{"id":380,"date":"2021-02-04T01:19:51","date_gmt":"2021-02-04T01:19:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=380"},"modified":"2022-03-16T05:37:58","modified_gmt":"2022-03-16T05:37:58","slug":"finding-the-derivative-using-implicit-differentiation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/finding-the-derivative-using-implicit-differentiation\/","title":{"raw":"Finding the Derivative Using Implicit Differentiation","rendered":"Finding the Derivative Using 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;Recognize the difference between an implicit function and an explicit function&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6145,&quot;3&quot;:{&quot;1&quot;:0},&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Recognize the difference between an implicit function and an explicit function<\/span><\/li>\r\n \t<li>Find the derivative of a complicated function by using implicit differentiation<\/li>\r\n \t<li>Use implicit differentiation to determine the equation of a tangent line<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1169737797432\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">What is Implicit Differentiation?<\/span><\/p>\r\n\r\n<div id=\"fs-id1169737702731\" class=\"bc-section section\">\r\n<p id=\"fs-id1169737919331\">In most discussions of math, if the dependent variable [latex]y[\/latex] is a function of the independent variable [latex]x[\/latex], we express [latex]y[\/latex] in terms of [latex]x[\/latex]. If this is the case, we say that [latex]y[\/latex] is an <em>explicit function<\/em> of [latex]x[\/latex]. For example, when we write the equation [latex]y=x^2+1[\/latex], we are defining [latex]y[\/latex] <em>explicitly<\/em> in terms of [latex]x[\/latex]. On the other hand, if the relationship between the function [latex]y[\/latex] and the variable [latex]x[\/latex] is expressed by an equation where [latex]y[\/latex] is not expressed entirely in terms of [latex]x[\/latex], we say that the equation defines [latex]y[\/latex] <em>implicitly<\/em> in terms of [latex]x[\/latex]. For example, the equation [latex]y-x^2=1[\/latex] defines the function [latex]y=x^2+1[\/latex] implicitly.<\/p>\r\n<p id=\"fs-id1169737766039\"><strong>Implicit differentiation<\/strong> allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of [latex]y[\/latex] are functions that satisfy the given equation, but that [latex]y[\/latex] is not actually a function of [latex]x[\/latex].<\/p>\r\n<p id=\"fs-id1169738018789\">In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly. For example, the functions<\/p>\r\n<p id=\"fs-id1169737771295\" style=\"text-align: center;\">[latex]y=\\sqrt{25-x^2}[\/latex],\u00a0 [latex]y = -\\sqrt{25-x^2}[\/latex], and [latex]y=\\begin{cases} \\sqrt{25-x^2} &amp; \\text{ if } \\, -5 \\le x &lt; 0 \\\\ -\\sqrt{25-x^2} &amp; \\text{ if } \\, 0 \\le x \\le 5 \\end{cases}[\/latex],<\/p>\r\nwhich are illustrated in Figure 1, are just three of the many functions defined implicitly by the equation [latex]x^2+y^2=25[\/latex].\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"875\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205502\/CNX_Calc_Figure_03_08_001.jpg\" alt=\"The circle with radius 5 and center at the origin is graphed fully in one picture. Then, only its segments in quadrants I and II are graphed. Then, only its segments in quadrants III and IV are graphed. Lastly, only its segments in quadrants II and IV are graphed.\" width=\"875\" height=\"988\" \/> Figure 1. The equation [latex]{x}^{2}+{y}^{2}=25[\/latex] defines many functions implicitly.[\/caption]\r\n<p id=\"fs-id1169737773822\">If we want to find the slope of the line tangent to the graph of [latex]x^2+y^2=25[\/latex] at the point [latex](3,4)[\/latex], we could evaluate the derivative of the function [latex]y=\\sqrt{25-x^2}[\/latex] at [latex]x=3[\/latex]. On the other hand, if we want the slope of the tangent line at the point [latex](3,-4)[\/latex], we could use the derivative of [latex]y=\u2212\\sqrt{25-x^2}[\/latex]. However, it is not always easy to solve for a function defined implicitly by an equation. Fortunately, the technique of implicit differentiation allows us to find the derivative of an implicitly defined function without ever solving for the function explicitly. The process of finding [latex]\\frac{dy}{dx}[\/latex] using implicit differentiation is described in the following problem-solving strategy.