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Homework Help: How to find the derivative of y= x^-1/e^-x

  1. Jan 31, 2009 #1
    1. The problem statement, all variables and given/known data
    y = ((x)^−1) (e^−x)

    3. The attempt at a solution

    Solving this out i got
    [(x^-1) - (x^-2)] e^-x

    however the book is telling me the answer is
    −(x^−1 + x^−2)e^−x

    wouldnt that give me (-x^−1 - x^−2)e^−x ??
    what did i do wrong??
  2. jcsd
  3. Jan 31, 2009 #2


    Staff: Mentor

    Re: Differentiate

    Apparently you lost a sign somewhere. I'm guessing that you forgot the factor of (-1) when you differentiated e^(-x).
    [tex]d/dx(x^{-1}e^{-x}) = x^{-1} e^{-x} (-1) - x^{-2}e^{-x}[/tex]
    [tex]= -x^{-1}e^{-x} - x^{-2} e^{-x}[/tex]
    which agrees with the book's answer.

    BTW, you're not "solving this out;" you're differentiating this function or calculating the derivative of the given function.
  4. Jan 31, 2009 #3
    Re: Differentiate

    Thanks i thought the derivative of e^-x stayed the same
  5. Jan 31, 2009 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Differentiate

    [tex]\frac{d}{dx}e^x = e^x[/tex]

    However, in general (using the chain rule)

    [tex]\frac{d}{dx} e^{f\left(x\right)} = f^\prime\left(x\right)e^{f\left(x\right)}[/tex]

    In your case

    [tex]f\left(x\right) = -x[/tex]


    [tex]\frac{d}{dx}e^{-x} = \frac{d}{dx}\left(-x\right)e^{-x} = - e^{-x}[/tex]

    Do you follow?
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