Faà di Bruno’s formula

Faà di Bruno’s formula is a generalization of the chain rule to higher order derivatives which expresses the derivative of a composition 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}\neq 0}\left(g^{(j)}(x)\right)^{m_{j}}

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

References

1 Faà di Bruno, C. F.. “Sullo sviluppo delle funzione.” Ann. di Scienze 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