<\/p>\r\n\r\n<div id=\"fs-id1169737850132\" class=\"textbox examples\">\r\n<h3>Problem-Solving Strategy: Implicit Differentiation<\/h3>\r\n<p id=\"fs-id1169737749590\">To perform implicit differentiation on an equation that defines a function [latex]y[\/latex] implicitly in terms of a variable [latex]x[\/latex], use the following steps:<\/p>\r\n\r\n<ol id=\"fs-id1169737815995\">\r\n \t<li>Take the derivative of both sides of the equation. Keep in mind that [latex]y[\/latex] is a function of [latex]x[\/latex]. Consequently, whereas [latex]\\frac{d}{dx}(\\sin x)= \\cos x, \\, \\frac{d}{dx}(\\sin y)= \\cos y\\frac{dy}{dx}[\/latex] because we must use the Chain Rule to differentiate [latex]\\sin y[\/latex] with respect to [latex]x[\/latex].<\/li>\r\n \t<li>Rewrite the equation so that all terms containing [latex]\\frac{dy}{dx}[\/latex] are on the left and all terms that do not contain [latex]\\frac{dy}{dx}[\/latex] are on the right.<\/li>\r\n \t<li>Factor out [latex]\\frac{dy}{dx}[\/latex] on the left.<\/li>\r\n \t<li>Solve for [latex]\\frac{dy}{dx}[\/latex] by dividing both sides of the equation by an appropriate algebraic expression.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1169737819953\" class=\"textbook exercises\">\r\n<h3>Example: Using Implicit Differentiation<\/h3>\r\n<p id=\"fs-id1169737931738\">Assuming that [latex]y[\/latex] is defined implicitly by the equation [latex]x^2+y^2=25[\/latex], find [latex]\\frac{dy}{dx}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737948455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737948455\"]\r\n<p id=\"fs-id1169737948455\">Follow the steps in the problem-solving strategy.<\/p>\r\n\r\n<div id=\"fs-id1169737840968\" class=\"equation unnumbered\">[latex]\\begin{array}{llll} \\frac{d}{dx}(x^2+y^2) = \\frac{d}{dx}(25) &amp; &amp; &amp; \\text{Step 1. Differentiate both sides of the equation.} \\\\ \\frac{d}{dx}(x^2)+\\frac{d}{dx}(y^2) = 0 &amp; &amp; &amp; \\begin{array}{l}\\text{Step 1.1. Use the sum rule on the left.} \\\\ \\text{On the right,} \\, \\frac{d}{dx}(25)=0. \\end{array} \\\\ 2x+2y\\frac{dy}{dx} = 0 &amp; &amp; &amp; \\begin{array}{l}\\text{Step 1.2. Take the derivatives, so} \\, \\frac{d}{dx}(x^2)=2x \\\\ \\text{and} \\, \\frac{d}{dx}(y^2)=2y\\frac{dy}{dx}. \\end{array} \\\\ 2y\\frac{dy}{dx} = -2x &amp; &amp; &amp; \\begin{array}{l}\\text{Step 2. Keep the terms with} \\, \\frac{dy}{dx} \\, \\text{on the left.} \\\\ \\text{Move the remaining terms to the right.} \\end{array} \\\\ \\frac{dy}{dx} = -\\frac{x}{y} &amp; &amp; &amp; \\begin{array}{l}\\text{Step 4. Divide both sides of the equation by} \\\\ 2y. \\, \\text{(Step 3 does not apply in this case.)} \\end{array} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1169737758515\">Note that the resulting expression for [latex]\\frac{dy}{dx}[\/latex] is in terms of both the independent variable [latex]x[\/latex] and the dependent variable [latex]y[\/latex]. Although in some cases it may be possible to express [latex]\\frac{dy}{dx}[\/latex] in terms of [latex]x[\/latex] only, it is generally not possible to do so.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738068274\" class=\"textbook exercises\">\r\n<h3>Example: Using Implicit Differentiation and the Product Rule<\/h3>\r\n<p id=\"fs-id1169737948472\">Assuming that [latex]y[\/latex] is defined implicitly by the equation [latex]x^3 \\sin y+y=4x+3[\/latex], find [latex]\\frac{dy}{dx}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738041617\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738041617\"]\r\n<div id=\"fs-id1169737814728\" class=\"equation unnumbered\">[latex]\\begin{array}{llll}\\frac{d}{dx}(x^3 \\sin y+y) = \\frac{d}{dx}(4x+3) &amp; &amp; &amp; \\text{Step 1: Differentiate both sides of the equation.} \\\\ \\frac{d}{dx}(x^3 \\sin y)+\\frac{d}{dx}(y) = 4 &amp; &amp; &amp; \\begin{array}{l}\\text{Step 1.1: Apply the sum rule on the left.} \\\\ \\text{On the right,} \\, \\frac{d}{dx}(4x+3)=4. \\end{array} \\\\ (\\frac{d}{dx}(x^3) \\cdot \\sin y+\\frac{d}{dx}(\\sin y) \\cdot x^3) + \\frac{dy}{dx} = 4 &amp; &amp; &amp; \\begin{array}{l}\\text{Step 1.2: Use the product rule to find} \\\\ \\frac{d}{dx}(x^3 \\sin y). \\, \\text{Observe that} \\, \\frac{d}{dx}(y)=\\frac{dy}{dx}. \\end{array} \\\\ 3x^2 \\sin y+(\\cos y\\frac{dy}{dx}) \\cdot x^3 + \\frac{dy}{dx} = 4 &amp; &amp; &amp; \\begin{array}{l}\\text{Step 1.3: We know} \\, \\frac{d}{dx}(x^3)=3x^2. \\, \\text{Use the} \\\\ \\text{chain rule to obtain} \\, \\frac{d}{dx}(\\sin y)= \\cos y\\frac{dy}{dx}. \\end{array} \\\\ x^3 \\cos y\\frac{dy}{dx}+\\frac{dy}{dx} = 4-3x^2 \\sin y &amp; &amp; &amp; \\begin{array}{l}\\text{Step 2: Keep all terms containing} \\, \\frac{dy}{dx} \\, \\text{on the} \\\\ \\text{left. Move all other terms to the right.} \\end{array} \\\\ \\frac{dy}{dx}(x^3 \\cos y+1) = 4-3x^2 \\sin y &amp; &amp; &amp; \\text{Step 3: Factor out} \\, \\frac{dy}{dx} \\, \\text{on the left.