{"id":1869,"date":"2018-01-11T20:55:26","date_gmt":"2018-01-11T20:55:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/implicit-differentiation\/"},"modified":"2018-01-31T20:48:33","modified_gmt":"2018-01-31T20:48:33","slug":"implicit-differentiation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/implicit-differentiation\/","title":{"raw":"3.8 Implicit Differentiation","rendered":"3.8 Implicit Differentiation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\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\">We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define [latex]y[\/latex] implicitly in terms of [latex]x.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169737702731\" class=\"bc-section section\">\r\n<h1>Implicit Differentiation<\/h1>\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 explicit function of [latex]x.[\/latex] For example, when we write the equation [latex]y={x}^{2}+1,[\/latex] we are defining [latex]y[\/latex] explicitly 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] implicitly 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\">[latex]y=\\sqrt{25-{x}^{2}}[\/latex] and [latex]y=\\bigg\\{\\begin{array}{c}\\sqrt{25-{x}^{2}}\\text{ if }-25\\le x&lt;0\\\\ \\text{\u2212}\\sqrt{25-{x}^{2}}\\text{ if }0\\le x\\le 25\\end{array},[\/latex] which are illustrated in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_001\">(Figure)<\/a>, are just three of the many functions defined implicitly by the equation [latex]{x}^{2}+{y}^{2}=25.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_03_08_001\" class=\"wp-caption aligncenter\">[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\" \/> <strong>Figure 1.<\/strong> The equation [latex]{x}^{2}+{y}^{2}=25[\/latex] defines many functions implicitly.[\/caption]<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\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=\\text{\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 key-takeaways problem-solving\">\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=\"textbox examples\">\r\n<h3>Using Implicit Differentiation<\/h3>\r\n<div id=\"fs-id1169738015312\" class=\"exercise\">\r\n<div id=\"fs-id1169738018639\" class=\"textbox\">\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\r\n<\/div>\r\n<div class=\"textbox shaded\">[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}{cccccc}\\hfill \\frac{d}{dx}({x}^{2}+{y}^{2})&amp; =\\hfill &amp; \\frac{d}{dx}(25)\\hfill &amp; &amp; &amp; \\text{Step 1. Differentiate both sides of the equation.}\\hfill \\\\ \\hfill \\frac{d}{dx}({x}^{2})+\\frac{d}{dx}({y}^{2})&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 1.1. Use the sum rule on the left.}\\hfill \\\\ \\text{On the right}\\frac{d}{dx}(25)=0.\\hfill \\end{array}\\hfill \\\\ \\hfill 2x+2y\\frac{dy}{dx}&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 1.2. Take the derivatives, so}\\frac{d}{dx}({x}^{2})=2x\\hfill \\\\ \\text{ and }\\frac{d}{dx}({y}^{2})=2y\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill 2y\\frac{dy}{dx}&amp; =\\hfill &amp; -2x\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 2. Keep the terms with}\\frac{dy}{dx}\\text{on the left.}\\hfill \\\\ \\text{Move the remaining terms to the right.}\\hfill \\end{array}\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; -\\frac{x}{y}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 4. Divide both sides of the equation by}\\hfill \\\\ 2y.\\text{(Step 3 does not apply in this case.)}\\hfill \\end{array}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738146824\" class=\"commentary\">\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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738068274\" class=\"textbox examples\">\r\n<h3>Using Implicit Differentiation and the Product Rule<\/h3>\r\n<div id=\"fs-id1169738149878\" class=\"exercise\">\r\n<div id=\"fs-id1169737946720\" class=\"textbox\">\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\r\n<\/div>\r\n<div id=\"fs-id1169738041617\" class=\"textbox shaded\">[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}{cccccc}\\hfill \\frac{d}{dx}({x}^{3} \\sin y+y)&amp; =\\hfill &amp; \\frac{d}{dx}(4x+3)\\hfill &amp; &amp; &amp; \\text{Step 1: Differentiate both sides of the equation.}\\hfill \\\\ \\hfill \\frac{d}{dx}({x}^{3} \\sin y)+\\frac{d}{dx}(y)&amp; =\\hfill &amp; 4\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 1.1: Apply the sum rule on the left.}\\hfill \\\\ \\text{On the right,}\\frac{d}{dx}(4x+3)=4.\\hfill \\end{array}\\hfill \\\\ \\hfill (\\frac{d}{dx}({x}^{3})\u00b7 \\sin y+\\frac{d}{dx}( \\sin y)\u00b7{x}^{3})+\\frac{dy}{dx}&amp; =\\hfill &amp; 4\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 1.2: Use the product rule to find}\\hfill \\\\ \\frac{d}{dx}({x}^{3} \\sin y).\\text{Observe that}\\frac{d}{dx}(y)=\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill 3{x}^{2} \\sin y+( \\cos y\\frac{dy}{dx})\u00b7{x}^{3}+\\frac{dy}{dx}&amp; =\\hfill &amp; 4\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 1.3: We know}\\frac{d}{dx}({x}^{3})=3{x}^{2}.\\text{Use the}\\hfill \\\\ \\text{chain rule to obtain}\\frac{d}{dx}( \\sin y)= \\cos y\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill {\\text{x}}^{3} \\cos y\\frac{dy}{dx}+\\frac{dy}{dx}&amp; =\\hfill &amp; 4-3{x}^{2} \\sin y\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 2: Keep all terms containing}\\frac{dy}{dx}\\text{on the}\\hfill \\\\ \\text{left. Move all other terms to the right.}\\hfill \\end{array}\\hfill \\\\ \\hfill \\frac{dy}{dx}({\\text{x}}^{3} \\cos y+1)&amp; =\\hfill &amp; 4-3{x}^{2} \\sin y\\hfill &amp; &amp; &amp; \\text{Step 3: Factor out}\\frac{dy}{dx}\\text{on the left.}\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; \\frac{4-3{x}^{2} \\sin y}{{x}^{3} \\cos y+1}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Step 4: Solve for}\\frac{dy}{dx}\\text{by dividing both sides of}\\hfill \\\\ \\text{the equation by}{\\text{x}}^{3} \\cos y+1.\\hfill \\end{array}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738217476\" class=\"textbox examples\">\r\n<h3>Using Implicit Differentiation to Find a Second Derivative<\/h3>\r\n<div id=\"fs-id1169738185350\" class=\"exercise\">\r\n<div id=\"fs-id1169738185353\" class=\"textbox\">\r\n<p id=\"fs-id1169738211084\">Find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] if [latex]{x}^{2}+{y}^{2}=25.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738226743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226743\"]\r\n<p id=\"fs-id1169738226743\">In <a class=\"autogenerated-content\" href=\"#fs-id1169737819953\">(Figure)<\/a>, 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}{d{x}^{2}}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169737766542\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{{d}^{2}y}{d{x}^{2}}&amp; =\\frac{d}{dy}(-\\frac{x}{y})\\hfill &amp; &amp; &amp; \\text{Differentiate both sides of}\\frac{dy}{dx}=-\\frac{x}{y}.