{"id":33865,"date":"2022-04-24T20:33:22","date_gmt":"2022-04-25T00:33:22","guid":{"rendered":"https:\/\/nstem.org\/staging\/?p=33865"},"modified":"2022-04-24T20:33:22","modified_gmt":"2022-04-25T00:33:22","slug":"ap-calculus-ab-implicit-differentiation","status":"publish","type":"post","link":"https:\/\/nstem.org\/staging\/2022\/04\/ap-calculus-ab-implicit-differentiation\/","title":{"rendered":"AP Calculus AB: Implicit Differentiation"},"content":{"rendered":"

As the Advanced Placement (AP) testing season nears this spring, it is important to familiarize ourselves with the content matter. According to College Board, differentiation: <\/span>implicit<\/span><\/i>, composite, and inverse functions will be 9-13% of the test score in 2022.<\/span><\/p>\n

To develop a better understanding of implicit differentiation, we need a brief semantic breakdown of the two types of functions: implicit and explicit. We must recognize that \u201cimplicit\u201d and \u201cexplicit\u201d are antonyms, while \u201cexplicit\u201d means \u201cclearly defined or expressed,\u201d \u201cimplicit\u201d means the latter. In the following sections, we will touch upon concepts like functions, derivatives, and the chain rule, all of which are pertinent to the AP Calculus AB Exam.<\/span><\/p>\n

What do we do when <\/span>y<\/span><\/i> isn\u2019t clearly expressed as a function of <\/span>x,<\/span><\/i> or vice versa, much like in implicit functions? Well, there are three main steps to successfully differentiate an equation implicitly. Firstly, we take the derivative. Secondly, we gather all terms with <\/span>dy\/dx<\/span><\/i> onto the left side of the equation. Thirdly, we factor out <\/span>dy\/dx<\/span><\/i> if necessary, to create a single <\/span>dy\/dx<\/span><\/i> term. Fourthly, we solve for <\/span>dy\/dx<\/span><\/i>.<\/span><\/p>\n

Let\u2019s have a look at a sample problem: \u201cfind<\/span> dy\/dx <\/span><\/i>for<\/span> y\u00b2 – 5x\u00b3 = 3x.<\/span><\/i>\u201d<\/span><\/p>\n

In step 1, we take the derivative.<\/span><\/p>\n

2y dy\/dx – 15x\u00b2 = 3 dy\/dx<\/span><\/p>\n

 <\/p>\n

In step 2, <\/span>we gather all terms with <\/span>dy\/dx<\/span><\/i> onto the left side of the equation.<\/span><\/p>\n

2y dy\/dx – 3 dy\/dx = 15x\u00b2<\/span><\/p>\n

 <\/p>\n

In step 3, <\/span>we factor out <\/span>dy\/dx<\/span><\/i> if necessary, to create a single <\/span>dy\/dx<\/span><\/i> term.\u00a0<\/span><\/p>\n

dy\/dx (2y – 3) = 15x\u00b2<\/span><\/p>\n

 <\/p>\n

In step 4, <\/span>we solve for <\/span>dy\/dx<\/span><\/i>.<\/span><\/p>\n

dy\/dx = 15x\u00b2\/2y – 3<\/span><\/p>\n

 <\/p>\n

Hopefully, this has facilitated your use of implicit differentiation as a method to differentiate implicitly defined functions, and you are now able to complete your exam with a firm resolve. <\/span><\/p>\n

 <\/p>\n

By: Estefania Olaiz<\/span><\/p>\n

March 20, 2022<\/span><\/p>\n

Sources:<\/b><\/p>\n

    \n
  1. https:\/\/www.cliffsnotes.com\/study-guides\/calculus\/calculus\/the-derivative\/implicit-differentiation<\/span><\/a>\u00a0<\/span><\/li>\n
  2. https:\/\/tutorial.math.lamar.edu\/classes\/calci\/implicitdiff.aspx<\/span><\/a><\/li>\n
  3. https:\/\/www.mathsisfun.com\/calculus\/implicit-differentiation.html<\/span><\/a><\/li>\n
  4. https:\/\/www.mathsisfun.com\/calculus\/implicit-differentiation.html<\/span><\/a><\/li>\n
  5. https:\/\/library.fiveable.me\/ap-calc\/unit-3\/implicit-differentiation\/study-guide\/k43S7kJDyGg9NFUm78Uw<\/span><\/a>\u00a0<\/span><\/li>\n
  6. https:\/\/ivytutorsnetwork.com\/blog\/how-to-study-for-the-ap-calculus-ab-exam<\/span><\/a><\/li>\n<\/ol>\n

