# 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:

$$\frac{{d}^{n}}{d{x}^{n}}f(g(x))=\sum _{{\sum}_{k=0}^{n}k{m}_{k}=n}\frac{n!}{{m}_{1}!{m}_{2}!{m}_{3}!\mathrm{\cdots}{1!}^{{m}_{1}}{\mathrm{\hspace{0.17em}2}!}^{{m}_{2}}{\mathrm{\hspace{0.17em}3}!}^{{m}_{3}}\mathrm{\cdots}}{f}^{({m}_{1}+\mathrm{\cdots}+{m}_{n})}(g(x))\prod _{j:{m}_{j}\ne 0}{\left({g}^{(j)}(x)\right)}^{{m}_{j}}$$ |

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

Title | Faà di Bruno’s formula |
---|---|

Canonical name | FaaDiBrunosFormula |

Date of creation | 2013-03-22 16:38:57 |

Last modified on | 2013-03-22 16:38:57 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 16W30 |

Synonym | Faa di Bruno’s formula |

Synonym | Faà di Bruno formula |

Synonym | Faa di Bruno formula |