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Derivative of an Nasty Exponential Function

  1. Jan 15, 2017 #1

    Drakkith

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    Staff: Mentor

    1. The problem statement, all variables and given/known data
    The question is given just like this:
    ##\frac{d}{dx}(exp\int_1^x P(s)\ ds)## = ?
    I assume they want me to find the derivative of the whole thing.

    2. Relevant equations


    3. The attempt at a solution

    I'm thinking the first step is:
    ##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x P(s)\ ds)(\frac{d}{dx}\int_1^x P(s)\ ds)##

    Unfortunately I don't know how to deal with the derivative of the integral now since the X is one of the limits of integration and not part of the function. I'm not sure I've ever seen a problem like this before. o_O
     
  2. jcsd
  3. Jan 15, 2017 #2

    fresh_42

    Staff: Mentor

    I've used something similar in one of @micromass ' challenges. I think it helps to write the integral according to the fundamental theorem as ##\int_1^x f = F(x) - F(1)## but I'm not sure.
     
  4. Jan 15, 2017 #3

    stevendaryl

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    Well, too much of a hint will give away the answer. But to get an idea of how this works, try a couple of examples for [itex]P(s)[/itex].
    For example,
    • Try [itex]P(s) = 1[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
    • Try [itex]P(s) = s[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
    • Try [itex]P(s) = s^2[/itex].
    Do you see the pattern?
     
  5. Jan 15, 2017 #4

    Mark44

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    The first part of the Fundamental Theorem of Calculus says, in essence, that ##\frac d{dx} \left(\int_a^x f(t)dt\right) = f(x)##.
     
  6. Jan 15, 2017 #5

    Mark44

    Staff: Mentor

    I should add that things have to be exactly as I wrote them. I.e., if we're differentiating with respect to x, x has to be the upper limit of integration. Likewise, if the differentiation is with respect to, say, r, r has to be the upper limit of integration.

    Also, although the integral appears to be a definite integral (which would imply that the integral represents a fixed number), it's really a function of x. Different values of x produce different values for the integral.
     
  7. Jan 15, 2017 #6

    Drakkith

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    Indeed. When I put those values in for P(s) it appears that the derivative to each integral is simply P(s) (or P(x) rather).

    So it does. I certainly feel silly now. o:)


    So that would mean that: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x P(s)\ ds)(\frac{d}{dx}\int_1^x P(s)\ ds) =(exp\int_1^x P(s)\ ds)P(x) ##
     
  8. Jan 15, 2017 #7

    Mark44

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    Looks good.
     
  9. Jan 15, 2017 #8

    Drakkith

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    Staff: Mentor

    Thanks all!
     
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