{"id":1688,"date":"2024-04-24T16:35:58","date_gmt":"2024-04-24T16:35:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1688"},"modified":"2024-08-05T12:44:22","modified_gmt":"2024-08-05T12:44:22","slug":"techniques-for-differentiation-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/techniques-for-differentiation-background-youll-need-1\/","title":{"raw":"Techniques for Differentiation: Background You'll Need 1","rendered":"Techniques for Differentiation: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify whether a function is given directly (explicit) or needs solving (implicit)&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Identify whether a function is given directly (explicit) or needs solving (implicit)<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Understanding Explicit and Implicit Functions<\/h2>\r\n<p>In mathematical analysis, the distinction between explicit and implicit functions is pivotal in understanding how variables interact within an equation. This understanding is fundamental when approaching calculus, as it affects how we might differentiate or integrate expressions with respect to a given variable.<\/p>\r\n<h3>Explicit Functions<\/h3>\r\n<p>An explicit function is one where the dependent variable, typically denoted as [latex]y[\/latex], is expressed directly in terms of the independent variable [latex]x[\/latex]. In simpler terms, [latex]y[\/latex] is isolated on one side of the equation. This direct expression allows us to readily compute the value of [latex]y[\/latex] for any given value of [latex]x[\/latex] without the need for additional manipulation.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>explicit functions<\/h3>\r\n<p>An <strong>explicit function<\/strong> clearly expresses the dependent variable, such as [latex]y[\/latex], in terms of the independent variable, such as [latex]x[\/latex]. It takes the form [latex]y=f(x)[\/latex], allowing for direct computation of [latex]y[\/latex] for any given [latex]x[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider the function:<\/p>\r\n<center>[latex]y = 3x^2 +2x-5[\/latex]<\/center>\r\n<p>For this quadratic equation, the value of [latex]y[\/latex] is defined explicitly for each [latex]x[\/latex], allowing for straightforward evaluation.<\/p>\r\n<\/section>\r\n<h3>Implicit Functions<\/h3>\r\n<p>Conversely, an implicit function is one where the relationship between [latex]y[\/latex] and [latex]x[\/latex] is implied within an equation. [latex]y[\/latex] is not isolated, and the equation must be manipulated to solve for [latex]y[\/latex] in terms of [latex]x[\/latex] if it's even possible. Implicit functions often arise in situations where two or more variables maintain a relationship, but one cannot be neatly expressed in terms of the others.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>implicit functions<\/h3>\r\n<p><strong>Implicit functions<\/strong> are those in which the relationship between variables is expressed indirectly. The dependent variable is not isolated on one side but is mixed with the independent variable within an equation. Solving for one variable in terms of the others may not be straightforward or sometimes even possible.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>The equation below defines a relationship between [latex]x[\/latex] and [latex]y[\/latex] where neither variable is isolated as the subject of the formula.<\/p>\r\n<p style=\"text-align: center;\">[latex]x^3+y^3=6xy[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p>To determine if a function is explicit or implicit, look for the dependent variable ([latex]y[\/latex]) and assess whether it is written on its own with respect to [latex]x[\/latex].<\/p>\r\n<ul>\r\n\t<li>If [latex]y[\/latex] is by itself on one side of the equation, then it's an <strong>explicit function<\/strong>.<\/li>\r\n\t<li>If [latex]y[\/latex] is mingled with [latex]x[\/latex] and cannot be easily isolated, then it's an <strong>implicit function<\/strong>.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Determine whether each of the following functions is explicit or implicit:<\/p>\r\n<ol>\r\n\t<li>[latex]y = 3x^2 - 7[\/latex]<\/li>\r\n\t<li>[latex]x^2 + y^2 = 16[\/latex]<\/li>\r\n\t<li>[latex]y^3 + 3y = x[\/latex]<\/li>\r\n\t<li>[latex]e^y = x + y[\/latex]<\/li>\r\n\t<li>[latex]\\ln(x) + \\ln(y) = 1[\/latex]<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"751669\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"751669\"]<\/p>\r\n<ol>\r\n\t<li>The function [latex]y = 3x^2 - 7[\/latex] is explicit because [latex]y[\/latex] is isolated on one side and expressed directly in terms of [latex]x[\/latex].<\/li>\r\n\t<li>The function [latex]x^2 + y^2 = 16[\/latex] is implicit because it represents a relationship between [latex]x[\/latex] and [latex]y[\/latex] that doesn't explicitly solve for either variable.<\/li>\r\n\t<li>The function [latex]y^3 + 3y = x[\/latex] is considered implicit. Although one might attempt to express [latex]y[\/latex] in terms of [latex]x[\/latex] by manipulating the equation, the term [latex]y[\/latex] cannot be easily isolated on one side due to its presence both as a cubic term and a linear term. This intermingling of [latex]y[\/latex] in multiple terms without a straightforward method to solve for [latex]y[\/latex] directly keeps the function in an implicit form.<\/li>\r\n\t<li>The function [latex]e^y = x + y[\/latex] is implicit. Although it might appear possible to solve for [latex]y[\/latex], the presence of [latex]y[\/latex] on both sides of the equation, especially as part of an exponent and a linear term, indicates that the function is not explicitly solved for [latex]y[\/latex].<\/li>\r\n\t<li>The function [latex]\\ln(x) + \\ln(y) = 1[\/latex] is implicit because both [latex]x[\/latex] and [latex]y[\/latex] are intertwined within the equation and there is no direct expression of [latex]y[\/latex] solely in terms of [latex]x[\/latex].