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Math Help - Derivative of the inverse

  1. #1
    TWN
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    Derivative of the inverse

    f(x) = x^3 + x

    h(x) is the inverse of f(x)

    Find the derivative of h(2), step by step please.

    Thanks in advance!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Derivative of the inverse

    f(x)=2\Leftrightarrow x^3+x=2\Leftrightarrow x=1 (only one real root). Now, using a well known theorem:

    h'(2)=\frac{1}{f'(1)}=\ldots=\frac{1}{4}
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  3. #3
    TWN
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    Re: Derivative of the inverse

    Quote Originally Posted by FernandoRevilla View Post
    f(x)=2\Leftrightarrow x^3+x=2\Leftrightarrow x=1 (only one real root). Now, using a well known theorem:

    h'(2)=\frac{1}{f'(1)}=\ldots=\frac{1}{4}
    Thank you, but I'm not sure I understand yet. Why did you set f(x) = 2 for the first step? What process are you using?
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    MHF Contributor Reckoner's Avatar
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    Re: Derivative of the inverse

    Quote Originally Posted by TWN View Post
    Thank you, but I'm not sure I understand yet. Why did you set f(x) = 2 for the first step? What process are you using?
    He is using the inverse function theorem,

    \left(f^{-1}\right)'(x) = \frac1{f'\left(f^{-1}(x)\right)},

    which is indeed very well known. You should familiarize yourself with it.

    To find \left(f^{-1}\right)'(2), we want to know f^{-1}(2). So we set f(x) = 2\Rightarrow x=1 so f(1)=2\Rightarrow f^{-1}(2) = 1. Now make the substitutions, and finish it.
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