     <\/p>\n","protected":false},"excerpt":{"rendered":"

    As the Advanced Placement (AP) testing season nears this spring, it is important to familiarize ourselves with the content matter. According to College Board, differentiation: implicit, composite, and inverse functions will be 9-13% of the test score in 2022. To develop a better understanding of implicit differentiation, we need a brief semantic breakdown of the […]<\/p>\n","protected":false},"author":9,"featured_media":33866,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pmpro_default_level":"","_kad_blocks_custom_css":"","_kad_blocks_head_custom_js":"","_kad_blocks_body_custom_js":"","_kad_blocks_footer_custom_js":"","footnotes":""},"categories":[198],"tags":[595,596,597,598,600,601,599,435,213],"class_list":["post-33865","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blog","tag-advanced-placement","tag-ap-calculus-ab","tag-ap-calculus-ab-exam","tag-calculus","tag-derivatives","tag-functions","tag-implicit-differentiation","tag-high-school","tag-mathematics","pmpro-has-access"],"taxonomy_info":{"category":[{"value":198,"label":"Blog"}],"post_tag":[{"value":595,"label":"#Advanced Placement"},{"value":596,"label":"#AP Calculus AB"},{"value":597,"label":"#AP Calculus AB Exam"},{"value":598,"label":"#Calculus"},{"value":600,"label":"#Derivatives"},{"value":601,"label":"#Functions"},{"value":599,"label":"#Implicit Differentiation"},{"value":435,"label":"high school"},{"value":213,"label":"mathematics"}]},"featured_image_src_large":["https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/ap_calculus.jpg",554,309,false],"author_info":{"display_name":"Eric Magers","author_link":"https:\/\/nstem.org\/staging\/author\/magerse\/"},"comment_info":0,"category_info":[{"term_id":198,"name":"Blog","slug":"blog","term_group":0,"term_taxonomy_id":198,"taxonomy":"category","description":"","parent":0,"count":274,"filter":"raw","cat_ID":198,"category_count":274,"category_description":"","cat_name":"Blog","category_nicename":"blog","category_parent":0}],"tag_info":[{"term_id":595,"name":"#Advanced Placement","slug":"advanced-placement","term_group":0,"term_taxonomy_id":595,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":596,"name":"#AP Calculus AB","slug":"ap-calculus-ab","term_group":0,"term_taxonomy_id":596,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":597,"name":"#AP Calculus AB Exam","slug":"ap-calculus-ab-exam","term_group":0,"term_taxonomy_id":597,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":598,"name":"#Calculus","slug":"calculus","term_group":0,"term_taxonomy_id":598,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":600,"name":"#Derivatives","slug":"derivatives","term_group":0,"term_taxonomy_id":600,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":601,"name":"#Functions","slug":"functions","term_group":0,"term_taxonomy_id":601,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":599,"name":"#Implicit Differentiation","slug":"implicit-differentiation","term_group":0,"term_taxonomy_id":599,"taxonomy":"post_tag","description":"","parent":0,"count":1,"filter":"raw"},{"term_id":435,"name":"high school","slug":"high-school","term_group":0,"term_taxonomy_id":435,"taxonomy":"post_tag","description":"","parent":0,"count":4,"filter":"raw"},{"term_id":213,"name":"mathematics","slug":"mathematics","term_group":0,"term_taxonomy_id":213,"taxonomy":"post_tag","description":"","parent":0,"count":2,"filter":"raw"}],"featured_image_src":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/ap_calculus.jpg","featured_image_src_square":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/ap_calculus.jpg","_links":{"self":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts\/33865","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/comments?post=33865"}],"version-history":[{"count":0,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts\/33865\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/media\/33866"}],"wp:attachment":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/media?parent=33865"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/categories?post=33865"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/tags?post=33865"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}