Geodesic
Formula for the differentiation of operator-valued functions depending on a parameter
Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 207-218.

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A study of the convergence of the differentiation formula $$ (f(A))'=f'(A)A'+\frac{f''(A)}{2!}[A'A]+\frac{f'''(A)}{3!}[[A'A]A]+\dots, $$ where $[XY]=XY-YX$, and $A=A(t)$ is a function of the real variable $t$ with values in a Banach algebra. 
@article{MZM_1971_10_2_a10,
     author = {V. I. Burenkov},
     title = {Formula for the differentiation of operator-valued functions depending on a parameter},
     journal = {Matemati\v{c}eskie zametki},
     pages = {207--218},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a10/}
}
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V. I. Burenkov. Formula for the differentiation of operator-valued functions depending on a parameter. Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 207-218. http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a10/