Calculate the derivarive of $E\left(\int_t^T S_udu|\mathcal{F}_t\right)$ in function of $t$ $$\frac{d}{dt}E\left(\int_t^T S_udu|\mathcal{F}_t\right)$$ where $S_t$ is a stochastic process( I try to avoid specifying its form).
My attempt: We have $$E\left(\int_t^T S_udu|\mathcal{F}_t\right) = \int_t^T E\left(S_u|\mathcal{F}_t\right)du$$ then $$\frac{d}{dt}E\left(\int_t^T S_udu|\mathcal{F}_t\right) = \frac{d}{dt}\left(\int_t^T E\left(S_u|\mathcal{F}_t\right)du\right) $$ But after that, I don't know how to calculate $\frac{d}{dt}\left(\int_t^T E\left(S_u|\mathcal{F}_{\color{red}{t}}\right)du\right) $ because of the subscript in red color.
Why do you think it's derivable ? Take for example $(S_t)$ being a Brownian motion. Then $$\mathbb E\left[\int_t^T S_u\,\mathrm d u\mid \mathcal F_t\right]=S_t(T-t),$$
which is not derivable.