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Math Help - Simple exponential derivative question

  1. #1
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    Simple exponential derivative question

    I know this is simple, but I'm blanking - I think it's time for bed.

    I need to find the derivative or the following.

    (e^w + 1) / e^w

    I would think it would go something like this...

    (e^w)(e^w) - (e^w + 1) (e^w)

    But I don't think I should be doing that - and if I do, I'm not sure what the next step is.
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  2. #2
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    Re: Simple exponential derivative question

    I think I see where I messed up.. I didn't put (e^w)^2 in the denominator. I'm assuming that essentially cancels the expressions in the numerator, right?

    That would leave the answer at...

    -(1/e^w)

    Is this correct, and if so, have I gone about it the right way?
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  3. #3
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    Re: Simple exponential derivative question

    Quote Originally Posted by astuart View Post
    I know this is simple, but I'm blanking - I think it's time for bed.

    I need to find the derivative or the following.

    (e^w + 1) / e^w

    I would think it would go something like this...

    (e^w)(e^w) - (e^w + 1) (e^w)

    But I don't think I should be doing that - and if I do, I'm not sure what the next step is.
    It's easier if you write \displaystyle \begin{align*} \frac{e^w + 1}{e^w} = 1 + \frac{1}{e^w} = 1 + e^{-w} \end{align*}.
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    Re: Simple exponential derivative question

    Quote Originally Posted by Prove It View Post
    It's easier if you write \displaystyle \begin{align*} \frac{e^w + 1}{e^w} = 1 + \frac{1}{e^w} = 1 + e^{-w} \end{align*}.
    I'm studying all of this via distance education, so sometimes it can take a little bit to get the idea to set in, so this'll probably be another silly question.

    Where you've written 1 + 1 / e^w how did you get to this point?

    The only way I can see it being done, is similar to the manner I got it there (chain rule), but with the fact that it was (e^w)^2 in the denominator that makes it stay as that expression (not squared). When compared to the numerator, which only had (e^w), this can simply be represented by 1 - almost like factorization.

    Any chance you could clear that up?
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    Re: Simple exponential derivative question

    Quote Originally Posted by astuart View Post
    I'm studying all of this via distance education, so sometimes it can take a little bit to get the idea to set in, so this'll probably be another silly question.

    Where you've written 1 + 1 / e^w how did you get to this point?

    The only way I can see it being done, is similar to the manner I got it there (chain rule), but with the fact that it was (e^w)^2 in the denominator that makes it stay as that expression (not squared). When compared to the numerator, which only had (e^w), this can simply be represented by 1 - almost like factorization.
    \frac{e^x+1}{e^x}=\frac{e^x}{e^x}+\frac{1}{e^x}=1+  \frac{1}{e^x}=1+e^{-x}

    D_x(1+e^{-x})=-e^{-x}=\frac{-1}{e^x}
    Last edited by Plato; August 4th 2012 at 02:37 PM.
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