\\hfill \\\\ &amp; =-\\frac{(1\u00b7y-x\\frac{dy}{dx})}{{y}^{2}}\\hfill &amp; &amp; &amp; \\text{Use the quotient rule to find}\\frac{d}{dy}(-\\frac{x}{y}).\\hfill \\\\ &amp; =\\frac{\\text{\u2212}y+x\\frac{dy}{dx}}{{y}^{2}}\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\\\ &amp; =\\frac{\\text{\u2212}y+x(-\\frac{x}{y})}{{y}^{2}}\\hfill &amp; &amp; &amp; \\text{Substitute}\\frac{dy}{dx}=-\\frac{x}{y}.\\hfill \\\\ &amp; =\\frac{\\text{\u2212}{y}^{2}-{x}^{2}}{{y}^{3}}\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169738044967\">At this point we have found an expression for [latex]\\frac{{d}^{2}y}{d{x}^{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}{d{x}^{2}}=-\\frac{25}{{y}^{3}}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737772805\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169738220221\" class=\"exercise\">\r\n<div id=\"fs-id1169738220223\" class=\"textbox\">\r\n<p id=\"fs-id1169738220225\">Find [latex]\\frac{dy}{dx}[\/latex] for [latex]y[\/latex] defined implicitly by the equation [latex]4{x}^{5}+ \\tan y={y}^{2}+5x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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}=\\frac{5-20{x}^{4}}{{ \\sec }^{2}y-2y}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738222203\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169738221942\">Follow the problem solving strategy, remembering to apply the chain rule to differentiate [latex] \\tan [\/latex] and [latex]{y}^{2}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737140783\" class=\"bc-section section\">\r\n<h1>Finding Tangent Lines Implicitly<\/h1>\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=\"textbox examples\">\r\n<h3>Finding a Tangent Line to a Circle<\/h3>\r\n<div id=\"fs-id1169737906658\" class=\"exercise\">\r\n<div id=\"fs-id1169737906660\" class=\"textbox\">\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\r\n<\/div>\r\n<div class=\"textbox shaded\">[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 <a class=\"autogenerated-content\" href=\"#fs-id1169737819953\">(Figure)<\/a>, 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}|\\begin{array}{l}\\\\ {}_{(3,-4)}\\end{array}=-\\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 obtain the equation [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_002\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_03_08_002\" class=\"wp-caption aligncenter\">[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\" \/> <strong> Figure 2.<\/strong> 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 (3, \u22124). [\/hidden-answer][\/caption]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1169737142258\" class=\"exercise\">\r\n<div id=\"fs-id1169737142260\" class=\"textbox\">\r\n<h3>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](\\frac{3}{2},\\frac{3}{2})[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_003\">(Figure)<\/a>). This curve is known as the folium (or leaf) of Descartes.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_03_08_003\" class=\"wp-caption aligncenter\">[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\" \/> <strong>Figure 3.<\/strong> Finding the tangent line to the folium of Descartes at [latex](\\frac{3}{2},\\frac{3}{2}).[\/latex][\/caption]<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]\\begin{array}{ccc}\\hfill \\frac{d}{dx}({y}^{3}+{x}^{3}-3xy)&amp; =\\hfill &amp; \\frac{d}{dx}(0)\\hfill \\\\ \\hfill 3{y}^{2}\\frac{dy}{dx}+3{x}^{2}-(3y+\\frac{dy}{dx}3x)&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; \\frac{3y-3{x}^{2}}{3{y}^{2}-3x}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169737904704\">Next, substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3{x}^{2}}{3{y}^{2}-3x}[\/latex] to find the slope of the tangent line:<\/p>\r\n\r\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}|\\begin{array}{l}\\\\ {}_{(\\frac{3}{2},\\frac{3}{2})}\\end{array}=-1.[\/latex]<\/div>\r\n<p id=\"fs-id1169737950853\">Finally, substitute into the point-slope equation of the line to obtain<\/p>\r\n\r\n<div id=\"fs-id1169737950856\" class=\"equation unnumbered\">[latex]y=\\text{\u2212}x+3.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737935216\" class=\"textbox examples\">\r\n<h3>Applying Implicit Differentiation<\/h3>\r\n<div id=\"fs-id1169737935219\" class=\"exercise\">\r\n<div id=\"fs-id1169737935221\" class=\"textbox\">\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]4{x}^{2}+25{y}^{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](3,\\frac{8}{3}),[\/latex] where will it intersect the [latex]x[\/latex]-axis?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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]4{x}^{2}+25{y}^{2}=100[\/latex] at [latex](3,\\frac{8}{3})[\/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\">[latex]8x+50y\\frac{dy}{dx}=0.[\/latex]<\/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\">[latex]\\frac{dy}{dx}=-\\frac{4x}{25y}.[\/latex]<\/div>\r\n<p id=\"fs-id1169738219286\">The slope of the tangent line is [latex]\\frac{dy}{dx}|{}_{(3,\\frac{8}{3})}=-\\frac{9}{50}.[\/latex] The equation of the tangent line is [latex]y=-\\frac{9}{50}x+\\frac{183}{200}.[\/latex] To determine where the line intersects the [latex]x[\/latex]-axis, solve [latex]0=-\\frac{9}{50}x+\\frac{183}{200}.[\/latex] The solution is [latex]x=\\frac{61}{3}.[\/latex] The missile intersects the [latex]x[\/latex]-axis at the point [latex](\\frac{61}{3},0).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738186785\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169738186788\" class=\"exercise\">\r\n<div id=\"fs-id1169738186791\" class=\"textbox\">\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\r\n<\/div>\r\n<div id=\"fs-id1169737145207\" class=\"textbox shaded\">[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]<\/div>\r\n<div id=\"fs-id1169737145234\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169738240184\">[latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738240212\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1169738240218\">\r\n \t<li>We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).