{"title":"n阶导数","authors":"Mhenni M. Benghorbal, Robert M Corless","doi":"10.1145/565145.565149","DOIUrl":null,"url":null,"abstract":"The following problem is one that many first year calculus students find quite difficult:Given a formula for a function <i>f</i> in a variable <i>x,</i> find a formula for its <i>n</i>th derivative.<b>Example 1.1:</b> [1, p. 229] If<i>f</i>(<i>x</i>) = <i>x<sup>m</sup>,</i> (1)then its <i>n</i>th derivative is<i>f</i><sup>(<i>n</i>)</sup> (<i>x</i>) = <i>m-<sup>n</sup>x<sup>m-n</sup>,</i> (2)where<i>m-<sup>n</sup> = m</i>(<i>m</i> - 1) (<i>m</i> - 2) ṡṡṡ (<i>m - n</i> + 1).The difficulties for students include, first, the discovery of a formula valid for all integers <i>n</i> 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 <i>n</i>th 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.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2002-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"The nth derivative\",\"authors\":\"Mhenni M. Benghorbal, Robert M Corless\",\"doi\":\"10.1145/565145.565149\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The following problem is one that many first year calculus students find quite difficult:Given a formula for a function <i>f</i> in a variable <i>x,</i> find a formula for its <i>n</i>th derivative.<b>Example 1.1:</b> [1, p. 229] If<i>f</i>(<i>x</i>) = <i>x<sup>m</sup>,</i> (1)then its <i>n</i>th derivative is<i>f</i><sup>(<i>n</i>)</sup> (<i>x</i>) = <i>m-<sup>n</sup>x<sup>m-n</sup>,</i> (2)where<i>m-<sup>n</sup> = m</i>(<i>m</i> - 1) (<i>m</i> - 2) ṡṡṡ (<i>m - n</i> + 1).The difficulties for students include, first, the discovery of a formula valid for all integers <i>n</i> 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 <i>n</i>th 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.\",\"PeriodicalId\":314801,\"journal\":{\"name\":\"SIGSAM Bull.\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2002-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIGSAM Bull.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/565145.565149\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIGSAM Bull.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/565145.565149","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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) ṡṡṡ (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.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
