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Showing the properties of differentiating an integral

  1. Jan 18, 2013 #1
    How do we show that

    [tex]\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}[/tex]
  2. jcsd
  3. Jan 18, 2013 #2


    Staff: Mentor

    Is this a homework problem?
  4. Jan 18, 2013 #3


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    Science Advisor

    Are we to assume, here, that y and x are functions of t? If we assume that y is a function of x only (with no "t" that is not in the "x") and x is a function of t, then we an write y(x(t)).

    Of course, then [tex]F(x)= \int y dt[/tex] is the function such that dF/dx= y. Given that, we have that [itex]d/dt(\int y dx)= dF/dt= (dF/dx)(dx/dt)= y(x)(dx/dt)[/itex] by the chain rule.
  5. Jan 18, 2013 #4
    Nope. Homework questions are usually standard, and answers are all in the textbooks.
    I came up with this problem just out of curiosity.

    Anyway, thanks for the solution HallsofIvy
    Last edited: Jan 18, 2013
  6. Jan 18, 2013 #5
    I wish my textbooks had the answers!
  7. Jan 22, 2013 #6
    not the exact answers, but they all follow the same template
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