{"id":3891,"date":"2021-05-20T18:29:55","date_gmt":"2021-05-20T18:29:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-implicit-differentiation\/"},"modified":"2021-07-03T18:56:00","modified_gmt":"2021-07-03T18:56:00","slug":"review-for-implicit-differentiation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-implicit-differentiation\/","title":{"raw":"Skills Review for Implicit Differentiation","rendered":"Skills Review for Implicit Differentiation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Isolate a specific variable within a formula&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Isolate a specific variable within a formula or equation<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the\u00a0<em>Implicit Differentiation<\/em>\u00a0section, you will learn how to take the derivative of functions that have both [latex]x[\/latex] and [latex]y[\/latex] on the same side of the equal sign. Here we will review solving an equation for a specified variable which will assist you when using differentiating certain types of functions implicitly.\r\n<h2>Isolate a Specified Variable in an Equation<\/h2>\r\nWe will sometimes work with equations that contain more than one variable. You will sometimes have to solve the equation for one of the variables.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solve An Equation for A Specified Variable<\/h3>\r\nLet [latex]x^{2}y-7x+y=3[\/latex]. Solve for\u00a0[latex]y[\/latex].\r\n\r\n[reveal-answer q=\"967601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"967601\"]\r\n\r\nFirst, isolate all terms that contain\u00a0[latex]y[\/latex].\r\n\r\nWe will first add\u00a0[latex]7x[\/latex] to both sides of the equation:\u00a0[latex]x^{2}y+y=3+7x[\/latex]\r\n\r\nThen, factor\u00a0[latex]y[\/latex] from both terms on the left side:\u00a0[latex]y(x^{2}+1)=3+7x[\/latex]\r\n\r\nNow, divide both sides by\u00a0[latex]x^{2}+1[\/latex]:\u00a0[latex]y=\\frac{3+7x}{x^{2}+1}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7638[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]8497[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solve An Equation for A Specified Variable<\/h3>\r\nLet [latex]3x^{2}+y\\cos{x}+x^{3}y=4[\/latex]. Solve for\u00a0[latex]y[\/latex].\r\n\r\n[reveal-answer q=\"967602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"967602\"]\r\n\r\nFirst, isolate all terms that contain\u00a0[latex]y[\/latex].\r\n\r\nWe will first subtract [latex]3x^{2}[\/latex] from both sides of the equation: [latex]y\\cos{x}+x^{3}y=4-3x^{2}[\/latex]\r\n\r\nThen, factor\u00a0[latex]y[\/latex] from both terms on the left side:\u00a0[latex]y(\\cos{x}+x^{3})=4-3x^{2}[\/latex]\r\n\r\nNow, divide both sides by\u00a0[latex]\\cos{x}+x^{3}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{4-3x^{2}}{(\\cos{x}+x^{3})}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Isolate a specific variable within a formula&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Isolate a specific variable within a formula or equation<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In the\u00a0<em>Implicit Differentiation<\/em>\u00a0section, you will learn how to take the derivative of functions that have both [latex]x[\/latex] and [latex]y[\/latex] on the same side of the equal sign. Here we will review solving an equation for a specified variable which will assist you when using differentiating certain types of functions implicitly.<\/p>\n<h2>Isolate a Specified Variable in an Equation<\/h2>\n<p>We will sometimes work with equations that contain more than one variable. You will sometimes have to solve the equation for one of the variables.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solve An Equation for A Specified Variable<\/h3>\n<p>Let [latex]x^{2}y-7x+y=3[\/latex]. Solve for\u00a0[latex]y[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967601\">Show Solution<\/span><\/p>\n<div id=\"q967601\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, isolate all terms that contain\u00a0[latex]y[\/latex].<\/p>\n<p>We will first add\u00a0[latex]7x[\/latex] to both sides of the equation:\u00a0[latex]x^{2}y+y=3+7x[\/latex]<\/p>\n<p>Then, factor\u00a0[latex]y[\/latex] from both terms on the left side:\u00a0[latex]y(x^{2}+1)=3+7x[\/latex]<\/p>\n<p>Now, divide both sides by\u00a0[latex]x^{2}+1[\/latex]:\u00a0[latex]y=\\frac{3+7x}{x^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7638\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7638&theme=oea&iframe_resize_id=ohm7638&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm8497\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=8497&theme=oea&iframe_resize_id=ohm8497&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solve An Equation for A Specified Variable<\/h3>\n<p>Let [latex]3x^{2}+y\\cos{x}+x^{3}y=4[\/latex]. Solve for\u00a0[latex]y[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967602\">Show Solution<\/span><\/p>\n<div id=\"q967602\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, isolate all terms that contain\u00a0[latex]y[\/latex].<\/p>\n<p>We will first subtract [latex]3x^{2}[\/latex] from both sides of the equation: [latex]y\\cos{x}+x^{3}y=4-3x^{2}[\/latex]<\/p>\n<p>Then, factor\u00a0[latex]y[\/latex] from both terms on the left side:\u00a0[latex]y(\\cos{x}+x^{3})=4-3x^{2}[\/latex]<\/p>\n<p>Now, divide both sides by\u00a0[latex]\\cos{x}+x^{3}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{4-3x^{2}}{(\\cos{x}+x^{3})}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3891\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen 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