} \\\\ \\frac{dy}{dx} = \\large \\frac{4-3x^2 \\sin y}{x^3 \\cos y+1} &amp; &amp; &amp; \\begin{array}{l}\\text{Step 4: Solve for} \\, \\frac{dy}{dx} \\, \\text{by dividing both sides of} \\\\ \\text{the equation by} \\, x^3 \\cos y+1. \\end{array} \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738217476\" class=\"textbook exercises\">\r\n<h3>Example: Using Implicit Differentiation to Find a Second Derivative<\/h3>\r\n<p id=\"fs-id1169738211084\">Find [latex]\\frac{d^2 y}{dx^2}[\/latex] if [latex]x^2+y^2=25[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738226743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226743\"]\r\n<p id=\"fs-id1169738226743\">In the first example, we showed that [latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex]. We can take the derivative of both sides of this equation to find [latex]\\frac{d^2 y}{dx^2}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169737766542\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll} \\frac{d^2 y}{dx^2} &amp; =\\large \\frac{d}{dx}(-\\frac{x}{y}) &amp; &amp; &amp; \\text{Differentiate both sides of} \\, \\frac{dy}{dx}=-\\frac{x}{y}. \\\\ &amp; = \\large -\\frac{(1 \\cdot y-x\\frac{dy}{dx})}{y^2} &amp; &amp; &amp; \\text{Use the quotient rule to find} \\, \\frac{d}{dx}(-\\frac{x}{y}). \\\\ &amp; = \\large \\frac{\u2212y+x\\frac{dy}{dx}}{y^2} &amp; &amp; &amp; \\text{Simplify.} \\\\ &amp; = \\large \\frac{\u2212y+x(-\\frac{x}{y})}{y^2} &amp; &amp; &amp; \\text{Substitute} \\, \\frac{dy}{dx}=-\\frac{x}{y}. \\\\ &amp; = \\large \\frac{\u2212y^2-x^2}{y^3} &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738044967\">At this point we have found an expression for [latex]\\frac{d^2 y}{dx^2}[\/latex]. If we choose, we can simplify the expression further by recalling that [latex]x^2+y^2=25[\/latex] and making this substitution in the numerator to obtain [latex]\\frac{d^2 y}{dx^2}=-\\frac{25}{y^3}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Using Implicit Differentiation to Find a Second Derivative.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F1FFAg-XhlQ?controls=0&amp;start=385&amp;end=473&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.8ImplicitDifferentiation385to473_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.8 Implicit Differentiation\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169737772805\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"8993550\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8993550\"]\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[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169737953797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737953797\"]\r\n<p id=\"fs-id1169737953797\">[latex]\\frac{dy}{dx}=\\large \\frac{5-20x^4}{\\sec^2 y-2y}[\/latex]<\/p>\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]206109[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding Tangent Lines Implicitly<\/h2>\r\n<p id=\"fs-id1169737931596\">Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.<\/p>\r\n\r\n<div id=\"fs-id1169737931601\" class=\"textbook exercises\">\r\n<h3>Example: Finding a Tangent Line to a Circle<\/h3>\r\n<p id=\"fs-id1169737950789\">Find the equation of the line tangent to the curve [latex]x^2+y^2=25[\/latex] at the point [latex](3,-4)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737143580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737143580\"]\r\n<p id=\"fs-id1169737143580\">Although we could find this equation without using implicit differentiation, using that method makes it much easier. In an earlier example, we found [latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex].<\/p>\r\n<p id=\"fs-id1169737144361\">The slope of the tangent line is found by substituting [latex](3,-4)[\/latex] into this expression. Consequently, the slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,-4)} =-\\frac{3}{-4}=\\frac{3}{4}[\/latex].<\/p>\r\n<p id=\"fs-id1169738198658\">Using the point [latex](3,-4)[\/latex] and the slope [latex]\\frac{3}{4}[\/latex] in the point-slope equation of the line, we then solve for [latex]y[\/latex] to obtain the equation [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] (Figure 2).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205505\/CNX_Calc_Figure_03_08_002.jpg\" alt=\"The circle with radius 5 and center at the origin is graphed. A tangent line is drawn through the point (3, \u22124).