{"id":5673,"date":"2014-09-26T20:22:37","date_gmt":"2014-09-26T20:22:37","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=5673"},"modified":"2018-01-27T21:17:47","modified_gmt":"2018-01-27T21:17:47","slug":"calculus-the-derivative-of-ax","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/calculus-the-derivative-of-ax\/","title":{"rendered":"Calculus:  the derivative of a^x"},"content":{"rendered":"<h1>Tutoring calculus, you might be asked about the derivative of a^x from time to time.  The math tutor shows the trick to this old favourite.<\/h1>\n<p><strong>Example:  Find the derivative of a^x<\/strong><\/p>\n<p>Solution:<\/p>\n<p>As in so many cases, the trick here is to rewrite the expression to a form more easily penetrable by derivative techniques.  With this example, we&#8217;ll realize that e^x and lnx are inverses.  Therefore,<\/p>\n<p>a^x=e^(lna^x)<\/p>\n<p>Using the &#8220;bring the exponent down in front of a log&#8221; rule, we can proceed to<\/p>\n<p>e^(lna^x)=e^(xlna)<\/p>\n<p>Now, we recall the derivate rule<\/p>\n<p>d\/dx e^(f(x))=f'(x)e^(f(x))<\/p>\n<p>Since lna is a constant, the derivate of xlna is just lna.  Therefore,<\/p>\n<p>d\/dx e^(xlna)=(lna)e^(xlna)<\/p>\n<p>Therefore, <\/p>\n<p>d\/dx a^x=(lna)e^(xlna)=(lna)a^x<\/p>\n<p>I&#8217;ll be talking more about derivatives in future posts.  Cheers:)<\/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>Tutoring calculus, you might be asked about the derivative of a^x from time to time. The math tutor shows the trick to this old favourite. Example: Find the derivative of a^x Solution: As in so many cases, the trick here &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/calculus-the-derivative-of-ax\/\"> <span class=\"screen-reader-text\">Calculus:  the derivative of a^x<\/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":[235],"class_list":["post-5673","post","type-post","status-publish","format-standard","hentry","category-calculus","tag-derivative-of-exponent-with-constant-base"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/5673","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=5673"}],"version-history":[{"count":43,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/5673\/revisions"}],"predecessor-version":[{"id":28818,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/5673\/revisions\/28818"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=5673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=5673"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=5673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}