I would like to calculate something like $$K\left(z\right)=\dfrac{dF\left(A\left(z\right)+B\left(z\right)\right)}{dz}$$ i.e. the derivative of a function $F$ of a sum of two operators $A$ and $B$ with respect to the variable $z$, when $A$ and $B$ do not commute with each other: $\left[A,B\right]\neq0$ and when $$\dfrac{dA}{dz}=A^{\prime}\;;\;\dfrac{dB}{dz}=B^{\prime}$$ are known. $F$, $A$ and $B$ have all good properties for physics (expandable in series around every point of interest).
I thought it exists an integral representation for such a derivative (which I may have read once in a paper by Feynman or Glauber) but I'm totally unable to found it since a few days, so perhaps I was just dreaming one more time :-(
Actually, my problem is simpler, since $F$ is an exponential $F=e^{iz}$, and $A$ and $B$ are simple functions of $z$: $dA/dz=0$ and $B=B_{0}z$, with $dB_{0}/dz=0$ and so I have $\left[A,B_{0}\right]\neq0$. I can obviously expand for small $A$ and $B_{0}$ which gives me a hint of what I'm looking for, but I would like to know if I can do something more general. Thanks in advance.
PS: I ask this question here instead of Mathematic-SE since I believe it is not of great interest for mathematicians. Feel free to transfer this question to Math-SE.
