Finding antiderivative without integration

[PhizKid]

Homework Statement

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

Homework Equations

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) ?

 

[jbunniii]

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).

 

[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

 

[jbunniii]

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).

 

[PhizKid]

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

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

 

[Mark44]

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[/color]) * 2x

 

[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?

 

[jbunniii]

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.

 

[PhizKid]

How do you know that u = x^2 ?

 

[jbunniii]

How do you know that u = x^2 ?

What does f'(x^2) mean? It means "evaluate the function f' at x^2." We have a formula for f'(u), i.e. a formula to "evaluate the function f' at u." If I want to evaluate f'(5), then I replace u with 5. If I want to evaluate f'(x^2), then I replace u with x^2.