\" width=\"487\" height=\"358\" \/> Figure 2. The line [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] is tangent to [latex]x^2+y^2=25[\/latex] at the point [latex](3,\u22124)[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbook exercises\">\r\n<h3>Example: Finding the Equation of the Tangent Line to a Curve<\/h3>\r\n<p id=\"fs-id1169737143587\">Find the equation of the line tangent to the graph of [latex]y^3+x^3-3xy=0[\/latex] at the point [latex]\\left(\\frac{3}{2},\\frac{3}{2}\\right)[\/latex] (Figure 3). This curve is known as the folium (or leaf) of Descartes.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205509\/CNX_Calc_Figure_03_08_003.jpg\" alt=\"A folium is shown, which is a line that creates a loop that crosses over itself. In this graph, it crosses over itself at (0, 0). Its tangent line from (3\/2, 3\/2) is shown.\" width=\"487\" height=\"433\" \/> Figure 3. Finding the tangent line to the folium of Descartes at [latex](\\frac{3}{2},\\frac{3}{2})[\/latex].[\/caption]\r\n<div class=\"wp-caption-text\"><\/div>\r\n[reveal-answer q=\"fs-id1169738217007\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738217007\"]\r\n<p id=\"fs-id1169738217007\">Begin by finding [latex]\\frac{dy}{dx}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169738185005\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{d}{dx}(y^3+x^3-3xy) = \\frac{d}{dx}(0) \\\\ \\\\ 3y^2\\frac{dy}{dx}+3x^2-(3y+3x\\frac{dy}{dx}) = 0 \\\\ \\\\ \\frac{dy}{dx} = \\frac{3y-3x^2}{3y^2-3x}. \\\\ \\\\ \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169737904704\">Next, substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex] to find the slope of the tangent line:<\/p>\r\n\r\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}|_{(\\frac{3}{2},\\frac{3}{2})}=-1[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169737950853\">Finally, substitute into the point-slope equation of the line and solve for [latex]y[\/latex] to obtain<\/p>\r\n\r\n<div id=\"fs-id1169737950856\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\u2212x+3[\/latex]<\/div>\r\n&nbsp;\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737935216\" class=\"textbook exercises\">\r\n<h3>Example: Applying Implicit Differentiation<\/h3>\r\n<p id=\"fs-id1169737935226\">In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation [latex]4x^2+25y^2=100[\/latex]. The rocket can fire missiles along lines tangent to its path. The object of the game is to destroy an incoming asteroid traveling along the positive [latex]x[\/latex]-axis toward [latex](0,0)[\/latex]. If the rocket fires a missile when it is located at [latex]\\left(3,\\frac{8}{5}\\right)[\/latex], where will it intersect the [latex]x[\/latex]-axis?<\/p>\r\n[reveal-answer q=\"fs-id1169738099354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738099354\"]\r\n<p id=\"fs-id1169738099354\">To solve this problem, we must determine where the line tangent to the graph of<\/p>\r\n<p id=\"fs-id1169738099357\">[latex]4x^2+25y^2=100[\/latex] at [latex](3,\\frac{8}{5})[\/latex] intersects the [latex]x[\/latex]-axis. Begin by finding [latex]\\frac{dy}{dx}[\/latex] implicitly.<\/p>\r\n<p id=\"fs-id1169738214601\">Differentiating, we have<\/p>\r\n\r\n<div id=\"fs-id1169737144258\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]8x+50y\\frac{dy}{dx}=0[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169737144293\">Solving for [latex]\\frac{dy}{dx}[\/latex], we have<\/p>\r\n\r\n<div id=\"fs-id1169738223546\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{dy}{dx}=-\\dfrac{4x}{25y}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169738219286\">The slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,\\frac{8}{5})}=-\\frac{3}{10}[\/latex]. The equation of the tangent line is [latex]y=-\\frac{3}{10}x+\\frac{5}{2}[\/latex]. To determine where the line intersects the [latex]x[\/latex]-axis, solve [latex]0=-\\frac{3}{10}x+\\frac{5}{2}[\/latex]. The solution is [latex]x=\\frac{25}{3}[\/latex]. The missile intersects the [latex]x[\/latex]-axis at the point [latex](\\frac{25}{3},0)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding the Equation of the Tangent Line to a Curve.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F1FFAg-XhlQ?controls=0&amp;start=553&amp;end=718&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.8ImplicitDifferentiation535to718_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.8 Implicit Differentiation\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169738186785\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"8005423\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8005423\"]\r\n<p id=\"fs-id1169738240184\">Using implicit differentiation, you should find that [latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169737145207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737145207\"][latex]y=\\frac{5}{3}x-\\frac{16}{3}[\/latex]\r\n\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;Recognize