<\/li>\r\n \t<li>By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169738240232\" class=\"textbox exercises\">\r\n<p id=\"fs-id1169738240235\">For the following exercises, use implicit differentiation to find [latex]\\frac{dy}{dx}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169738212639\" class=\"exercise\">\r\n<div id=\"fs-id1169738212641\" class=\"textbox\">\r\n<p id=\"fs-id1169738212643\">[latex]{x}^{2}-{y}^{2}=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738106195\" class=\"exercise\">\r\n<div id=\"fs-id1169738106197\" class=\"textbox\">\r\n<p id=\"fs-id1169738106200\">[latex]6{x}^{2}+3{y}^{2}=12[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738106228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738106228\"]\r\n<p id=\"fs-id1169738106228\">[latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738106257\" class=\"exercise\">\r\n<div id=\"fs-id1169738106259\" class=\"textbox\">\r\n<p id=\"fs-id1169738106261\">[latex]{x}^{2}y=y-7[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738073286\" class=\"exercise\">\r\n<div id=\"fs-id1169738073289\" class=\"textbox\">\r\n<p id=\"fs-id1169738073291\">[latex]3{x}^{3}+9x{y}^{2}=5{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738223311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738223311\"]\r\n<p id=\"fs-id1169738223311\">[latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738223350\" class=\"exercise\">\r\n<div id=\"fs-id1169738223352\" class=\"textbox\">\r\n<p id=\"fs-id1169738223354\">[latex]xy- \\cos (xy)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738237485\" class=\"exercise\">\r\n<div id=\"fs-id1169738237487\" class=\"textbox\">\r\n<p id=\"fs-id1169738237489\">[latex]y\\sqrt{x+4}=xy+8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738237518\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738237518\"]\r\n<p id=\"fs-id1169738237518\">[latex]\\frac{dy}{dx}=\\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738073148\" class=\"exercise\">\r\n<div id=\"fs-id1169738073151\" class=\"textbox\">\r\n<p id=\"fs-id1169738073153\">[latex]\\text{\u2212}xy-2=\\frac{x}{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737911358\" class=\"exercise\">\r\n<div id=\"fs-id1169737911360\" class=\"textbox\">\r\n<p id=\"fs-id1169737911362\">[latex]y \\sin (xy)={y}^{2}+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169737911399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737911399\"]\r\n<p id=\"fs-id1169737911399\">[latex]\\frac{dy}{dx}=\\frac{{y}^{2} \\cos (xy)}{2y- \\sin (xy)-xy \\cos xy}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738212451\" class=\"exercise\">\r\n<div id=\"fs-id1169738212454\" class=\"textbox\">\r\n<p id=\"fs-id1169738212456\">[latex]{(xy)}^{2}+3x={y}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738187223\" class=\"exercise\">\r\n<div id=\"fs-id1169738187226\" class=\"textbox\">\r\n<p id=\"fs-id1169738187228\">[latex]{x}^{3}y+x{y}^{3}=-8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]\\frac{dy}{dx}=\\frac{-3{x}^{2}y-{y}^{3}}{{x}^{3}+3x{y}^{2}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169738184790\">For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.<\/p>\r\n\r\n<div id=\"fs-id1169738184795\" class=\"exercise\">\r\n<div id=\"fs-id1169738184797\" class=\"textbox\">\r\n<p id=\"fs-id1169738184799\"><strong>[T]<\/strong>[latex]{x}^{4}y-x{y}^{3}=-2,(-1,-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738184892\" class=\"exercise\">\r\n<div id=\"fs-id1169738184894\" class=\"textbox\">\r\n<p id=\"fs-id1169738184896\"><strong>[T]<\/strong>[latex]{x}^{2}{y}^{2}+5xy=14,(2,1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738226137\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738226137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226137\"]<span id=\"fs-id1169738226145\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205513\/CNX_Calc_Figure_03_08_202.jpg\" alt=\"The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope \u22121\/2 and y intercept 2.\" \/><\/span>\r\n[latex]y=\\frac{-1}{2}x+2[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738226177\" class=\"exercise\">\r\n<div id=\"fs-id1169738226179\" class=\"textbox\">\r\n<p id=\"fs-id1169738226181\"><strong>[T]<\/strong>[latex] \\tan (xy)=y,(\\frac{\\pi }{4},1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738238122\" class=\"exercise\">\r\n<div id=\"fs-id1169738238124\" class=\"textbox\">\r\n<p id=\"fs-id1169738238126\"><strong>[T]<\/strong>[latex]x{y}^{2}+ \\sin (\\pi y)-2{x}^{2}=10,(2,-3)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738234517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738234517\"]<span id=\"fs-id1169738234524\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205517\/CNX_Calc_Figure_03_08_204.jpg\" alt=\"The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1\/(\u03c0 + 12) and y intercept \u2212(3\u03c0 + 38)\/(\u03c0 + 12).\" \/><\/span>\r\n[latex]y=\\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738234576\" class=\"exercise\">\r\n<div id=\"fs-id1169738234578\" class=\"textbox\">\r\n<p id=\"fs-id1169738234580\"><strong>[T]<\/strong>[latex]\\frac{x}{y}+5x-7=-\\frac{3}{4}y,(1,2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738221157\" class=\"exercise\">\r\n<div id=\"fs-id1169738221159\" class=\"textbox\">\r\n<p id=\"fs-id1169738221161\"><strong>[T]<\/strong>[latex]xy+ \\sin (x)=1,(\\frac{\\pi }{2},0)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738221212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738221212\"]<span id=\"fs-id1169738221218\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205521\/CNX_Calc_Figure_03_08_206.jpg\" alt=\"The graph starts in the third quadrant near (\u22125, 0), remains near 0 until x = \u22124, at which point it decreases until it reaches near (0, \u22125). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0.\" \/><\/span>\r\n[latex]y=0[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n<strong>[T]<\/strong> The graph of a folium of Descartes with equation [latex]2{x}^{3}+2{y}^{3}-9xy=0[\/latex] is given in the following graph.\r\n\r\n<span id=\"fs-id1169737145046\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205525\/CNX_Calc_Figure_03_08_207.jpg\" alt=\"A folium is graphed which has equation 2x3 + 2y3 \u2013 9xy = 0. It crosses over itself at (0, 0).\" \/><\/span>\r\n<ol id=\"fs-id1169737145054\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the equation of the tangent line at the point [latex](2,1).[\/latex] Graph the tangent line along with the folium.<\/li>\r\n \t<li>Find the equation of the normal line to the tangent line in a. at the point [latex](2,1).[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738244350\" class=\"exercise\">\r\n<div id=\"fs-id1169738244352\" class=\"textbox\">\r\n<p id=\"fs-id1169738244354\">For the equation [latex]{x}^{2}+2xy-3{y}^{2}=0,[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1169738244389\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the equation of the normal to the tangent line at the point [latex](1,1).[\/latex]<\/li>\r\n \t<li>At what other point does the normal line in a. intersect the graph of the equation?