{"id":5688,"date":"2022-05-02T21:28:01","date_gmt":"2022-05-03T04:28:01","guid":{"rendered":"https:\/\/studysoup.com\/blog\/?p=5688"},"modified":"2022-05-02T21:28:01","modified_gmt":"2022-05-03T04:28:01","slug":"implicit-differentiation-6-steps","status":"publish","type":"post","link":"https:\/\/newblog.studysoup.com\/?p=5688","title":{"rendered":"Implicit Differentiation in 6 Steps"},"content":{"rendered":"\n<p>Sometimes in calculus, equations can\u2019t be solved easily since x and y don\u2019t explicitly define y as a function x. <\/p>\n\n\n\n<p>In such cases, we imply that a function  <em>y= f(x)<\/em>to satisfy the given equation. In short words this is implicit differentiation, giving you the ability to find the derivative of y with respect to x without solving the given equation for y.&nbsp; <\/p>\n\n\n\n<p>Let\u2019s look at an example problem solved that implements this concept.<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved: Implicit Differentiation Problem&nbsp;<\/h2>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Implicit Differentiation to Determine the Equation of  Tangent and Normal Line\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cshRFSvKJXk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/studysoup.com\/blog\/wp-content\/uploads\/2022\/05\/Screen-Shot-2022-05-02-at-8.21.07-AM-1024x113.png\" alt=\"implicit differentiation problem\" class=\"wp-image-5694\" width=\"649\" height=\"73\"\/><\/figure>\n\n\n\n<p>This problem has a few stages which we have to consider. First thing we have to do is perform implicit differentiation to determine the following using the given equation:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/pCbEjJILCHyDJHA1bQvAs6anf_hZ2iu3OCQI4TSfnuE6Jf1gJ7LMfduSQO6nNk0OX9chZxUcJVy1mNSkqaQtku56P_fBueDicLD1CF9RVqHGsu0HvHd3qKwR8uVDNT5bXXIqrOXK\" alt=\"implicit differentiation problem\" width=\"130\" height=\"58\"\/><\/figure><\/div>\n\n\n\n<p>After that we\u2019ll determine the value of <a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=y'%3D%5Cfrac%7Bdy%7D%7Bdx%7D#0\"><\/a> at the given point mentioned above which will give us the slope of the tangent line.<br><\/p>\n\n\n\n<p>From there, we can determine the equation of the tangent line and then determine the equation of the normal line. So let\u2019s break it down each part of the problem step by step:<br><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><a href=\"https:\/\/studysoup.com\/tsg\/index?utm_source=bannerad&amp;utm_medium=blog&amp;utm_campaign=jamessousaimplicitdifferentiation\" target=\"_blank\" rel=\"noreferrer noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/studysoup.com\/blog\/wp-content\/uploads\/2022\/04\/StudySoup-Textbook-Survival-Guides.png\" alt=\"textbooksurvivalguides\" class=\"wp-image-5657\" width=\"793\" height=\"264\"\/><\/a><\/figure><\/div>\n\n\n\n<p>1)Let\u2019s start by differentiating both sides of the equation with respect to x. If we have a function of y, we have to apply the chain rule and we\u2019ll have an extra factor of the following:<a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=%5Cfrac%7Bdy%7D%7Bdx%7D#0\"><\/a><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/7I8IcbbCZ8s3G2Ba5v4CbvwuthkwrWXZ8FGDzuudinQT6h4NzSAThUNbeOZtwaKhewQg3SQ76E4pkEDILEQ6f5x7lOoEU1EriCPxjkM2KClytEZtpl8z4oI-l_E7Lz7c8vvZufW-\" alt=\"implicit differentiation\" width=\"36\" height=\"58\"\/><\/figure><\/div>\n\n\n\n<p>Differentiating both sides with respect to