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Math Help - Inverse Derivative Help

  1. #1
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    Question Inverse Derivative Help

    I cannot figure out what to do!!
    Given the function f(x)=5 x^3+2 x+5 Let g be the inverse function of f. i.e. g(x)=f^{-1}(x).
    g^{\prime}(12)=

    I can't manage to find g(x) let alone g'(x)
    Thank you for any help.
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  2. #2
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    Quote Originally Posted by Krooger View Post
    I cannot figure out what to do!!
    Given the function f(x)=5 x^3+2 x+5 Let g be the inverse function of f. i.e. g(x)=f^{-1}(x).
    g^{\prime}(12)=

    I can't manage to find g(x) let alone g'(x)
    Thank you for any help.
    note that g'(x) = \frac 1{f'(g(x))} .....not hard to derive. your textbook should do this

    now, by definition, g(a) = x \implies f(x) = a

    thus, g(12) is the x-value so that f(x) = 12

    by inspection, this is 1, thus g(12) = 1


    thus, g'(12) = \frac 1{f'(1)}

    i leave the rest to you
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    note that g'(x) = \frac 1{f'(g(x))} .....not hard to derive. your textbook should do this

    now, by definition, g(a) = x \implies f(x) = a

    thus, g(12) is the x-value so that f(x) = 12

    by inspection, this is 1, thus g(12) = 1


    thus, g'(12) = \frac 1{f'(1)}

    i leave the rest to you
    f^{-1}(x) \neq \frac{1}{f(x)}.

    Edit: Never mind, I figured out what you were doing. I worked it out using implicit differentiation, but your way works too.
    Last edited by Prove It; January 25th 2009 at 02:52 AM.
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  4. #4
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    Quote Originally Posted by Prove It View Post
    f^{-1}(x) \neq \frac{1}{f(x)}.

    Edit: Never mind, I figured out what you were doing. I worked it out using implicit differentiation, but your way works too.
    The first formula Jhevon gives comes from implicit differentiation of f(g(x))=x

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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Prove It View Post
    f^{-1}(x) \neq \frac{1}{f(x)}.
    that's not the formula i have

    Edit: Never mind, I figured out what you were doing. I worked it out using implicit differentiation, but your way works too.
    yes, i worked it out that way as well...er, like Moo said
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  6. #6
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    I was unaware that:
    <br />
g'(x) = \frac 1{f'(g(x))}

    ...and to be honest upon first glace it still dosen't make sense to me haha. When I get a minute I will have to look it up. Thank You all for the help.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krooger View Post
    I was unaware that:
    <br />
g'(x) = \frac 1{f'(g(x))}

    ...and to be honest upon first glace it still dosen't make sense to me haha. When I get a minute I will have to look it up. Thank You all for the help.
    since f(x) and g(x) are inverse functions,

    f(g(x)) = x

    now differentiate both sides with respect to x (note you will need the chain rule to differentiate the left side) and solve for g'(x)
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