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Math Help - Derivative of Inverse

  1. #1
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    Derivative of Inverse

    The question asks

    If f(X) = x+2e^x, find the value of g'(1+2e), where g(x) = inverse of f(x) for all x

    I have no idea where to start. I tried to find the inverse of f(x) but that only led to x=y+2e^y and I'm stuck there.

    Please help.
    Last edited by vnsin; October 28th 2009 at 12:51 AM.
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  2. #2
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    Quote Originally Posted by vnsin View Post
    The question asks

    If f(X) = x+2e^x, find the value of g'(1+2e), where g(x) = inverse of f(x) for all x

    I have no idea where to start. I tried to find the inverse of f(x) but that only led to x=y+2e^y and I'm stuck there.

    Please help.
    Let y = f^{-1}(x) \Rightarrow f(y) = x. Therefore:

    y + 2 e^y = x

    \Rightarrow \frac{dy}{dx} + 2e^y \frac{dy}{dx}  = 1 \Rightarrow \frac{dy}{dx} (1 + 2 e^y) = 1 \Rightarrow \frac{dy}{dx} = \frac{1}{1 + 2 e^y}.

    Now note that f(x) = 1 + 2 e \Rightarrow x = 1. Therefore y = f^{-1}(1 + 2e) = 1.

    Substitute y = 1 into \frac{dy}{dx} = \frac{1}{1 + 2 e^y}:

    \frac{dy}{dx} = \frac{1}{1 + 2e}.
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