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Math Help - derivative of e

  1. #1
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    derivative of e

    Firstly, isn't the rule that subtraction of powers is division not multiplication?

    derivative of e-ql_4e2312437c4e3c4537d41da80397d509_l3.png

    Then on finding the integral

    derivative of e-ql_9b2f7c21e23574eb10ad8610db2b2fa7_l3.png

    why has the e to 3 come out as negative?

    Thanks.
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  2. #2
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    Re: derivative of e

    e^{3-x} = \frac{e^{3}}{e^{x}} = e^{3}e^{-x}

    It is all the same. Please ponder it until you see that it is not scary.

    \int a\cdot e^{-x}dx = a\cdot\int e^{-x}dx

    Feel free simply to remove constants from the integral argument. e^{3} is a constant.

    By the way, "the derivative of e" is zero. Be more careful with words.
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  3. #3
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    Re: derivative of e

    perhaps it will make more sense if you realize that \frac{d}{dx}(e^{-x}) = -e^{-x}.

    this is a consequence of the chain rule: e^{-x} is really the composition:

    exp\circ g(x), where g(x)= -x = (-1)x.

    so when we integrate, we need a minus sign to go inside the integral, and another minus sign to go outside (to balance, (-1)(-1) = 1). the minus sign inside the integral disappears, since when we integrate:

    \int \ -e^{-x} dx = e^{-x} + C.

    the minus sign outside the integral stays, however.
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  4. #4
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    Re: derivative of e

    Got it. Great explanations.

    Thanks guys.
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