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Math Help - trouble taking a derivative

  1. #1
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    trouble taking a derivative

    I need help taking the following derivatives:


    d/dt of (e^-2)*(e^(1+e^x))*(e^t)

    and


    d/dt of (e^-2)*(e^(1+e^t))*(e^x)


    thanks

    Fred1956


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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by Fred1956 View Post
    I need help taking the following derivatives:


    d/dt of (e^-2)*(e^(1+e^x))*(e^t)
    If x \neq f(t) let u =  e^{-2}(e^{1+e^x}) which is a constant which gives:

    \frac{d}{dt}(ue^t) where u is a constant, should be easy enough.




    d/dt of (e^-2)*(e^(1+e^t))*(e^x)
    \frac{d}{dt}(e^{-2}e^x)(e^{1+e^t})

    Let y = (e^{-2}e^x)(e^{1+e^t})

    take logs:

    Spoiler:
    ln(y) = ln[(e^{-2}e^x)(e^{1+e^t})]

    (Because ln(abc) = ln(a)+ln(b)+ln(c))
    = ln(e^{-2}) + ln(e^x) + ln(e^{1+e^t})

    (Because ln(a^k) = k\,ln(a) and ln(e^k) = k)
    = -2 + x + 1+e^t = e^t+x-1


    ln(y) = e^t+x-1

    Differentiate implicitly:

    \frac{1}{y} \frac{dy}{dt} = e^t

    <br />
\frac{dy}{dt} = ye^t = e^t(e^{-2}e^x)(e^{1+e^t})
    Last edited by e^(i*pi); October 29th 2009 at 01:00 PM. Reason: Changed a minus sign to a plus. Doesn't make a difference when differentiating because the x term is treated as constant.
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