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Derivative of an accumulation function.

  1. Apr 22, 2013 #1
    1. The problem statement, all variables and given/known data

    [itex]F(x) = \int^{ln(x)}_{\pi}cos(e^t)\,dt[/itex]

    2. Relevant equations

    3. The attempt at a solution

    Following from a theorem given in the text im using:

    If f is continuous on an open interval I containing a, then, for every x in the interval,
    [itex]d/dx[\int^x_af(t)\,dt] = f(x)[/itex]

    I thought it would be as simple as

    [itex]F'(x) = d/dx[sin(e^{ln(x)})] = d/dx[sin(x)] = cos(x)[/itex]

    But according to the text the answer is [itex]cos(x)/x[/itex]

    So what am i doing wrong?
  2. jcsd
  3. Apr 22, 2013 #2


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    Science Advisor
    Homework Helper

    It's not quite that simple if your limit isn't simply x. [itex]d/dx[\int^{g(x)}_af(t)\,dt] = f(g(x))g'(x)[/itex]. In general you have to use http://en.wikipedia.org/wiki/Leibniz_integral_rule
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