Can the derivative the map
$$ f(A) = e^A, A \in \mathbb{R}^{n \times n} $$
be defined as
$$ \lim_{h\to 0} \frac{e^{A+h\Delta} -e^{A}}{h} =\lim_{h\to 0} e^{A} \frac{e^{h\Delta} - I}{h} $$
where $\Delta$ is an "increment matrix" of some form? Or maybe
$$ \lim_{\lVert \Delta \rVert \to 0} \frac{\lVert e^{A+\Delta} -e^{A} - f'(A) \rVert}{\lVert \Delta \rVert} $$
I think the latter might make more sense (though not 100% sure)