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Derivative of an inverse question

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data

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


    2. Relevant equations

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

    3. 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?
     
  2. jcsd
  3. Nov 23, 2009 #2
    It might be difficult finding the inverse function, but it's easy to see that f(1)=2, so h(2)=1.
     
  4. Nov 23, 2009 #3

    Mark44

    Staff: Mentor

    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).
     
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