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738244421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738244421\"]\r\n<p id=\"fs-id1169738244421\">a. [latex]y=\\text{\u2212}x+2[\/latex] b. [latex](3,-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738244457\" class=\"exercise\">\r\n<div id=\"fs-id1169738244460\" class=\"textbox\">\r\n<p id=\"fs-id1169738244462\">Find all points on the graph of [latex]{y}^{3}-27y={x}^{2}-90[\/latex] at which the tangent line is vertical.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738211822\" class=\"exercise\">\r\n<div id=\"fs-id1169738211824\" class=\"textbox\">\r\n<p id=\"fs-id1169738211826\">For the equation [latex]{x}^{2}+xy+{y}^{2}=7,[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1169738211857\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the [latex]x[\/latex]-intercept(s).<\/li>\r\n \t<li>Find the slope of the tangent line(s) at the [latex]x[\/latex]-intercept(s).<\/li>\r\n \t<li>What does the value(s) in b. indicate about the tangent line(s)?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738211884\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738211884\"]\r\n<p id=\"fs-id1169738211884\">a. [latex](\u00b1\\sqrt{7},0)[\/latex] b. -2 c. They are parallel since the slope is the same at both intercepts.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738211914\" class=\"exercise\">\r\n<div id=\"fs-id1169738211916\" class=\"textbox\">\r\n<p id=\"fs-id1169738211918\">Find the equation of the tangent line to the graph of the equation [latex]{ \\sin }^{-1}x+{ \\sin }^{-1}y=\\frac{\\pi }{6}[\/latex] at the point [latex](0,\\frac{1}{2}).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738201886\" class=\"exercise\">\r\n<div id=\"fs-id1169738201888\" class=\"textbox\">\r\n<p id=\"fs-id1169738201890\">Find the equation of the tangent line to the graph of the equation [latex]{ \\tan }^{-1}(x+y)={x}^{2}+\\frac{\\pi }{4}[\/latex] at the point [latex](0,1).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738201952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738201952\"]\r\n<p id=\"fs-id1169738201952\">[latex]y=\\text{\u2212}x+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738186907\" class=\"exercise\">\r\n<div id=\"fs-id1169738186909\" class=\"textbox\">\r\n<p id=\"fs-id1169738186911\">Find [latex]{y}^{\\prime }[\/latex] and [latex]y\\text{\u2033}[\/latex] for [latex]{x}^{2}+6xy-2{y}^{2}=3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737934269\" class=\"exercise\">\r\n<div id=\"fs-id1169737934271\" class=\"textbox\">\r\n<p id=\"fs-id1169737934273\"><strong>[T]<\/strong> The number of cell phones produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]60{x}^{3\\text{\/}4}{y}^{1\\text{\/}4}=3240.[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1169737934320\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find [latex]\\frac{dy}{dx}[\/latex] and evaluate at the point [latex](81,16).[\/latex]<\/li>\r\n \t<li>Interpret the result of a.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738185028\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738185028\"]\r\n<p id=\"fs-id1169738185028\">a. -0.5926 b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738185040\" class=\"exercise\">\r\n<div id=\"fs-id1169738185042\" class=\"textbox\">\r\n<p id=\"fs-id1169738185044\"><strong>[T]<\/strong> The number of cars produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]30{x}^{1\\text{\/}3}{y}^{2\\text{\/}3}=360.[\/latex]<\/p>\r\n<p id=\"fs-id1169738185091\">(Both [latex]x[\/latex] and [latex]y[\/latex] are measured in thousands of dollars.)<\/p>\r\n\r\n<ol id=\"fs-id1169738185102\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find [latex]\\frac{dy}{dx}[\/latex] and evaluate at the point [latex](27,8).[\/latex]<\/li>\r\n \t<li>Interpret the result of a.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738185162\" class=\"exercise\">\r\n<div id=\"fs-id1169738185164\" class=\"textbox\">\r\n<p id=\"fs-id1169738185166\">The volume of a right circular cone of radius [latex]x[\/latex] and height [latex]y[\/latex] is given by [latex]V=\\frac{1}{3}\\pi {x}^{2}y.[\/latex] Suppose that the volume of the cone is [latex]85\\pi {\\text{cm}}^{3}.[\/latex] Find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=4[\/latex] and [latex]y=16.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169737954152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737954152\"]\r\n<p id=\"fs-id1169737954152\">-8<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169737954161\">For the following exercises, consider a closed rectangular box with a square base with side [latex]x[\/latex] and height [latex]y.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169737954174\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\nFind an equation for the surface area of the rectangular box, [latex]S(x,y).[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169737954242\" class=\"exercise\">\r\n<div id=\"fs-id1169737954244\" class=\"textbox\">\r\n<p id=\"fs-id1169737954246\">If the surface area of the rectangular box is 78 square feet, find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=3[\/latex] feet and [latex]y=5[\/latex] feet.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738071332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738071332\"]\r\n<p id=\"fs-id1169738071332\">-2.67<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169738071340\">For the following exercises, use implicit differentiation to determine [latex]{y}^{\\prime }.[\/latex] Does the answer agree with the formulas we have previously determined?<\/p>\r\n\r\n<div id=\"fs-id1169738071354\" class=\"exercise\">\r\n<div id=\"fs-id1169738071356\" class=\"textbox\">\r\n<p id=\"fs-id1169738071358\">[latex]x= \\sin y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738071406\" class=\"exercise\">\r\n<div id=\"fs-id1169738071409\" class=\"textbox\">\r\n<p id=\"fs-id1169738071411\">[latex]x= \\cos y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738071429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738071429\"]\r\n<p id=\"fs-id1169738071429\">[latex]{y}^{\\prime }=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738071461\" class=\"exercise\">\r\n<div id=\"fs-id1169738071463\" class=\"textbox\">\r\n<p id=\"fs-id1169738071465\">[latex]x= \\tan y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169738191096\" class=\"definition\">\r\n \t<dt>implicit differentiation<\/dt>\r\n \t<dd id=\"fs-id1169738191101\">is a technique for computing [latex]\\frac{dy}{dx}[\/latex] for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable [latex]y[\/latex] as a function) and solving for [latex]\\frac{dy}{dx}[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\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\">We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define [latex]y[\/latex] implicitly in terms of [latex]x.