The nth derivative

  • Authors:

  • Mhenni Benghorbal;Robert M. Corless

  • Affiliations:

  • The University of Western Ontario, London, Ontario, Canada;The University of Western Ontario, London, Ontario, Canada

  • Venue:
  • ACM SIGSAM Bulletin
  • Year:
  • 2002

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Abstract

The following problem is one that many first year calculus students find quite difficult:Given a formula for a function f in a variable x, find a formula for its nth derivative.Example 1.1: [1, p. 229] Iff(x) = xm, (1)then its nth derivative isf(n) (x) = m-nxm-n, (2)wherem-n = m(m - 1) (m - 2) ⋅⋅⋅ (m - n + 1).The difficulties for students include, first, the discovery of a formula valid for all integers n and, second, the proof (for example, by induction) that the formula is correct. Can computer algebra systems do better?It is certain that Macsyma could (we remember commands for infinite Taylor series expansion of elementary functions, and that necessarily involves discovering a correct formula for the nth derivative of the input). Currently, Maple cannot---at least, not without help, just with one command, now that the Formal Power Series package in the share library is no longer supported.