{"id":3360,"date":"2024-06-20T15:25:40","date_gmt":"2024-06-20T15:25:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3360"},"modified":"2024-08-05T02:02:38","modified_gmt":"2024-08-05T02:02:38","slug":"implicit-differentiation-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/implicit-differentiation-fresh-take\/","title":{"raw":"Implicit Differentiation: Fresh Take","rendered":"Implicit Differentiation: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\">Use implicit differentiation to find derivatives and the equations for tangent lines<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>What is Implicit Differentiation?<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Implicit differentiation is a method to find derivatives of functions that are not explicitly defined in terms of one variable.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Explicit vs. Implicit Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Explicit: [latex]y = f(x)[\/latex] (e.g., [latex]y = x^2 + 1[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Implicit: Relationship between [latex]x[\/latex] and [latex]y[\/latex] (e.g., [latex]x^2 + y^2 = 25[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Treat [latex]y[\/latex] as a function of [latex]x[\/latex] and use the chain rule when differentiating with respect to [latex]x[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Process:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Differentiate both sides of the equation with respect to [latex]x[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Group terms with [latex]\\frac{dy}{dx}[\/latex] on one side.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738220225\">Find [latex]\\frac{dy}{dx}[\/latex] for [latex]y[\/latex] defined implicitly by the equation [latex]4x^5+ \\tan y=y^2+5x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8993550\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8993550\"]<\/p>\r\n<p id=\"fs-id1169738221942\">Follow the problem solving strategy, remembering to apply the chain rule to differentiate [latex]\\tan y[\/latex] and [latex]y^2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737953797\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737953797\"]<\/p>\r\n<p id=\"fs-id1169737953797\">[latex]\\frac{dy}{dx}=\\large \\frac{5-20x^4}{\\sec^2 y-2y}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find [latex]\\frac{dy}{dx}[\/latex] for the equation [latex]x^3 + y^3 = 6xy[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"671838\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"671838\"]<\/p>\r\n<p>Differentiate both sides with respect to [latex]x[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(x^3 + y^3) = \\frac{d}{dx}(6xy)[\/latex]<\/p>\r\n<p>Apply the chain rule and differentiate:<\/p>\r\n<p style=\"text-align: center;\">[latex]3x^2 + 3y^2\\frac{dy}{dx} = 6y + 6x\\frac{dy}{dx}[\/latex]<\/p>\r\n<p>Group terms with [latex]\\frac{dy}{dx}[\/latex] on one side:<\/p>\r\n<p style=\"text-align: center;\">[latex]3y^2\\frac{dy}{dx} - 6x\\frac{dy}{dx} = 6y - 3x^2[\/latex]<\/p>\r\n<p>Factor out [latex]\\frac{dy}{dx}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex](3y^2 - 6x)\\frac{dy}{dx} = 6y - 3x^2[\/latex]<\/p>\r\n<p>Solve for [latex]\\frac{dy}{dx}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{6y - 3x^2}{3y^2 - 6x}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Finding Tangent Lines Implicitly<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Application of Implicit Differentiation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Used to find tangent lines for curves defined by complex equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Process:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find [latex]\\frac{dy}{dx}[\/latex] using implicit differentiation<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Evaluate [latex]\\frac{dy}{dx}[\/latex] at the given point to find the slope<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use point-slope form to find the equation of the tangent line<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Advantages:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Can handle curves not easily expressed as explicit functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplifies calculations for certain types of curves (e.g., conics)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Equations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Slope-intercept form: [latex]y = mx + b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738186793\">Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[\/latex] at the point [latex](5,3)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8005423\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8005423\"]<\/p>\r\n<p id=\"fs-id1169738240184\">Using implicit differentiation, you should find that [latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737145207\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737145207\"][latex]y=\\frac{5}{3}x-\\frac{16}{3}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the equation of the tangent line to the curve [latex]xy + y^2 = 4[\/latex] at the point [latex](1, 1)[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"288841\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"288841\"]<\/p>\r\n<p>Use implicit differentiation to find [latex]\\frac{dy}{dx}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>\r\n\\frac{d}{dx}(xy + y^2) &amp;=&amp; \\frac{d}{dx}(4) \\\\<br \/>\r\ny + x\\frac{dy}{dx} + 2y\\frac{dy}{dx} &amp;=&amp; 0 \\\\<br \/>\r\n\\frac{dy}{dx}(x + 2y) &amp;=&amp; -y \\\\<br \/>\r\n\\frac{dy}{dx} &amp;=&amp; -\\frac{y}{x + 2y}<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p>Evaluate [latex]\\frac{dy}{dx}[\/latex] at [latex](1, 1)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx}\\big|_{(1,1)} = -\\frac{1}{1 + 2(1)} = -\\frac{1}{3}[\/latex]<\/p>\r\n<p>Use point-slope form to find the equation of the tangent line:<\/p>\r\n<p style=\"text-align: center;\">[latex]y - 1 = -\\frac{1}{3}(x - 1)[\/latex]<\/p>\r\n<p>Convert to slope-intercept form:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>\r\ny - 1 &amp;=&amp; -\\frac{1}{3}x + \\frac{1}{3} \\\\<br \/>\r\ny &amp;=&amp; -\\frac{1}{3}x + \\frac{4}{3}<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p>Therefore, the equation of the tangent line is [latex]y = -\\frac{1}{3}x + \\frac{4}{3}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use implicit differentiation to find derivatives and the equations for tangent lines<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>What is Implicit Differentiation?