner function is h(x) = 4x - 1. The chain rule tells us that the derivative of f(x) with respect to x is the derivative of g with respect to u (the outer function), multiplied by the derivative of h with respect to x (the inner function).
The derivative of the outer function is g'(u) = 3u^2 and the derivative of the inner function is h'(x) = 4. Applying the chain rule, we get:
f'(x) = g'(h(x)) \cdot h'(x) = 3(4x - 1)^2 \cdot 4
To simplify, we multiply the constant factor 3 by 4 to get 12:
f'(x) = 12(4x - 1)^2