If d/dx [f(3x)]=x^2, then determine f'(x).

Question

a.) x²/27

b.) x²/9

c.) x²/3

d.) x²

e.) 3x²

Answer 1

We know f'(3x) is x^2/3 from the chain rule. let's let y=3x, and x=y/3 so f'(y) is y^2/27. That means f'(x) is x^2/27 as well. Answer choice A.

Let's check that. f(x) is thus the integral of x^2/27 which would be x^3/81. f(3x) is 27x^3/81 = x^3/3, and the derivative of f(3x) is thus x^2, which we wanted to show.