the difference between an implicit function and an explicit function&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6145,&quot;3&quot;:{&quot;1&quot;:0},&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Recognize the difference between an implicit function and an explicit function<\/span><\/li>\n<li>Find the derivative of a complicated function by using implicit differentiation<\/li>\n<li>Use implicit differentiation to determine the equation of a tangent line<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1169737797432\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">What is Implicit Differentiation?<\/span><\/p>\n<div id=\"fs-id1169737702731\" class=\"bc-section section\">\n<p id=\"fs-id1169737919331\">In most discussions of math, if the dependent variable [latex]y[\/latex] is a function of the independent variable [latex]x[\/latex], we express [latex]y[\/latex] in terms of [latex]x[\/latex]. If this is the case, we say that [latex]y[\/latex] is an <em>explicit function<\/em> of [latex]x[\/latex]. For example, when we write the equation [latex]y=x^2+1[\/latex], we are defining [latex]y[\/latex] <em>explicitly<\/em> in terms of [latex]x[\/latex]. On the other hand, if the relationship between the function [latex]y[\/latex] and the variable [latex]x[\/latex] is expressed by an equation where [latex]y[\/latex] is not expressed entirely in terms of [latex]x[\/latex], we say that the equation defines [latex]y[\/latex] <em>implicitly<\/em> in terms of [latex]x[\/latex]. For example, the equation [latex]y-x^2=1[\/latex] defines the function [latex]y=x^2+1[\/latex] implicitly.<\/p>\n<p id=\"fs-id1169737766039\"><strong>Implicit differentiation<\/strong> allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of [latex]y[\/latex] are functions that satisfy the given equation, but that [latex]y[\/latex] is not actually a function of [latex]x[\/latex].<\/p>\n<p id=\"fs-id1169738018789\">In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly. For example, the functions<\/p>\n<p id=\"fs-id1169737771295\" style=\"text-align: center;\">[latex]y=\\sqrt{25-x^2}[\/latex],\u00a0 [latex]y = -\\sqrt{25-x^2}[\/latex], and [latex]y=\\begin{cases} \\sqrt{25-x^2} & \\text{ if } \\, -5 \\le x < 0 \\\\ -\\sqrt{25-x^2} & \\text{ if } \\, 0 \\le x \\le 5 \\end{cases}[\/latex],<\/p>\n<p>which are illustrated in Figure 1, are just three of the many functions defined implicitly by the equation [latex]x^2+y^2=25[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div style=\"width: 885px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205502\/CNX_Calc_Figure_03_08_001.jpg\" alt=\"The circle with radius 5 and center at the origin is graphed fully in one picture. Then, only its segments in quadrants I and II are graphed. Then, only its segments in quadrants III and IV are graphed. Lastly, only its segments in quadrants II and IV are graphed.\" width=\"875\" height=\"988\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The equation [latex]{x}^{2}+{y}^{2}=25[\/latex] defines many functions implicitly.<\/p>\n<\/div>\n<p id=\"fs-id1169737773822\">If we want to find the slope of the line tangent to the graph of [latex]x^2+y^2=25[\/latex] at the point [latex](3,4)[\/latex], we could evaluate the derivative of the function [latex]y=\\sqrt{25-x^2}[\/latex] at [latex]x=3[\/latex]. On the other hand, if we want the slope of the tangent line at the point [latex](3,-4)[\/latex], we could use the derivative of [latex]y=\u2212\\sqrt{25-x^2}[\/latex]. However, it is not always easy to solve for a function defined implicitly by an equation. Fortunately, the technique of implicit differentiation allows us to find the derivative of an implicitly defined function without ever solving for the function explicitly. The process of finding [latex]\\frac{dy}{dx}[\/latex] using implicit differentiation is described in the following problem-solving strategy.<\/p>\n<div id=\"fs-id1169737850132\" class=\"textbox examples\">\n<h3>Problem-Solving Strategy: Implicit Differentiation<\/h3>\n<p id=\"fs-id1169737749590\">To perform implicit differentiation on an equation that defines a function [latex]y[\/latex] implicitly in terms of a variable [latex]x[\/latex], use the following steps:<\/p>\n<ol id=\"fs-id1169737815995\">\n<li>Take the derivative of both sides of the equation. Keep in mind that [latex]y[\/latex] is a function of [latex]x[\/latex]. Consequently, whereas [latex]\\frac{d}{dx}(\\sin x)= \\cos x, \\, \\frac{d}{dx}(\\sin y)= \\cos y\\frac{dy}{dx}[\/latex] because we must use the Chain Rule to differentiate [latex]\\sin y[\/latex] with respect to [latex]x[\/latex].