From: Robin Chapman Newsgroups: sci.math Subject: Re: Kth derivative Of Exp[u[x]] Date: Mon, 01 Jun 1998 08:14:41 GMT In article <356EAA8C.3D41@student.utwente.nl>, Wilbert Dijkhof wrote: > > Robin Chapman wrote: > > > The formula for the n-th derivative of a composite f o g in terms > > of the derivatives of f and g is known as Faa di Bruno's formula. > > It's exercise 1.2.5.21 in Knuth's Fundamental Algorithms, 2nd ed., > > Addison-Wesley, 1973. In this case one of the functions is the exponential > > whose derivatives are easy to find :-) > > Would you post this formula? I wasn't intending to do so since it's semi-gruesome, but I must respond to this challenge. The Faa di Bruno formula for the n-th derivative of f o g is a sum over all partitions of n. Let f_j and g_j denote the j-th derivatives of f and g. A partition of n is a sequence of non-negative integers k_1,...,k_n with k_1 + 2k_2+...+ n k_n = n. The term corresponding to this partition is n!/(k_1! 1!^(k_1) k_2! 2!^(k_2) ... k_n! n!^(k_n) ) (f_r o g) g_1^(k_1) ... g_n^(k_n) where r = k_1 + k_2 + ... + k_n. (E&OE) To prove this: it's easy to see by induction that we must have a sum of terms like this and it remains to find the coefficients. To find them set f(x) = a_0 + a_1 x + a_2 x^2/2! + ... + a_n x^n/n! + ... and g(x) = b_1 x + b_2 x^2/2! + ... + b_n x^n/n! + ... and compute g(f(x)) by substituting. Now b_j = g_j(0) and a_j = (f_j o g)(0). The n-th derivative of f o g evaluated at zero is n! times the x^n coefficient of the series f(g(x)). Expanding this out gives Faa di Bruno's formula. Francesco Faa di Bruno (1825-1907) was canonized in 1971 see http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Faa_di_Bruno.html Robin Chapman + "They did not have proper Department of Mathematics - palms at home in Exeter." University of Exeter, EX4 4QE, UK + rjc@maths.exeter.ac.uk - Peter Carey, http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/ Now offering spam-free web-based newsreading