[\/latex]<\/p>\n<div id=\"fs-id1169737702731\" class=\"bc-section section\">\n<h1>Implicit Differentiation<\/h1>\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 explicit function of [latex]x.[\/latex] For example, when we write the equation [latex]y={x}^{2}+1,[\/latex] we are defining [latex]y[\/latex] explicitly 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] implicitly 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\">[latex]y=\\sqrt{25-{x}^{2}}[\/latex] and [latex]y=\\bigg\\{\\begin{array}{c}\\sqrt{25-{x}^{2}}\\text{ if }-25\\le x<0\\\\ \\text{\u2212}\\sqrt{25-{x}^{2}}\\text{ if }0\\le x\\le 25\\end{array},[\/latex] which are illustrated in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_001\">(Figure)<\/a>, are just three of the many functions defined implicitly by the equation [latex]{x}^{2}+{y}^{2}=25.[\/latex]<\/p>\n<div id=\"CNX_Calc_Figure_03_08_001\" class=\"wp-caption aligncenter\">\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\"><strong>Figure 1.<\/strong> The equation [latex]{x}^{2}+{y}^{2}=25[\/latex] defines many functions implicitly.<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-caption-text\"><\/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=\\text{\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 key-takeaways problem-solving\">\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=\"textbox examples\">\n<h3>Using Implicit Differentiation<\/h3>\n<div id=\"fs-id1169738015312\" class=\"exercise\">\n<div id=\"fs-id1169738018639\" class=\"textbox\">\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>\n<div class=\"textbox shaded\">\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}{cccccc}\\hfill \\frac{d}{dx}({x}^{2}+{y}^{2})& =\\hfill & \\frac{d}{dx}(25)\\hfill & & & \\text{Step 1. Differentiate both sides of the equation.}\\hfill \\\\ \\hfill \\frac{d}{dx}({x}^{2})+\\frac{d}{dx}({y}^{2})& =\\hfill & 0\\hfill & & & \\begin{array}{c}\\text{Step 1.1. Use the sum rule on the left.}\\hfill \\\\ \\text{On the right}\\frac{d}{dx}(25)=0.\\hfill \\end{array}\\hfill \\\\ \\hfill 2x+2y\\frac{dy}{dx}& =\\hfill & 0\\hfill & & & \\begin{array}{c}\\text{Step 1.2. Take the derivatives, so}\\frac{d}{dx}({x}^{2})=2x\\hfill \\\\ \\text{ and }\\frac{d}{dx}({y}^{2})=2y\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill 2y\\frac{dy}{dx}& =\\hfill & -2x\\hfill & & & \\begin{array}{c}\\text{Step 2. Keep the terms with}\\frac{dy}{dx}\\text{on the left.}\\hfill \\\\ \\text{Move the remaining terms to the right.}\\hfill \\end{array}\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & -\\frac{x}{y}\\hfill & & & \\begin{array}{c}\\text{Step 4. Divide both sides of the equation by}\\hfill \\\\ 2y.\\text{(Step 3 does not apply in this case.)}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738146824\" class=\"commentary\">\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 id=\"fs-id1169738068274\" class=\"textbox examples\">\n<h3>Using Implicit Differentiation and the Product Rule<\/h3>\n<div id=\"fs-id1169738149878\" class=\"exercise\">\n<div id=\"fs-id1169737946720\" class=\"textbox\">\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>\n<div id=\"fs-id1169738041617\" class=\"textbox shaded\">\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}{cccccc}\\hfill \\frac{d}{dx}({x}^{3} \\sin y+y)& =\\hfill & \\frac{d}{dx}(4x+3)\\hfill & & & \\text{Step 1: Differentiate both sides of the equation.}\\hfill \\\\ \\hfill \\frac{d}{dx}({x}^{3} \\sin y)+\\frac{d}{dx}(y)& =\\hfill & 4\\hfill & & & \\begin{array}{c}\\text{Step 1.1: Apply the sum rule on the left.}\\hfill \\\\ \\text{On the right,}\\frac{d}{dx}(4x+3)=4.\\hfill \\end{array}\\hfill \\\\ \\hfill (\\frac{d}{dx}({x}^{3})\u00b7 \\sin y+\\frac{d}{dx}( \\sin y)\u00b7{x}^{3})+\\frac{dy}{dx}& =\\hfill & 4\\hfill & & & \\begin{array}{c}\\text{Step 1.2: Use the product rule to find}\\hfill \\\\ \\frac{d}{dx}({x}^{3} \\sin y).\\text{Observe that}\\frac{d}{dx}(y)=\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill 3{x}^{2} \\sin y+( \\cos y\\frac{dy}{dx})\u00b7{x}^{3}+\\frac{dy}{dx}& =\\hfill & 4\\hfill & & & \\begin{array}{c}\\text{Step 1.3: We know}\\frac{d}{dx}({x}^{3})=3{x}^{2}.\\text{Use the}\\hfill \\\\ \\text{chain rule to obtain}\\frac{d}{dx}( \\sin y)= \\cos y\\frac{dy}{dx}.\\hfill \\end{array}\\hfill \\\\ \\hfill {\\text{x}}^{3} \\cos y\\frac{dy}{dx}+\\frac{dy}{dx}& =\\hfill & 4-3{x}^{2} \\sin y\\hfill & & & \\begin{array}{c}\\text{Step 2: Keep all terms containing}\\frac{dy}{dx}\\text{on the}\\hfill \\\\ \\text{left. Move all other terms to the right.}\\hfill \\end{array}\\hfill \\\\ \\hfill \\frac{dy}{dx}({\\text{x}}^{3} \\cos y+1)& =\\hfill & 4-3{x}^{2} \\sin y\\hfill & & & \\text{Step 3: Factor out}\\frac{dy}{dx}\\text{on the left.}\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & \\frac{4-3{x}^{2} \\sin y}{{x}^{3} \\cos y+1}\\hfill & & & \\begin{array}{c}\\text{Step 4: Solve for}\\frac{dy}{dx}\\text{by dividing both sides of}\\hfill \\\\ \\text{the equation by}{\\text{x}}^{3} \\cos y+1.\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738217476\" class=\"textbox examples\">\n<h3>Using Implicit Differentiation to Find a Second Derivative<\/h3>\n<div id=\"fs-id1169738185350\" class=\"exercise\">\n<div id=\"fs-id1169738185353\" class=\"textbox\">\n<p id=\"fs-id1169738211084\">Find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] if [latex]{x}^{2}+{y}^{2}=25.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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 <a class=\"autogenerated-content\" href=\"#fs-id1169737819953\">(Figure)<\/a>, 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}{d{x}^{2}}.[\/latex]<\/p>\n<div id=\"fs-id1169737766542\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{{d}^{2}y}{d{x}^{2}}& =\\frac{d}{dy}(-\\frac{x}{y})\\hfill & & & \\text{Differentiate both sides of}\\frac{dy}{dx}=-\\frac{x}{y}.\\hfill \\\\ & =-\\frac{(1\u00b7y-x\\frac{dy}{dx})}{{y}^{2}}\\hfill & & & \\text{Use the quotient rule to find}\\frac{d}{dy}(-\\frac{x}{y}).\\hfill \\\\ & =\\frac{\\text{\u2212}y+x\\frac{dy}{dx}}{{y}^{2}}\\hfill & & & \\text{Simplify.}\\hfill \\\\ & =\\frac{\\text{\u2212}y+x(-\\frac{x}{y})}{{y}^{2}}\\hfill & & & \\text{Substitute}\\frac{dy}{dx}=-\\frac{x}{y}.