Total derivative of an integration

**anicole** · September 25th 2011, 12:14 PM

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.
$\frac{\partial F\left ( t \right )}{\partial A\left ( t \right )}=-B\left ( t \right )C\left ( t \right )$

**FernandoRevilla** · September 25th 2011, 06:06 PM

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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