I have the following equation -
$$I = \exp\left(\int_{t_0}^{t}\frac{-B-A|u(t)|^2}{AB}\ \mathrm{d}t_1\right)\cdot\left(1 + \int_{t_0}^{t}\frac{\exp\left(-\displaystyle\int_{t_0}^{t}\frac{-B-A|u(t)|^2}{AB}\ \mathrm{d}t_2\right)}{A}\ \mathrm{d}t_3\right).$$
where $u(t) = r(t) + i.m(t)$ with $i$ representing the usual $\sqrt{-1}$.
I am not sure if I am right in using $dt_1,dt_2,dt_3$ to represent the three different $dt$s, but I hope you understand what I am trying to say here.
I need to calculate the following partial derivatives of this function: $\frac{\partial I}{\partial r}$ and $\frac{\partial I}{\partial m}$.
I obtained the expression of $I$ from Mathematica and this is correct. But I have no idea how to calculate the partial derivatives. Asking around on the Mathematica stackexchange site did not help. I am open to doing this calculation on pen-and-paper if required, but I am quite simply stuck. I am not sure how to apply the chain rule if I have to.
As an engineer, I guess what I am asking for is the easiest way for me to calculate the required derivatives. Any help would be really appreciated.