The problem involves finding the derivative of a function defined as g(x) = f(x^2), where f'(u) = 1 / (1 + u^3). The discussion centers around the application of the chain rule and the evaluation of derivatives without directly finding antiderivatives.
Participants are actively questioning the application of the chain rule and the interpretation of derivatives. Some guidance has been offered regarding the evaluation of f'(x^2) by substituting u with x^2, but there is still uncertainty about the necessity of further application of the chain rule.
There is a noted confusion regarding the definitions and relationships between the functions involved, particularly concerning the need to find antiderivatives and how to correctly apply the chain rule in this context.
PhizKid said: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
Since g(x) = f(x2), then g'(x) = f'(x2[/color]) * 2xPhizKid said:We learned the Chain Rule but not sure how to apply it here. My best guess would be:
g'(x) = f'(x) * 2x
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 said: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?
PhizKid said:How do you know that u = x^2 ?