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Math Help - Total derivative of an integration

  1. #1
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    Total derivative of an integration

    I've tried several days for this problem. Please help~

    F\left ( t \right )=\int_{t}^{T}A\left ( u \right )B\left ( u \right )C\left ( u \right )du

    dF\left ( t \right )=\frac{\partial F\left ( t \right )}{\partial A\left ( t \right )}dA\left ( t \right )+\frac{\partial F\left ( t \right )}{\partial B\left ( t \right )}dB\left ( t \right )+\frac{\partial F\left ( t \right )}{\partial C\left ( t \right )}dC\left ( t \right )

    How to compute this term, \frac{\partial F\left ( t \right )}{\partial A\left ( t \right )} ?

    I was doing following. But it looks wrong.
    \frac{\partial F\left ( t \right )}{\partial A\left ( t \right )}=-B\left ( t \right )C\left ( t \right )
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Total derivative of an integration

    Using the First Fundamental Theorem of Calculus we directly obtain dF(t)=F'(t)dt=-A(t)B(t)C(t)dt
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