{"id":19294,"date":"2016-12-25T22:15:27","date_gmt":"2016-12-25T22:15:27","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=19294"},"modified":"2016-12-25T22:15:27","modified_gmt":"2016-12-25T22:15:27","slug":"calculus-the-derivative-of-an-inverse-function","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/calculus-the-derivative-of-an-inverse-function\/","title":{"rendered":"Calculus:  the derivative of an inverse function"},"content":{"rendered":"<h1>The tutor shows the development of a formula for the derivative of an inverse.<\/h1>\n<p>Let&#8217;s imagine m(x) is a function with inverse m<sup>-1<\/sup>(x).  Then<\/p>\n<p style=\"text-align:center\">m(m<sup>-1<\/sup>(x)) = x<\/p>\n<p>By implicit differentiation,<\/p>\n<p style=\"text-align:center\">[m(m<sup>-1<\/sup>(x))]&#8217; = 1<\/p>\n<p>By the chain rule,<\/p>\n<p style=\"text-align:center\">[m(m<sup>-1<\/sup>(x))]&#8217; = m'(m<sup>-1<\/sup>(x))*(m<sup>-1<\/sup>(x))&#8217;<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align:center\">m'(m<sup>-1<\/sup>(x))*(m<sup>-1<\/sup>(x))&#8217; = 1<\/p>\n<p>Dividing both sides by m'(m<sup>-1<\/sup>(x)) yields<\/p>\n<p style=\"text-align:center\">(m<sup>-1<\/sup>(x))&#8217; = 1\/m'(m<sup>-1<\/sup>(x))<\/p>\n<p>In a coming post I&#8217;ll show an example of using this formula to find the derivative of a specific inverse function.<\/p>\n<p>HTH:)<\/p>\n<p>Source:<\/p>\n<p>Larson, Roland E. and Robert P. Hostetler.  <u>Calculus<\/u>, 3rd ed.  Toronto:  D C Heath and Company, 1989.<\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The tutor shows the development of a formula for the derivative of an inverse. Let&#8217;s imagine m(x) is a function with inverse m-1(x). Then m(m-1(x)) = x By implicit differentiation, [m(m-1(x))]&#8217; = 1 By the chain rule, [m(m-1(x))]&#8217; = m'(m-1(x))*(m-1(x))&#8217; &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/calculus-the-derivative-of-an-inverse-function\/\"> <span class=\"screen-reader-text\">Calculus:  the derivative of an inverse function<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[234],"tags":[507,2078,450],"class_list":["post-19294","post","type-post","status-publish","format-standard","hentry","category-calculus","tag-chain-rule","tag-derivative-of-inverse-function","tag-implicit-differentiation"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19294","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=19294"}],"version-history":[{"count":13,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19294\/revisions"}],"predecessor-version":[{"id":19307,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19294\/revisions\/19307"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=19294"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=19294"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=19294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}