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Math Help - how would I find a general formula for f^(-1)''(x) ?

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    how would I find a general formula for f^(-1)''(x) ?

    how would I find a general formula for f^(-1)''(x) ?

    Thanks in advance.
    Last edited by mr fantastic; November 18th 2009 at 02:22 PM. Reason: Included post title in main body of post
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dgmath View Post
    The title explains it all. Thanks in advance.
    I assume the following is what you are asking for. I will give you the outline without all the neccessary filler

    Problem: Find a general formula for \left\{f^{-1}(x)\right\}''.


    Lemma: \left\{f^{-1}(x)\right\}'=\frac{1}{f'\left(f^{-1}(x)\right)}

    Proof: Notice that f\left(f^{-1}(x)\right)=x so that f'\left(f^{-1}(x)\right)\cdot \left\{f^{-1}(x)\right\}'=1. The conclusion follows. \blacksquare

    Proof:So then \left\{f^{-1}(x)\right\}''=\left\{\frac{1}{f'\left(f^{-1}(x)\right)}\right\}' =\frac{-1}{\left(f'\left(f^{-1}(x)\right)\right)^2}\cdot\left\{f'\left(f^{-1}(x)\right)\right\}'=\frac{-1}{\left(f'\left(f^{-1}(x)\right)\right)^2}\cdot f''\left(f^{-1}(x)\right)\cdot\frac{1}{f'\left(f^{-1}(x)\right)} =\frac{-f''\left(f^{-1}(x)\right)}{\left(f'\left(f^{-1}(x)\right)\right)^3}
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