<\/li>\n<li>Rewrite the equation so that all terms containing [latex]\\frac{dy}{dx}[\/latex] are on the left and all terms that do not contain [latex]\\frac{dy}{dx}[\/latex] are on the right.<\/li>\n<li>Factor out [latex]\\frac{dy}{dx}[\/latex] on the left.<\/li>\n<li>Solve for [latex]\\frac{dy}{dx}[\/latex] by dividing both sides of the equation by an appropriate algebraic expression.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1169737819953\" class=\"textbook exercises\">\n<h3>Example: Using Implicit Differentiation<\/h3>\n<p id=\"fs-id1169737931738\">Assuming that [latex]y[\/latex] is defined implicitly by the equation [latex]x^2+y^2=25[\/latex], find [latex]\\frac{dy}{dx}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737948455\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737948455\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737948455\">Follow the steps in the problem-solving strategy.<\/p>\n<div id=\"fs-id1169737840968\" class=\"equation unnumbered\">[latex]\\begin{array}{llll} \\frac{d}{dx}(x^2+y^2) = \\frac{d}{dx}(25) & & & \\text{Step 1. Differentiate both sides of the equation.} \\\\ \\frac{d}{dx}(x^2)+\\frac{d}{dx}(y^2) = 0 & & & \\begin{array}{l}\\text{Step 1.1. Use the sum rule on the left.} \\\\ \\text{On the right,} \\, \\frac{d}{dx}(25)=0. \\end{array} \\\\ 2x+2y\\frac{dy}{dx} = 0 & & & \\begin{array}{l}\\text{Step 1.2. Take the derivatives, so} \\, \\frac{d}{dx}(x^2)=2x \\\\ \\text{and} \\, \\frac{d}{dx}(y^2)=2y\\frac{dy}{dx}. \\end{array} \\\\ 2y\\frac{dy}{dx} = -2x & & & \\begin{array}{l}\\text{Step 2. Keep the terms with} \\, \\frac{dy}{dx} \\, \\text{on the left.} \\\\ \\text{Move the remaining terms to the right.} \\end{array} \\\\ \\frac{dy}{dx} = -\\frac{x}{y} & & & \\begin{array}{l}\\text{Step 4. Divide both sides of the equation by} \\\\ 2y. \\, \\text{(Step 3 does not apply in this case.)} \\end{array} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<h4>Analysis<\/h4>\n<p id=\"fs-id1169737758515\">Note that the resulting expression for [latex]\\frac{dy}{dx}[\/latex] is in terms of both the independent variable [latex]x[\/latex] and the dependent variable [latex]y[\/latex]. Although in some cases it may be possible to express [latex]\\frac{dy}{dx}[\/latex] in terms of [latex]x[\/latex] only, it is generally not possible to do so.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738068274\" class=\"textbook exercises\">\n<h3>Example: Using Implicit Differentiation and the Product Rule<\/h3>\n<p id=\"fs-id1169737948472\">Assuming that [latex]y[\/latex] is defined implicitly by the equation [latex]x^3 \\sin y+y=4x+3[\/latex], find [latex]\\frac{dy}{dx}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738041617\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738041617\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169737814728\" class=\"equation unnumbered\">[latex]\\begin{array}{llll}\\frac{d}{dx}(x^3 \\sin y+y) = \\frac{d}{dx}(4x+3) & & & \\text{Step 1: Differentiate both sides of the equation.} \\\\ \\frac{d}{dx}(x^3 \\sin y)+\\frac{d}{dx}(y) = 4 & & & \\begin{array}{l}\\text{Step 1.1: Apply the sum rule on the left.} \\\\ \\text{On the right,} \\, \\frac{d}{dx}(4x+3)=4. \\end{array} \\\\ (\\frac{d}{dx}(x^3) \\cdot \\sin y+\\frac{d}{dx}(\\sin y) \\cdot x^3) + \\frac{dy}{dx} = 4 & & & \\begin{array}{l}\\text{Step 1.2: Use the product rule to find} \\\\ \\frac{d}{dx}(x^3 \\sin y). \\, \\text{Observe that} \\, \\frac{d}{dx}(y)=\\frac{dy}{dx}. \\end{array} \\\\ 3x^2 \\sin y+(\\cos y\\frac{dy}{dx}) \\cdot x^3 + \\frac{dy}{dx} = 4 & & & \\begin{array}{l}\\text{Step 1.3: We know} \\, \\frac{d}{dx}(x^3)=3x^2. \\, \\text{Use the} \\\\ \\text{chain rule to obtain} \\, \\frac{d}{dx}(\\sin y)= \\cos y\\frac{dy}{dx}. \\end{array} \\\\ x^3 \\cos y\\frac{dy}{dx}+\\frac{dy}{dx} = 4-3x^2 \\sin y & & & \\begin{array}{l}\\text{Step 2: Keep all terms containing} \\, \\frac{dy}{dx} \\, \\text{on the} \\\\ \\text{left. Move all other terms to the right.} \\end{array} \\\\ \\frac{dy}{dx}(x^3 \\cos y+1) = 4-3x^2 \\sin y & & & \\text{Step 3: Factor out} \\, \\frac{dy}{dx} \\, \\text{on the left.} \\\\ \\frac{dy}{dx} = \\large \\frac{4-3x^2 \\sin y}{x^3 \\cos y+1} & & & \\begin{array}{l}\\text{Step 4: Solve for} \\, \\frac{dy}{dx} \\, \\text{by dividing both sides of} \\\\ \\text{the equation by} \\, x^3 \\cos y+1. \\end{array} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738217476\" class=\"textbook exercises\">\n<h3>Example: Using Implicit Differentiation to Find a Second Derivative<\/h3>\n<p id=\"fs-id1169738211084\">Find [latex]\\frac{d^2 y}{dx^2}[\/latex] if [latex]x^2+y^2=25[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738226743\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738226743\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738226743\">In the first example, we showed that [latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex]. We can take the derivative of both sides of this equation to find [latex]\\frac{d^2 y}{dx^2}[\/latex].