\\hfill \\\\ & =\\frac{\\text{\u2212}{y}^{2}-{x}^{2}}{{y}^{3}}\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169738044967\">At this point we have found an expression for [latex]\\frac{{d}^{2}y}{d{x}^{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}{d{x}^{2}}=-\\frac{25}{{y}^{3}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737772805\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169738220221\" class=\"exercise\">\n<div id=\"fs-id1169738220223\" class=\"textbox\">\n<p id=\"fs-id1169738220225\">Find [latex]\\frac{dy}{dx}[\/latex] for [latex]y[\/latex] defined implicitly by the equation [latex]4{x}^{5}+ \\tan y={y}^{2}+5x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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}=\\frac{5-20{x}^{4}}{{ \\sec }^{2}y-2y}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169738222203\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169738221942\">Follow the problem solving strategy, remembering to apply the chain rule to differentiate [latex]\\tan[\/latex] and [latex]{y}^{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737140783\" class=\"bc-section section\">\n<h1>Finding Tangent Lines Implicitly<\/h1>\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=\"textbox examples\">\n<h3>Finding a Tangent Line to a Circle<\/h3>\n<div id=\"fs-id1169737906658\" class=\"exercise\">\n<div id=\"fs-id1169737906660\" class=\"textbox\">\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>\n<div class=\"textbox shaded\">\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 <a class=\"autogenerated-content\" href=\"#fs-id1169737819953\">(Figure)<\/a>, 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}|\\begin{array}{l}\\\\ {}_{(3,-4)}\\end{array}=-\\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 obtain the equation [latex]y=\\frac{3}{4}x-\\frac{25}{4}[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_002\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_03_08_002\" 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\" \/> <strong> Figure 2.<\/strong> 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 (3, \u22124). <\/div>\n<\/div>\n<p>[\/caption]<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1169737142258\" class=\"exercise\">\n<div id=\"fs-id1169737142260\" class=\"textbox\">\n<h3>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](\\frac{3}{2},\\frac{3}{2})[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_08_003\">(Figure)<\/a>). This curve is known as the folium (or leaf) of Descartes.<\/p>\n<div id=\"CNX_Calc_Figure_03_08_003\" class=\"wp-caption aligncenter\">\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\"><strong>Figure 3.<\/strong> Finding the tangent line to the folium of Descartes at [latex](\\frac{3}{2},\\frac{3}{2}).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<div class=\"textbox shaded\">\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\">[latex]\\begin{array}{ccc}\\hfill \\frac{d}{dx}({y}^{3}+{x}^{3}-3xy)& =\\hfill & \\frac{d}{dx}(0)\\hfill \\\\ \\hfill 3{y}^{2}\\frac{dy}{dx}+3{x}^{2}-(3y+\\frac{dy}{dx}3x)& =\\hfill & 0\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & \\frac{3y-3{x}^{2}}{3{y}^{2}-3x}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169737904704\">Next, substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3{x}^{2}}{3{y}^{2}-3x}[\/latex] to find the slope of the tangent line:<\/p>\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}|\\begin{array}{l}\\\\ {}_{(\\frac{3}{2},\\frac{3}{2})}\\end{array}=-1.[\/latex]<\/div>\n<p id=\"fs-id1169737950853\">Finally, substitute into the point-slope equation of the line to obtain<\/p>\n<div id=\"fs-id1169737950856\" class=\"equation unnumbered\">[latex]y=\\text{\u2212}x+3.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737935216\" class=\"textbox examples\">\n<h3>Applying Implicit Differentiation<\/h3>\n<div id=\"fs-id1169737935219\" class=\"exercise\">\n<div id=\"fs-id1169737935221\" class=\"textbox\">\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]4{x}^{2}+25{y}^{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](3,\\frac{8}{3}),[\/latex] where will it intersect the [latex]x[\/latex]-axis?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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]4{x}^{2}+25{y}^{2}=100[\/latex] at [latex](3,\\frac{8}{3})[\/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\">[latex]8x+50y\\frac{dy}{dx}=0.[\/latex]<\/div>\n<p id=\"fs-id1169737144293\">Solving for [latex]\\frac{dy}{dx},[\/latex] we have<\/p>\n<div id=\"fs-id1169738223546\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=-\\frac{4x}{25y}.[\/latex]<\/div>\n<p id=\"fs-id1169738219286\">The slope of the tangent line is [latex]\\frac{dy}{dx}|{}_{(3,\\frac{8}{3})}=-\\frac{9}{50}.[\/latex] The equation of the tangent line is [latex]y=-\\frac{9}{50}x+\\frac{183}{200}.[\/latex] To determine where the line intersects the [latex]x[\/latex]-axis, solve [latex]0=-\\frac{9}{50}x+\\frac{183}{200}.[\/latex] The solution is [latex]x=\\frac{61}{3}.[\/latex] The missile intersects the [latex]x[\/latex]-axis at the point [latex](\\frac{61}{3},0).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738186785\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169738186788\" class=\"exercise\">\n<div id=\"fs-id1169738186791\" class=\"textbox\">\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>\n<div id=\"fs-id1169737145207\" class=\"textbox shaded\">\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]<\/div>\n<div id=\"fs-id1169737145234\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169738240184\">[latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738240212\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1169738240218\">\n<li>We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).<\/li>\n<li>By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169738240232\" class=\"textbox exercises\">\n<p id=\"fs-id1169738240235\">For the following exercises, use implicit differentiation to find [latex]\\frac{dy}{dx}.[\/latex]<\/p>\n<div id=\"fs-id1169738212639\" class=\"exercise\">\n<div id=\"fs-id1169738212641\" class=\"textbox\">\n<p id=\"fs-id1169738212643\">[latex]{x}^{2}-{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738106195\" class=\"exercise\">\n<div id=\"fs-id1169738106197\" class=\"textbox\">\n<p id=\"fs-id1169738106200\">[latex]6{x}^{2}+3{y}^{2}=12[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738106228\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738106228\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738106228\">[latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738106257\" class=\"exercise\">\n<div id=\"fs-id1169738106259\" class=\"textbox\">\n<p id=\"fs-id1169738106261\">[latex]{x}^{2}y=y-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738073286\" class=\"exercise\">\n<div id=\"fs-id1169738073289\" class=\"textbox\">\n<p id=\"fs-id1169738073291\">[latex]3{x}^{3}+9x{y}^{2}=5{x}^{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738223311\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738223311\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738223311\">[latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738223350\" class=\"exercise\">\n<div