x:<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/m60-B9XgtdXLaAZeKumpMjabVmD095ClhA1liqRXjkA41rwbRGXlbnfRzdMRs9EWeYtorGM7yoqRiFdK_HTk9hrVTGkIesduudJvWPskmpGvxpTRrwKUXTmbStzoz6lIG8eCROuk\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p><br>Now we need to apply the product rule:<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/0ZJ3bnbJiBggYHd0Q_-6_THEAKM3jcvkEI1iuVId1ynWI5XTSGMAkS0RMVyeZC_h5MjpATrRpOof4cEGy0N8SAg05USO3o3jKDcaA0fuf3nGlTrn0MPiK_IdcBZDzBm1jZ3y7EDW\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/jqv1N6JgbMzvRF0oFdid8Wz7KkNyVlZOAtg9NSHFPdTAuW6SbsKWHnKY7LyfKxF61wAnWm7xxvBzm3CcEGPf94N0tIX9EML1plif8uDKmuRMmq_13Yg6JKh-3740_CHK18TKM5rC\" alt=\"implicit differentiation\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p>This is the derivative of the first function times the derivative of the second function.&nbsp;<br><\/p>\n\n\n\n<p>Now differentiating the second function:<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/eKyDAgn9nRMeYukf-H4SCQ0jD6np5E36xczhAciGbZr3B0uO2wwzRGdp5APIuLlGHwDI3LMzNSwNtBES2DBoJbZe3OnBtNYopo_i9Qo9z7BrCkxCwXhW0qfE4swQ_lkd48zAqLmB\" alt=\"implicit differentiation problem\" width=\"500\" height=\"55\"\/><\/figure>\n\n\n\n<p>3) Now let\u2019s solve this equation for dy\/dx. Since the dy\/dx terms are both negative, we can move them to the right of the equation.&nbsp;<br><\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/5j6Pw_p2i5umzDnQuS32JFuT9_uLbN3KCVMofHbLPgbbFR0kAM5nzIHHv6n011kE5W-f3p-hLxq5pl0pZRVtjQbWBolP_fnTn8G5bmUxVQufA4Nz2LgVUzZc6xG3IvIK9-IJ36gO\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p>4) We can now factor out dy\/dx on the right:<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/T6LOYdXr4-h0o5BhqEnKDOMFfNAtxVFWpauUzlYANf31PiJc58JDB_XB2xxQX4akwvmjmnkBtyBGLaWhks0fcqOe2VW8G8wpgC1dl2iKQengLQYU12J5ghciSWUvgGThJFmyk9nT\" alt=\"\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p>Taking <em>cos(y)<\/em> as common factor in both the sides,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/MOSS9E9Fc2_2SBCJCRBDUMI5U8XR7ZjZI4j326QJuSa-0Njfue8CvtkLFMitaoJUVUDR70s6A5XbX6T0kSRLYqjcW0CRoSbQ-v0Boo1MrFP30V4EalAEK3NdRBpndCX4vtFhv5gv\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p>Eliminating cosine terms in both the sides,&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/B6amruNDyKNEgJAMzAILDaWpkyH2leEl7msp-W4ewEuyL9fVA4T3JdEIQGyNxuik2xzYpy7Wd2YCDtzqhoUQxAYgMzUv4limKS-yYJFYcFJp6xVbrGSStdRt9TWIJEv02W7zTjMK\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure>\n\n\n\n<p>Now we will write the equation for <em>y\u00b9<\/em><a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=y'#0\"><\/a><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/AkJChJm2e3f9gc4CLfIWNdRZrrqe-W_X_OUskYOBx53xrH-H14l_shZ7RrCAd_IhP75N0VIvksUTdaEq6LGvxi7007jiYvIhGm5dDbcs1_G4oGiyCmyHFraDdEq8hgYv4_2kiDJe\" alt=\"implicit differentiation problem\" width=\"336\" height=\"38\"\/><\/figure><\/div>\n\n\n\n<p>5) Let\u2019s find the slope of the tangent line at the point  (0,2<a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=(0%2C2%5Cpi)#0\"><\/a>\u03c0) To do this, we need to evaluate y\u2019at the point. In this case we need to substitute 0 for x and <a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=2%5Cpi#0\"><\/a> 2\u03c0 for y and simplify:<br><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/YMjFrnJpKQIaUKp_eTurKE-A9CqRPiPjOKLqPhcQkigIWCrHcVb1Ns1uRTAqE52OcY-MqeZ8Rg1VB0DBjr3NCodfjqjJzHjdp0EbJBwDAfJjuRaHCyQdRl2G2ar6e3Wi2oeCy5E9\" alt=\"implicit differentiation problem\" width=\"403\" height=\"44\"\/><\/figure><\/div>\n\n\n\n<p>Now we know the slope of the tangent line is <strong>&nbsp;m=0<\/strong>. This also indicates that the tangent line is<em> y=2\u03c0<\/em><a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=y%3D2%5Cpi#0\"><\/a>.