If given $\mathbf{G}(t)=\exp(\mathbf{F}(t-t_0))\mathbf{A}\exp^T(\mathbf{F}(t-t_0)) + \int^t_{t_0}\exp(\mathbf{F}(t-s))\mathbf{M}\exp^T(\mathbf{F}(t-s))ds$
$\exp$ represents expotional matrix, and $\mathbf{F}, \mathbf{A}, \mathbf{M}$ are constant matrices
How to take derivative of $\mathbf{G}(t)$? w.r.t $t$?
$\frac{d\mathbf{G}(t)}{dt}=?$
I assume that $\exp^T(F(t-t_0))$ is the transpose of $U(t-t_0)=\exp((t-t_0)F)$, that is ${U(t-t_0)}^T$.
Then $G'(t)=FU(t-t_0)A{U(t-t_0)}^T+U(t-t_0)AF^T{U(t-t_0)}^T+M+\int_{t_0}^t (FU(t-s)MU(t-s)+U(t-s)MF^T{U(t-s)}^T) ds$.