Somewhere in my notes the following formula appears
$\int_0^1 e^{s R} \frac{d R}{dt} e^{(1-s)R} ds = \frac{d}{dt} e^{R}$
where $R$ depends on $t$, and has values in a Lie algebra [$\mathfrak{so}(3)$ is what I was dealing with at the time]. If I assume this formula is true, I can make great simplifications in my work, so it's very important to me. But I have two problems.
First, I can't remember where I got this. I've looked carefully through my references, but can't find it anywhere. Any good references for this formula?
Second, (though a good reference should cure this problem) I can't figure out why it's true. When I use the "Hadamard" Lemma, and expand both sides, I get vaguely similar-looking formulas, but can't quite get to equality. Any ideas?
[On a related note, it seems like this use of integrals from 0 to 1 is pretty common in analysis of Lie groups/algebras. Any good (but fairly basic) references that discuss this strategy per se, and its usefulness?]