id=\"fs-id1169738223352\" class=\"textbox\">\n<p id=\"fs-id1169738223354\">[latex]xy- \\cos (xy)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738237485\" class=\"exercise\">\n<div id=\"fs-id1169738237487\" class=\"textbox\">\n<p id=\"fs-id1169738237489\">[latex]y\\sqrt{x+4}=xy+8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738237518\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738237518\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738237518\">[latex]\\frac{dy}{dx}=\\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738073148\" class=\"exercise\">\n<div id=\"fs-id1169738073151\" class=\"textbox\">\n<p id=\"fs-id1169738073153\">[latex]\\text{\u2212}xy-2=\\frac{x}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737911358\" class=\"exercise\">\n<div id=\"fs-id1169737911360\" class=\"textbox\">\n<p id=\"fs-id1169737911362\">[latex]y \\sin (xy)={y}^{2}+2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737911399\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737911399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737911399\">[latex]\\frac{dy}{dx}=\\frac{{y}^{2} \\cos (xy)}{2y- \\sin (xy)-xy \\cos xy}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738212451\" class=\"exercise\">\n<div id=\"fs-id1169738212454\" class=\"textbox\">\n<p id=\"fs-id1169738212456\">[latex]{(xy)}^{2}+3x={y}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738187223\" class=\"exercise\">\n<div id=\"fs-id1169738187226\" class=\"textbox\">\n<p id=\"fs-id1169738187228\">[latex]{x}^{3}y+x{y}^{3}=-8[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]\\frac{dy}{dx}=\\frac{-3{x}^{2}y-{y}^{3}}{{x}^{3}+3x{y}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169738184790\">For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.<\/p>\n<div id=\"fs-id1169738184795\" class=\"exercise\">\n<div id=\"fs-id1169738184797\" class=\"textbox\">\n<p id=\"fs-id1169738184799\"><strong>[T]<\/strong>[latex]{x}^{4}y-x{y}^{3}=-2,(-1,-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738184892\" class=\"exercise\">\n<div id=\"fs-id1169738184894\" class=\"textbox\">\n<p id=\"fs-id1169738184896\"><strong>[T]<\/strong>[latex]{x}^{2}{y}^{2}+5xy=14,(2,1)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169738226137\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738226137\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738226137\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1169738226145\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205513\/CNX_Calc_Figure_03_08_202.jpg\" alt=\"The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope \u22121\/2 and y intercept 2.\" \/><\/span><br \/>\n[latex]y=\\frac{-1}{2}x+2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738226177\" class=\"exercise\">\n<div id=\"fs-id1169738226179\" class=\"textbox\">\n<p id=\"fs-id1169738226181\"><strong>[T]<\/strong>[latex]\\tan (xy)=y,(\\frac{\\pi }{4},1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738238122\" class=\"exercise\">\n<div id=\"fs-id1169738238124\" class=\"textbox\">\n<p id=\"fs-id1169738238126\"><strong>[T]<\/strong>[latex]x{y}^{2}+ \\sin (\\pi y)-2{x}^{2}=10,(2,-3)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738234517\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738234517\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1169738234524\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205517\/CNX_Calc_Figure_03_08_204.jpg\" alt=\"The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1\/(\u03c0 + 12) and y intercept \u2212(3\u03c0 + 38)\/(\u03c0 + 12).\" \/><\/span><br \/>\n[latex]y=\\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738234576\" class=\"exercise\">\n<div id=\"fs-id1169738234578\" class=\"textbox\">\n<p id=\"fs-id1169738234580\"><strong>[T]<\/strong>[latex]\\frac{x}{y}+5x-7=-\\frac{3}{4}y,(1,2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738221157\" class=\"exercise\">\n<div id=\"fs-id1169738221159\" class=\"textbox\">\n<p id=\"fs-id1169738221161\"><strong>[T]<\/strong>[latex]xy+ \\sin (x)=1,(\\frac{\\pi }{2},0)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738221212\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738221212\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1169738221218\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205521\/CNX_Calc_Figure_03_08_206.jpg\" alt=\"The graph starts in the third quadrant near (\u22125, 0), remains near 0 until x = \u22124, at which point it decreases until it reaches near (0, \u22125). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0.\" \/><\/span><br \/>\n[latex]y=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p><strong>[T]<\/strong> The graph of a folium of Descartes with equation [latex]2{x}^{3}+2{y}^{3}-9xy=0[\/latex] is given in the following graph.<\/p>\n<p><span id=\"fs-id1169737145046\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205525\/CNX_Calc_Figure_03_08_207.jpg\" alt=\"A folium is graphed which has equation 2x3 + 2y3 \u2013 9xy = 0. It crosses over itself at (0, 0).\" \/><\/span><\/p>\n<ol id=\"fs-id1169737145054\" style=\"list-style-type: lower-alpha\">\n<li>Find the equation of the tangent line at the point [latex](2,1).[\/latex] Graph the tangent line along with the folium.<\/li>\n<li>Find the equation of the normal line to the tangent line in a. at the point [latex](2,1).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738244350\" class=\"exercise\">\n<div id=\"fs-id1169738244352\" class=\"textbox\">\n<p id=\"fs-id1169738244354\">For the equation [latex]{x}^{2}+2xy-3{y}^{2}=0,[\/latex]<\/p>\n<ol id=\"fs-id1169738244389\" style=\"list-style-type: lower-alpha\">\n<li>Find the equation of the normal to the tangent line at the point [latex](1,1).[\/latex]<\/li>\n<li>At what other point does the normal line in a. intersect the graph of the equation?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738244421\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738244421\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738244421\">a. [latex]y=\\text{\u2212}x+2[\/latex] b. [latex](3,-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738244457\" class=\"exercise\">\n<div id=\"fs-id1169738244460\" class=\"textbox\">\n<p id=\"fs-id1169738244462\">Find all points on the graph of [latex]{y}^{3}-27y={x}^{2}-90[\/latex] at which the tangent line is vertical.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738211822\" class=\"exercise\">\n<div id=\"fs-id1169738211824\" class=\"textbox\">\n<p id=\"fs-id1169738211826\">For the equation [latex]{x}^{2}+xy+{y}^{2}=7,[\/latex]<\/p>\n<ol id=\"fs-id1169738211857\" style=\"list-style-type: lower-alpha\">\n<li>Find the [latex]x[\/latex]-intercept(s).