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

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

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

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

I was doing following. But it looks wrong.

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