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:

$${d^n \over dx^n} f(g(x))=\sum_{\sum_{k=0}^n k m_k = n} \frac{n!}{m_1!\,m_2!\,m_3!\,\cdots 1!^{m_1}\,2!^{m_2}\,3!^{m_3}\,\cdots} f^{(m_1 + \cdots + m_n)}(g(x)) \prod_{j\,:\,m_j\neq 0}\left(g^{(j)}(x)\right)^{m_j}$$

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

Bibliography

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