FaaDiBrunosFormula 2007-02-01 00:18:40 2007-02-01 00:37:26 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here {\em Fa\`a 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 Fa\`a di Bruno in the 1850s and can be proved by induction on the order of the derivative. \begin{thebibliography}{1} \bibitem{} Fa\`a di Bruno, C. F.. ``Sullo sviluppo delle funzione.'' {\it Ann. di Scienze Matem. et Fisiche di Tortoloni} {\bf 6} (1855): 479-480 \bibitem{} Fa\`a di Bruno, C. F.. ``Note sur un nouvelle formule de calcul diff\'erentiel.'' {\it Quart. J. Math.} {\bf 1} (1857): 359-360 \bibitem{} H. Figueroa \& J. M. Gracia-Bond\'ia, ``Combinatorial Hopf Algebras in Quantum Field Theory I'' {\it Rev. Math. Phys.} {\bf 17} (2005): 881 - 975 \end{thebibliography}