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Homework Help: Inverse Derivative Question

  1. Oct 28, 2007 #1
    1. The problem statement, all variables and given/known data

    Assume the function f(x)=x^3+x has an inverse in R. Determine d/dx(f^-1(x)) at x=2.

    2. Relevant equations

    (f^-1(x))'=1/f'(f^-1(x))

    3. The attempt at a solution

    y'=3x^2+1

    (f^-1(x))'=1/(3y^2+1)

    now i substitute f^-1(2), y=10, into the equation but do not get the correct answer. I suspect that the mistake i am making is becuase i do not fully understand the question. Any help is greatly appreciated.
     
  2. jcsd
  3. Oct 28, 2007 #2
    f^-1 is the functional inverse, not the reciprocal.
     
  4. Oct 30, 2007 #3
    [tex]
    \frac{dy}{dx} = 3 x^2 + 1
    [/tex]
    using implicit differentiaion
    [tex]
    \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{3 x^2 + 1}
    [/tex]
    accordingly replacing x by y
    [tex]
    \frac{dy}{dx} = \frac{1}{3 y^2 + 1}
    [/tex]
    i.e.
    [tex]
    \frac{dy}{dx} = \frac{1}{3 \left( {x^3 + x } \right)^2 + 1}
    [/tex]
    then you have only to substituting [tex] x=2 [/tex].
     
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