I'm trying to derive the following equation using a Riemann sum formulation only $$ \frac{d}{d t}e^{A(t)} = \int_0^1 ds \,\,\,\,e^{sA(t)}(\frac{d A(t)}{dt})e^{(1-s)A(t)}. $$
The book is using einstein notation on the indices. What I've done so far is make sure I understand how to take the derivative of $e^{A(t)}$ for matrix $A$ and continuous parameter $t$:
$$\frac{d}{dt}e^{A(t)} = \frac{d}{dt}(\lim_{N\to \infty} (1\;+\;\frac{A}{N}^N) = \lim_{N\to \infty} \, \sum_k^{N}\;\;(1 \; + \; \frac{A}{N})^{N-k} \; \frac{1}{N}\frac{dA}{dt} \; (1 \; + \; \frac{A}{N})^{k-1}.$$
Here's a wiki page that may be relevant.
Now, the formula above makes perfect sense; it's just a matter of accounting for the non-commutativity of $A$ and carrying out the derivative of the limit definition of the exponential correctly. I see how the limit and sum together are implicitly an integral, but I can't seem to figure out the substitution correctly. It's obvious that since we're taking the limit as $N$ goes to infinity that I need to divide $k$ into infinitesimal chunks, so define $s = k/N$. This would make the $1/N$ term in the derivative expression I have above become $\Delta s$ since $k$ is an integer and summing over it would make $\Delta k = 1$ for all $k$. From here, my trouble is in how to get the exponents in the exponential terms correct.
Here's some of my work. I'm not sure if this is correct or not. Even if this is correct, I'm not sure how to get the exponentials in swapped order to get the expression my book has. Thanks!
