# Finding antiderivative without integration

## Homework Statement

f'(u) = 1 / (1 + u^3)
g(x) = f(x^2)
Find g'(x) and g'(2)

## 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) ?

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jbunniii
Homework Helper
Gold Member
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).

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

jbunniii
Homework Helper
Gold Member
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).

We learned the Chain Rule but not sure how to apply it here. My best guess would be:

g'(x) = f'(x) * 2x

Mark44
Mentor
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

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?

jbunniii
Homework Helper
Gold Member
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.

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How do you know that u = x^2 ?

jbunniii