<\/li>\n<li>Find the slope of the tangent line(s) at the [latex]x[\/latex]-intercept(s).<\/li>\n<li>What does the value(s) in b. indicate about the tangent line(s)?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738211884\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738211884\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738211884\">a. [latex](\u00b1\\sqrt{7},0)[\/latex] b. -2 c. They are parallel since the slope is the same at both intercepts.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738211914\" class=\"exercise\">\n<div id=\"fs-id1169738211916\" class=\"textbox\">\n<p id=\"fs-id1169738211918\">Find the equation of the tangent line to the graph of the equation [latex]{ \\sin }^{-1}x+{ \\sin }^{-1}y=\\frac{\\pi }{6}[\/latex] at the point [latex](0,\\frac{1}{2}).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738201886\" class=\"exercise\">\n<div id=\"fs-id1169738201888\" class=\"textbox\">\n<p id=\"fs-id1169738201890\">Find the equation of the tangent line to the graph of the equation [latex]{ \\tan }^{-1}(x+y)={x}^{2}+\\frac{\\pi }{4}[\/latex] at the point [latex](0,1).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738201952\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738201952\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738201952\">[latex]y=\\text{\u2212}x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738186907\" class=\"exercise\">\n<div id=\"fs-id1169738186909\" class=\"textbox\">\n<p id=\"fs-id1169738186911\">Find [latex]{y}^{\\prime }[\/latex] and [latex]y\\text{\u2033}[\/latex] for [latex]{x}^{2}+6xy-2{y}^{2}=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737934269\" class=\"exercise\">\n<div id=\"fs-id1169737934271\" class=\"textbox\">\n<p id=\"fs-id1169737934273\"><strong>[T]<\/strong> The number of cell phones produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]60{x}^{3\\text{\/}4}{y}^{1\\text{\/}4}=3240.[\/latex]<\/p>\n<ol id=\"fs-id1169737934320\" style=\"list-style-type: lower-alpha\">\n<li>Find [latex]\\frac{dy}{dx}[\/latex] and evaluate at the point [latex](81,16).[\/latex]<\/li>\n<li>Interpret the result of a.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738185028\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738185028\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738185028\">a. -0.5926 b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738185040\" class=\"exercise\">\n<div id=\"fs-id1169738185042\" class=\"textbox\">\n<p id=\"fs-id1169738185044\"><strong>[T]<\/strong> The number of cars produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]30{x}^{1\\text{\/}3}{y}^{2\\text{\/}3}=360.[\/latex]<\/p>\n<p id=\"fs-id1169738185091\">(Both [latex]x[\/latex] and [latex]y[\/latex] are measured in thousands of dollars.)<\/p>\n<ol id=\"fs-id1169738185102\" style=\"list-style-type: lower-alpha\">\n<li>Find [latex]\\frac{dy}{dx}[\/latex] and evaluate at the point [latex](27,8).[\/latex]<\/li>\n<li>Interpret the result of a.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738185162\" class=\"exercise\">\n<div id=\"fs-id1169738185164\" class=\"textbox\">\n<p id=\"fs-id1169738185166\">The volume of a right circular cone of radius [latex]x[\/latex] and height [latex]y[\/latex] is given by [latex]V=\\frac{1}{3}\\pi {x}^{2}y.[\/latex] Suppose that the volume of the cone is [latex]85\\pi {\\text{cm}}^{3}.[\/latex] Find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=4[\/latex] and [latex]y=16.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737954152\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737954152\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737954152\">-8<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169737954161\">For the following exercises, consider a closed rectangular box with a square base with side [latex]x[\/latex] and height [latex]y.[\/latex]<\/p>\n<div id=\"fs-id1169737954174\" class=\"exercise\">\n<div class=\"textbox\">\n<p>Find an equation for the surface area of the rectangular box, [latex]S(x,y).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169737954242\" class=\"exercise\">\n<div id=\"fs-id1169737954244\" class=\"textbox\">\n<p id=\"fs-id1169737954246\">If the surface area of the rectangular box is 78 square feet, find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=3[\/latex] feet and [latex]y=5[\/latex] feet.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738071332\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738071332\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738071332\">-2.67<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169738071340\">For the following exercises, use implicit differentiation to determine [latex]{y}^{\\prime }.[\/latex] Does the answer agree with the formulas we have previously determined?<\/p>\n<div id=\"fs-id1169738071354\" class=\"exercise\">\n<div id=\"fs-id1169738071356\" class=\"textbox\">\n<p id=\"fs-id1169738071358\">[latex]x= \\sin y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738071406\" class=\"exercise\">\n<div id=\"fs-id1169738071409\" class=\"textbox\">\n<p id=\"fs-id1169738071411\">[latex]x= \\cos y[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738071429\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738071429\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738071429\">[latex]{y}^{\\prime }=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738071461\" class=\"exercise\">\n<div id=\"fs-id1169738071463\" class=\"textbox\">\n<p id=\"fs-id1169738071465\">[latex]x= \\tan y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169738191096\" class=\"definition\">\n<dt>implicit differentiation<\/dt>\n<dd id=\"fs-id1169738191101\">is a technique for computing [latex]\\frac{dy}{dx}[\/latex] for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable [latex]y[\/latex] as a function) and solving for [latex]\\frac{dy}{dx}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1869","chapter","type-chapter","status-publish","hentry"],"part":1777,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1869","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1869\/revisions"}],"predecessor-version":[{"id":2435,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1869\/revisions\/2435"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1777"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1869\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1869"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1869"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1869"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1869"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}