Derivative of matrix exponential w.r.t. to each element of the matrix I have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?

I arrive at the following. The ij element of $e^A$ is $$e^A_{ij} = \sum_{n=0}^\infty \frac{1}{n!} A_i^{k_1} ... A_{k_{n-1}j} ~.$$ Each matrix element can be seen as an independent variable, so the derivative toward $A^{kl}$ is $$e^A_{ij} = \sum_{n=0}^\infty \frac{1}{n!}  \sum_{p=0}^{n-1} A^p_{ik} A^{n-1-p}_{lj} ~.$$
For my purpose $t$ is less relevant. It can easily be added in without changing my result.
