# Finding antiderivative without integration

by PhizKid
Tags: antiderivative, integration
 P: 459 1. The problem statement, all variables and given/known data f'(u) = 1 / (1 + u^3) g(x) = f(x^2) Find g'(x) and g'(2) 2. Relevant equations 3. The attempt at a solution So the derivative of function f at u is: 1 / (1 + u^3) That means g'(x) would be f'(x^2), but to find the general derivative of f at u is 1 / (1 + u^3) so can I just plug in x^2 for u so I get: 1 / (1 + x^6) ?
 Sci Advisor HW Helper PF Gold P: 2,618 Your subject has nothing to do with this problem. There are no antiderivatives here. You need to use the chain rule to compute g'(x). It is NOT true that g'(x) = f'(x^2).
 P: 459 Don't you need to find f(u) which would be going backwards from 1 / (1 + u^3), which is an antiderivative? And why is g'(x) != f'(x^2)? I just took the derivative of both functions on each side with respect to x
HW Helper
PF Gold
P: 2,618

## Finding antiderivative without integration

 Quote by PhizKid Don't you need to find f(u) which would be going backwards from 1 / (1 + u^3), which is an antiderivative? And why is g'(x) != f'(x^2)? I just took the derivative of both functions on each side with respect to x
No, you don't need to find f(u). Regarding why g'(x) != f'(x^2), please apply the chain rule to differentiate both sides of g(x) = f(x^2).
 P: 459 We learned the Chain Rule but not sure how to apply it here. My best guess would be: g'(x) = f'(x) * 2x
Mentor
P: 19,860
 Quote by PhizKid We learned the Chain Rule but not sure how to apply it here. My best guess would be: g'(x) = f'(x) * 2x
Since g(x) = f(x2), then g'(x) = f'(x2) * 2x
 P: 459 Sorry, that's what I meant. I write it down on paper but I'm not very good at typing So for g'(x) I have f'(x^2) * 2x Do I have to somehow use f'(u) = 1 / (1 + u^3) in order to find the values for 'x' for f'(x^2) * 2x ? Do I need to use the Chain Rule again? If so, how would I apply it in this situation?
HW Helper
PF Gold
P: 2,618
 Quote by PhizKid Sorry, that's what I meant. I write it down on paper but I'm not very good at typing So for g'(x) I have f'(x^2) * 2x Do I have to somehow use f'(u) = 1 / (1 + u^3) in order to find the values for 'x' for f'(x^2) * 2x ? Do I need to use the Chain Rule again? If so, how would I apply it in this situation?
You don't need to use the chain rule again. If f'(u) = 1/(1 + u^3) then what is f'(x^2)? Just substitute u = x^2. You are simply evaluating the function defined by f'(u) = 1/(1 + u^3) at the particular value u = x^2.
 P: 459 How do you know that u = x^2 ?