Derivative of an inverse question

  • #1

Homework Statement



f(x)=(x^3)+x. If h(x) is the inverse of f(x), find h'(2).


Homework Equations



(F[tex]^{-1}[/tex])'(x)=[tex]\frac{1}{F'(F^{-1}x)}[/tex]

The Attempt at a Solution




I want to find h'(2)=(F[tex]^{-1}[/tex])'(2)=[tex]\frac{1}{F'(F^{-1}(2))}[/tex]

I know f'(x)=3(x^2)+1, so I just need to find h(2), but I don't know how to solve f(x)=(x^3)+x for its inverse. Is it possible to solve for x and then switch x and y with this type of function?
 

Answers and Replies

  • #2
73
1
It might be difficult finding the inverse function, but it's easy to see that f(1)=2, so h(2)=1.
 
  • #3
35,019
6,768
Since h is the inverse of f, h(f(x)) = x
Differentiating, you get h'(f(x))*f'(x) = 1, so h'(f(x)) = 1/f'(x).

Can you work in grief's comment that f(1) = 2 to find h'(2)? You will also need to find f'(x), and from that f'(1).
 

Related Threads on Derivative of an inverse question

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
871
  • Last Post
Replies
7
Views
787
  • Last Post
Replies
4
Views
888
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
7
Views
1K
Replies
3
Views
1K
  • Last Post
Replies
2
Views
838
Replies
1
Views
3K
  • Last Post
Replies
3
Views
1K
Top