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Math Help - Help computing derivative of a 2 variable integral

  1. #1
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    Help computing derivative of a 2 variable integral

    Im not sure if there is a typo, or I'm just not seeing how to solve this:
    help please (:


    Compute the derivative w.r.t. t of g(t) := ∫0t f(t,x) dx.

    sorry for the iffy notation
    Last edited by Maroka93; December 4th 2012 at 07:17 AM.
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  2. #2
    Senior Member TriKri's Avatar
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    Re: Help computing derivative of a 2 variable integral

    Use the definition of the derivative:

    \frac{\partial}{\partial\,t}\int_0^t f(t,\,x)\, \operatorname{d}x \,=\, \lim_{h\to 0} \frac{\int_0^{t+h} f(t+h,\,x)\, \operatorname{d}x - \int_0^t f(t,\,x)\, \operatorname{d}x}{h}

    You see that the first term in the fraction can be written as

    \int_0^{t+h} f(t+h,\,x)\, \operatorname{d}x \,=\, \int_0^t f(t+h,\,x)\, \operatorname{d}x + \int_t^{t+h} f(t+h,\,x)\, \operatorname{d}x

    =\, \int_0^t f(t+h,\,x)\, \operatorname{d}x + h\,f(t,\,t) + O(h^2)

    =\, \int_0^t \left(f(t,\,x) + h\,\frac{\partial}{\partial t}f(t,\,x) + O(h^2)\right) \operatorname{d}x + h\,f(t,\,t) + O(h^2)

    From there it should be an easy task to escape the limit formulation and obtain the derivative.

    Good luck!
    Last edited by TriKri; December 4th 2012 at 11:07 AM.
    Thanks from topsquark
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  3. #3
    Senior Member TriKri's Avatar
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    Re: Help computing derivative of a 2 variable integral

    Had any luck solving the problem yet?
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  4. #4
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    Re: Help computing derivative of a 2 variable integral

    I would recommend "Leibniz' rule" which generalizes the "Fundamental Theorem of Calculus":
    \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt= f(x, u(x))- f(x,v(x))+ \int_{u(x)}^{v(x)}\frac{\partial f}{\partial x}dx
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  5. #5
    Senior Member TriKri's Avatar
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    Re: Help computing derivative of a 2 variable integral

    I think that should be

    \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = \frac{dv(x)}{dx} f(x,v(x)) - \frac{du(x)}{dx} f(x, u(x)) + \int_{u(x)}^{v(x)}\frac{\partial f(x,t)}{\partial x}dt
    Last edited by TriKri; December 8th 2012 at 12:07 PM.
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  6. #6
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    Re: Help computing derivative of a 2 variable integral

    Yes, you are right. I forgot the derivatives of u and v. Thanks.
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