<\/p>\n<div id=\"fs-id1169737766542\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll} \\frac{d^2 y}{dx^2} & =\\large \\frac{d}{dx}(-\\frac{x}{y}) & & & \\text{Differentiate both sides of} \\, \\frac{dy}{dx}=-\\frac{x}{y}. \\\\ & = \\large -\\frac{(1 \\cdot y-x\\frac{dy}{dx})}{y^2} & & & \\text{Use the quotient rule to find} \\, \\frac{d}{dx}(-\\frac{x}{y}). \\\\ & = \\large \\frac{\u2212y+x\\frac{dy}{dx}}{y^2} & & & \\text{Simplify.} \\\\ & = \\large \\frac{\u2212y+x(-\\frac{x}{y})}{y^2} & & & \\text{Substitute} \\, \\frac{dy}{dx}=-\\frac{x}{y}. \\\\ & = \\large \\frac{\u2212y^2-x^2}{y^3} & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738044967\">At this point we have found an expression for [latex]\\frac{d^2 y}{dx^2}[\/latex]. If we choose, we can simplify the expression further by recalling that [latex]x^2+y^2=25[\/latex] and making this substitution in the numerator to obtain [latex]\\frac{d^2 y}{dx^2}=-\\frac{25}{y^3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Using Implicit Differentiation to Find a Second Derivative.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F1FFAg-XhlQ?controls=0&amp;start=385&amp;end=473&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.8ImplicitDifferentiation385to473_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.8 Implicit Differentiation&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169737772805\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8993550\">Hint<\/span><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737953797\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm206109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=206109&theme=oea&iframe_resize_id=ohm206109&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding Tangent Lines Implicitly<\/h2>\n<p id=\"fs-id1169737931596\">Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.<\/p>\n<div id=\"fs-id1169737931601\" class=\"textbook exercises\">\n<h3>Example: Finding a Tangent Line to a Circle<\/h3>\n<p id=\"fs-id1169737950789\">Find the equation of the line tangent to the curve [latex]x^2+y^2=25[\/latex] at the point [latex](3,-4)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737143580\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737143580\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737143580\">Although we could find this equation without using implicit differentiation, using that method makes it much easier. In an earlier example, we found [latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex].<\/p>\n<p id=\"fs-id1169737144361\">The slope of the tangent line is found by substituting [latex](3,-4)[\/latex] into this expression. Consequently, the slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,-4)} =-\\frac{3}{-4}=\\frac{3}{4}[\/latex].<\/p>\n<p id=\"fs-id1169738198658\">Using the point [latex](3,-4)[\/latex] and the slope [latex]\\frac{3}{4}[\/latex] in the point-slope equation of the line, we then solve for [latex]y[\/latex] to obtain the equation [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] (Figure 2).<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205505\/CNX_Calc_Figure_03_08_002.jpg\" alt=\"The circle with radius 5 and center at the origin is graphed. A tangent line is drawn through the point (3, \u22124).\" width=\"487\" height=\"358\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The line [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] is tangent to [latex]x^2+y^2=25[\/latex] at the point [latex](3,\u22124)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook exercises\">\n<h3>Example: Finding the Equation of the Tangent Line to a Curve<\/h3>\n<p id=\"fs-id1169737143587\">Find the equation of the line tangent to the graph of [latex]y^3+x^3-3xy=0[\/latex] at the point [latex]\\left(\\frac{3}{2},\\frac{3}{2}\\right)[\/latex] (Figure 3). This curve is known as the folium (or leaf) of Descartes.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205509\/CNX_Calc_Figure_03_08_003.jpg\" alt=\"A folium is shown, which is a line that creates a loop that crosses over itself. In this graph, it crosses over itself at (0, 0). Its tangent line from (3\/2, 3\/2) is shown.\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Finding the tangent line to the folium of Descartes at [latex](\\frac{3}{2},\\frac{3}{2})[\/latex].