Faà di Bruno’s formula

Faà di Bruno’s formula is a generalization of the chain ruleto higher order derivatives which expresses the derivative of acomposition of functions as a series of products of derivatives:

{d^{n}\\over dx^{n}}f(g(x))=\\sum_{\\sum_{k=0}^{n}km_{k}=n}\\frac{n!}{m_{1}!\\,m_{2%}!\\,m_{3}!\\,\\cdots 1!^{m_{1}}\\,2!^{m_{2}}\\,3!^{m_{3}}\\,\\cdots}f^{(m_{1}+\\cdots%+m_{n})}(g(x))\\prod_{j\\,:\\,m_{j}\eq 0}\\left(g^{(j)}(x)\\right)^{m_{j}}

This formula was discovered by Francesco Faà di Bruno in the 1850s and canbe proved by induction on the order of the derivative.

References

1 Faà di Bruno, C. F.. “Sullo sviluppo delle funzione.” Ann. diScienze Matem. et Fisiche di Tortoloni 6 (1855): 479-480
2 Faà di Bruno, C. F.. “Note sur un nouvelle formule de calcul différentiel.” Quart. J. Math. 1 (1857): 359-360
3 H. Figueroa & J. M. Gracia-Bondía, “Combinatorial Hopf Algebras in Quantum Field Theory I” Rev. Math. Phys. 17 (2005): 881 - 975