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Math Help - derivatives of inverse functions

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    derivatives of inverse functions

    this topic probably isn't supposed to be hard but i find it very confusing. my teacher went over it very briefly if not at all and my book doesn't help explain either. so it's my understanding that if you let y = f^-1 (x), then dy/dx = 1 / (dx/dy) and if you let y = f(x) and x = f^-1 (y), then dx/dy = 1 / (dy/dx). so derivatives of inverse functions can be written in 2 different ways? one of my problems asks to find the derivative of e^x (which i already know is e^x) by finding the derivative of the inverse of ln x. i did dy/dx = 1 / (1/y) = y. i also did dx/dy = 1 / (1/x) = x. how do i get e^x for either one? my equations are y = lnx and the inverse of that is x = lny.
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  2. #2
    Member Em Yeu Anh's Avatar
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    Red face

    Hmm, well you had x=lny and that \frac{dy}{dx} = \frac{1}{\frac{1}{y}} = y

    From the original function you can see that x = lny is equivalent to e^x = y so I think you may have done it correctly. However my prof barely touched upon this subject so I'm sort of unfamiliar with it as well.
    Last edited by Em Yeu Anh; November 21st 2009 at 08:04 PM.
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  3. #3
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    Quote Originally Posted by oblixps View Post
    this topic probably isn't supposed to be hard but i find it very confusing. my teacher went over it very briefly if not at all and my book doesn't help explain either. so it's my understanding that if you let y = f^-1 (x), then dy/dx = 1 / (dx/dy) and if you let y = f(x) and x = f^-1 (y), then dx/dy = 1 / (dy/dx). so derivatives of inverse functions can be written in 2 different ways? one of my problems asks to find the derivative of e^x (which i already know is e^x) by finding the derivative of the inverse of ln x. i did dy/dx = 1 / (1/y) = y. i also did dx/dy = 1 / (1/x) = x. how do i get e^x for either one? my equations are y = lnx and the inverse of that is x = lny.
    You are getting your notation a bit confused. When talking about general functions, we might talk about f(x) and f^{-1}(x) but when you have specific things like " y= e^x" it is best to write x= ln(y). From that, dx/dy= 1/y so [/tex]dy/dx= y= e^x[/tex]
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