<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738217007\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738217007\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738217007\">Begin by finding [latex]\\frac{dy}{dx}.[\/latex]<\/p>\n<div id=\"fs-id1169738185005\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{d}{dx}(y^3+x^3-3xy) = \\frac{d}{dx}(0) \\\\ \\\\ 3y^2\\frac{dy}{dx}+3x^2-(3y+3x\\frac{dy}{dx}) = 0 \\\\ \\\\ \\frac{dy}{dx} = \\frac{3y-3x^2}{3y^2-3x}. \\\\ \\\\ \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169737904704\">Next, substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex] to find the slope of the tangent line:<\/p>\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}|_{(\\frac{3}{2},\\frac{3}{2})}=-1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169737950853\">Finally, substitute into the point-slope equation of the line and solve for [latex]y[\/latex] to obtain<\/p>\n<div id=\"fs-id1169737950856\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\u2212x+3[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737935216\" class=\"textbook exercises\">\n<h3>Example: Applying Implicit Differentiation<\/h3>\n<p id=\"fs-id1169737935226\">In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation [latex]4x^2+25y^2=100[\/latex]. The rocket can fire missiles along lines tangent to its path. The object of the game is to destroy an incoming asteroid traveling along the positive [latex]x[\/latex]-axis toward [latex](0,0)[\/latex]. If the rocket fires a missile when it is located at [latex]\\left(3,\\frac{8}{5}\\right)[\/latex], where will it intersect the [latex]x[\/latex]-axis?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738099354\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738099354\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738099354\">To solve this problem, we must determine where the line tangent to the graph of<\/p>\n<p id=\"fs-id1169738099357\">[latex]4x^2+25y^2=100[\/latex] at [latex](3,\\frac{8}{5})[\/latex] intersects the [latex]x[\/latex]-axis. Begin by finding [latex]\\frac{dy}{dx}[\/latex] implicitly.<\/p>\n<p id=\"fs-id1169738214601\">Differentiating, we have<\/p>\n<div id=\"fs-id1169737144258\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]8x+50y\\frac{dy}{dx}=0[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1169737144293\">Solving for [latex]\\frac{dy}{dx}[\/latex], we have<\/p>\n<div id=\"fs-id1169738223546\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{dy}{dx}=-\\dfrac{4x}{25y}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1169738219286\">The slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,\\frac{8}{5})}=-\\frac{3}{10}[\/latex]. The equation of the tangent line is [latex]y=-\\frac{3}{10}x+\\frac{5}{2}[\/latex]. To determine where the line intersects the [latex]x[\/latex]-axis, solve [latex]0=-\\frac{3}{10}x+\\frac{5}{2}[\/latex]. The solution is [latex]x=\\frac{25}{3}[\/latex]. The missile intersects the [latex]x[\/latex]-axis at the point [latex](\\frac{25}{3},0)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding the Equation of the Tangent Line to a Curve.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F1FFAg-XhlQ?controls=0&amp;start=553&amp;end=718&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.8ImplicitDifferentiation535to718_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.8 Implicit Differentiation&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169738186785\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8005423\">Hint<\/span><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737145207\">Show Solution<\/span><\/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<\/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-380\">\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>3.8 Implicit Differentiation. <strong>Authored by<\/strong>: Ryan Melton. <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>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":34,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.8 Implicit Differentiation\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-380","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/380\/revisions"}],"predecessor-version":[{"id":4818,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/380\/revisions\/4818"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/380\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=380"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=380"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=380"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}