<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Implicit differentiation is a method to find derivatives of functions that are not explicitly defined in terms of one variable.<\/li>\n<li class=\"whitespace-normal break-words\">Explicit vs. Implicit Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Explicit: [latex]y = f(x)[\/latex] (e.g., [latex]y = x^2 + 1[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Implicit: Relationship between [latex]x[\/latex] and [latex]y[\/latex] (e.g., [latex]x^2 + y^2 = 25[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Treat [latex]y[\/latex] as a function of [latex]x[\/latex] and use the chain rule when differentiating with respect to [latex]x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">General Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Differentiate both sides of the equation with respect to [latex]x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Group terms with [latex]\\frac{dy}{dx}[\/latex] on one side.<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738220225\">Find [latex]\\frac{dy}{dx}[\/latex] for [latex]y[\/latex] defined implicitly by the equation [latex]4x^5+ \\tan y=y^2+5x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8993550\">Hint<\/button><\/p>\n<div id=\"q8993550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738221942\">Follow the problem solving strategy, remembering to apply the chain rule to differentiate [latex]\\tan y[\/latex] and [latex]y^2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737953797\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737953797\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737953797\">[latex]\\frac{dy}{dx}=\\large \\frac{5-20x^4}{\\sec^2 y-2y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find [latex]\\frac{dy}{dx}[\/latex] for the equation [latex]x^3 + y^3 = 6xy[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q671838\">Show Answer<\/button><\/p>\n<div id=\"q671838\" class=\"hidden-answer\" style=\"display: none\">\n<p>Differentiate both sides with respect to [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(x^3 + y^3) = \\frac{d}{dx}(6xy)[\/latex]<\/p>\n<p>Apply the chain rule and differentiate:<\/p>\n<p style=\"text-align: center;\">[latex]3x^2 + 3y^2\\frac{dy}{dx} = 6y + 6x\\frac{dy}{dx}[\/latex]<\/p>\n<p>Group terms with [latex]\\frac{dy}{dx}[\/latex] on one side:<\/p>\n<p style=\"text-align: center;\">[latex]3y^2\\frac{dy}{dx} - 6x\\frac{dy}{dx} = 6y - 3x^2[\/latex]<\/p>\n<p>Factor out [latex]\\frac{dy}{dx}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex](3y^2 - 6x)\\frac{dy}{dx} = 6y - 3x^2[\/latex]<\/p>\n<p>Solve for [latex]\\frac{dy}{dx}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{6y - 3x^2}{3y^2 - 6x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding Tangent Lines Implicitly<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Application of Implicit Differentiation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to find tangent lines for curves defined by complex equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find [latex]\\frac{dy}{dx}[\/latex] using implicit differentiation<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate [latex]\\frac{dy}{dx}[\/latex] at the given point to find the slope<\/li>\n<li class=\"whitespace-normal break-words\">Use point-slope form to find the equation of the tangent line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can handle curves not easily expressed as explicit functions<\/li>\n<li class=\"whitespace-normal break-words\">Simplifies calculations for certain types of curves (e.g., conics)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Slope-intercept form: [latex]y = mx + b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738186793\">Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[\/latex] at the point [latex](5,3)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8005423\">Hint<\/button><\/p>\n<div id=\"q8005423\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738240184\">Using implicit differentiation, you should find that [latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737145207\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737145207\" class=\"hidden-answer\" style=\"display: none\">[latex]y=\\frac{5}{3}x-\\frac{16}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the equation of the tangent line to the curve [latex]xy + y^2 = 4[\/latex] at the point [latex](1, 1)[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q288841\">Show Answer<\/button><\/p>\n<div id=\"q288841\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use implicit differentiation to find [latex]\\frac{dy}{dx}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>  \\frac{d}{dx}(xy + y^2) &=& \\frac{d}{dx}(4) \\\\<br \/>  y + x\\frac{dy}{dx} + 2y\\frac{dy}{dx} &=& 0 \\\\<br \/>  \\frac{dy}{dx}(x + 2y) &=& -y \\\\<br \/>  \\frac{dy}{dx} &=& -\\frac{y}{x + 2y}<br \/>  \\end{array}[\/latex]<\/p>\n<p>Evaluate [latex]\\frac{dy}{dx}[\/latex] at [latex](1, 1)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx}\\big|_{(1,1)} = -\\frac{1}{1 + 2(1)} = -\\frac{1}{3}[\/latex]<\/p>\n<p>Use point-slope form to find the equation of the tangent line:<\/p>\n<p style=\"text-align: center;\">[latex]y - 1 = -\\frac{1}{3}(x - 1)[\/latex]<\/p>\n<p>Convert to slope-intercept form:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>  y - 1 &=& -\\frac{1}{3}x + \\frac{1}{3} \\\\<br \/>  y &=& -\\frac{1}{3}x + \\frac{4}{3}<br \/>  \\end{array}[\/latex]<\/p>\n<p>Therefore, the equation of the tangent line is [latex]y = -\\frac{1}{3}x + \\frac{4}{3}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3360"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3360\/revisions"}],"predecessor-version":[{"id":3783,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3360\/revisions\/3783"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3360\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3360"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3360"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3360"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}