Formula (12) in the paper


* Bauer, M., Chetrite, R., Ebrahimi-Fard, K., & Patras, F. (2013).
Time-ordering and a generalized Magnus expansion. Letters in
Mathematical Physics, 103(3), 331-350.


expresses the derivative of an exponential of a parameter-dependent Lie algebra element. The authors call it 'Duhamel's formula', without giving a reference.


Where and in which context did Duhamel prove this formula? (I know the formula as Kubo's formula, from his work in statistical mechanics.)



Duhamel's formula
Duhamel's formula for the derivative of the exponent of a matrix refers to [Jean-Marie-Constant Duhamel](https://en.wikipedia.org/wiki/Jean-Marie_Duhamel), who described it in [Eléments de calcul infinitésimal](http://catalogue.bnf.fr/ark:/12148/cb3037024) (volume 2, 1856; page 36) as a method to obtain solutions to inhomogeneous linear differential equations in terms of the solution to the homogenous equation.  

In the context of Lie groups the [formula for the derivative of the exponential map](https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map) is attributed to Schur (1891).


A modern proof is in these [notes.](https://personal.math.ubc.ca/~feldman/m428/duhamel.pdf) or alternatively [here](http://scipp.ucsc.edu/~haber/webpage/MatrixExpLog.pdf) (theorem 3). A formal proof is given by
\begin{align}
 \frac{d}{dt}e^{A(t)}
 &= \lim\_{N \to \infty}\frac{d}{dt}\left(1 + \frac{A(t)}{N}\right)^N\\
 &= \frac{1}{N}\lim\_{N \to \infty}\sum\_{k=1}^N\left(1 + \frac{A(t)}{N}\right)^{k-1}A'(t)\left(1 + \frac{A(t)}{N}\right)^{N-k}~\\
&=\int\_0^1 e^{sA(t)}A'(t)e^{(1-s)A(t)}\,ds.
\end{align}


