I have the following matrix equation: $$\int_0^te^{A(t-x)}Be^{(A+B)x}\text{d}x$$
where $A$ and $B$ are matrices in $\mathbb{R}^{n \times n}$ and $x$ is a dummy variable
I want to find the derivative of the of the above integral equation with respect to $t$. i.e.: $$\frac{\text{d}}{\text{d}t}\left( \int_0^te^{A(t-x)}Be^{(A+B)x}\text{d}x \right)$$
Is it as simple as replacing $x$ with $t$ in the above equation and removing the integral? That is, $$e^{A(t-t)}Be^{(A+B)t}$$
Or do I have to integrate the equation out first (assuming I would have to use integration by parts), since it's in matrix form and then take the derivative?
Thanks for any hints!
Assuming the $t$ in the integrand is meant to be the same as the $t$ in the range of integration, no, it's not so simple. In general you should think of this in terms of the multivariate chain rule.
In the particular case here, it seems that the $e^{At}$ term may be factored out: $$e^{At} \cdot \int_0^t e^{-Ax} B e^{(A+B)x} \, \mathrm{d}x,$$ so you can use the product rule instead.