<br><\/p>\n\n\n\n<p>*We can also simplify to get the tangent line y=2\u03c0 and m=0 using point-slope form:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/YlicreIJ9TlLSl5udwi68qjTMz6cW8ndnAm_kbDbHv2ZZUXL-H3_YyLnnrVdP1XzhLLQDtEtT2kutNjw0FBPSEvIorX0QljCmq5-XMys73FRhHhf6TBS-9yOa7pSba5mFHzEYRUd\" alt=\"implicit differentiation problem\" width=\"269\" height=\"30\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/zWpQNQi0t18CJs2-mj7yOfeIPQCNqulP86Tj3aYc_gxR6v0EUoXS16EgccRDZHxmVStTHtK3gvei-qN8Sjsxv5q-N1Mkfuz8Xp1o74oMMCfcOitNrT03xsNE_Mw4sdEyB9WvTqTM\" alt=\"implicit differentiation problem\" width=\"253\" height=\"28\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/l7fiGeDYL65e-mXM5J5U55StEOh4jRssqMscnutdIheA3bB17IN2t-_ZcmjxRuTbnvZB6cE9Oj3apCqozbtVM0k49Fjsui0ym3v-ih-1u_3SfGqwBhFOf2MG1O4kz0vHTGvI9GWv\" alt=\"implicit differentiation problem\" width=\"186\" height=\"20\"\/><\/figure><\/div>\n\n\n\n<p>6) If the tangent line is a horizontal line and we know that the regular line is a vertical line. The vertical line passing through (0,22\u03c0) must be x=0. <br><\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Therefore our final answer is:<\/strong><\/h4>\n\n\n\n<p>Tangent line: <em>y=2<\/em>\u03c0<a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=y%3D2%5Cpi#0\"><\/a><\/p>\n\n\n\n<p>Normal line: <em>x=0<\/em><a href=\"https:\/\/www.codecogs.com\/eqnedit.php?latex=x%3D0#0\"><\/a><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">We can also look at it graphically:<br><\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/vEXfsUcrHEmfSTsGwCTmUu0BcmdwWFf2mAWiRk8nYxoJcmqf9Q7i7xyqlveNrG0wi4zfRTa3DUHbyWN8w6zScVJPlO5sqTHzDA091MrUyeku3hQ8pv-ifQzonDGzuapPlvL5D_TD\" alt=\"implicit differentiation\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Conclusion<\/strong><\/h2>\n\n\n\n<p>This was one example of how you can apply implicit differentiation in calculus. If you\u2019re struggling with it- keep practicing. Or better yet, take a look at a <a href=\"https:\/\/studysoup.com\/tsg\/index?utm_source=bannerad&amp;utm_medium=blog&amp;utm_campaign=jamessousaimplicitdifferentiation\">high quality survival guide<\/a> for your textbook along with on demand help with solutions.&nbsp;<br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sometimes in calculus, equations can\u2019t be solved easily since x and y don\u2019t explicitly define y as a function x. In such cases, we imply that a function y= f(x)to satisfy the given equation. In short words this is implicit differentiation, giving you the ability to find the derivative of y with respect to x [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":5701,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[14,17,16],"class_list":["post-5688","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math","tag-calculus","tag-implicit-differentiation","tag-pre-calculus"],"_links":{"self":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts\/5688","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5688"}],"version-history":[{"count":0,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts\/5688\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/"}],"wp:attachment":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5688"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5688"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5688"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}