<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]284663[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify whether a function is given directly (explicit) or needs solving (implicit)&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Identify whether a function is given directly (explicit) or needs solving (implicit)<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Understanding Explicit and Implicit Functions<\/h2>\n<p>In mathematical analysis, the distinction between explicit and implicit functions is pivotal in understanding how variables interact within an equation. This understanding is fundamental when approaching calculus, as it affects how we might differentiate or integrate expressions with respect to a given variable.<\/p>\n<h3>Explicit Functions<\/h3>\n<p>An explicit function is one where the dependent variable, typically denoted as [latex]y[\/latex], is expressed directly in terms of the independent variable [latex]x[\/latex]. In simpler terms, [latex]y[\/latex] is isolated on one side of the equation. This direct expression allows us to readily compute the value of [latex]y[\/latex] for any given value of [latex]x[\/latex] without the need for additional manipulation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>explicit functions<\/h3>\n<p>An <strong>explicit function<\/strong> clearly expresses the dependent variable, such as [latex]y[\/latex], in terms of the independent variable, such as [latex]x[\/latex]. It takes the form [latex]y=f(x)[\/latex], allowing for direct computation of [latex]y[\/latex] for any given [latex]x[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Consider the function:<\/p>\n<div style=\"text-align: center;\">[latex]y = 3x^2 +2x-5[\/latex]<\/div>\n<p>For this quadratic equation, the value of [latex]y[\/latex] is defined explicitly for each [latex]x[\/latex], allowing for straightforward evaluation.<\/p>\n<\/section>\n<h3>Implicit Functions<\/h3>\n<p>Conversely, an implicit function is one where the relationship between [latex]y[\/latex] and [latex]x[\/latex] is implied within an equation. [latex]y[\/latex] is not isolated, and the equation must be manipulated to solve for [latex]y[\/latex] in terms of [latex]x[\/latex] if it&#8217;s even possible. Implicit functions often arise in situations where two or more variables maintain a relationship, but one cannot be neatly expressed in terms of the others.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>implicit functions<\/h3>\n<p><strong>Implicit functions<\/strong> are those in which the relationship between variables is expressed indirectly. The dependent variable is not isolated on one side but is mixed with the independent variable within an equation. Solving for one variable in terms of the others may not be straightforward or sometimes even possible.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>The equation below defines a relationship between [latex]x[\/latex] and [latex]y[\/latex] where neither variable is isolated as the subject of the formula.<\/p>\n<p style=\"text-align: center;\">[latex]x^3+y^3=6xy[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p>To determine if a function is explicit or implicit, look for the dependent variable ([latex]y[\/latex]) and assess whether it is written on its own with respect to [latex]x[\/latex].<\/p>\n<ul>\n<li>If [latex]y[\/latex] is by itself on one side of the equation, then it&#8217;s an <strong>explicit function<\/strong>.<\/li>\n<li>If [latex]y[\/latex] is mingled with [latex]x[\/latex] and cannot be easily isolated, then it&#8217;s an <strong>implicit function<\/strong>.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Determine whether each of the following functions is explicit or implicit:<\/p>\n<ol>\n<li>[latex]y = 3x^2 - 7[\/latex]<\/li>\n<li>[latex]x^2 + y^2 = 16[\/latex]<\/li>\n<li>[latex]y^3 + 3y = x[\/latex]<\/li>\n<li>[latex]e^y = x + y[\/latex]<\/li>\n<li>[latex]\\ln(x) + \\ln(y) = 1[\/latex]<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q751669\">Show Answer<\/button><\/p>\n<div id=\"q751669\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The function [latex]y = 3x^2 - 7[\/latex] is explicit because [latex]y[\/latex] is isolated on one side and expressed directly in terms of [latex]x[\/latex].<\/li>\n<li>The function [latex]x^2 + y^2 = 16[\/latex] is implicit because it represents a relationship between [latex]x[\/latex] and [latex]y[\/latex] that doesn&#8217;t explicitly solve for either variable.<\/li>\n<li>The function [latex]y^3 + 3y = x[\/latex] is considered implicit. Although one might attempt to express [latex]y[\/latex] in terms of [latex]x[\/latex] by manipulating the equation, the term [latex]y[\/latex] cannot be easily isolated on one side due to its presence both as a cubic term and a linear term. This intermingling of [latex]y[\/latex] in multiple terms without a straightforward method to solve for [latex]y[\/latex] directly keeps the function in an implicit form.<\/li>\n<li>The function [latex]e^y = x + y[\/latex] is implicit. Although it might appear possible to solve for [latex]y[\/latex], the presence of [latex]y[\/latex] on both sides of the equation, especially as part of an exponent and a linear term, indicates that the function is not explicitly solved for [latex]y[\/latex].<\/li>\n<li>The function [latex]\\ln(x) + \\ln(y) = 1[\/latex] is implicit because both [latex]x[\/latex] and [latex]y[\/latex] are intertwined within the equation and there is no direct expression of [latex]y[\/latex] solely in terms of [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm284663\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=284663&theme=lumen&iframe_resize_id=ohm284663&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1688"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1688\/revisions"}],"predecessor-version":[{"id":4510,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1688\/revisions\/4510"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1688\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1688"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1688"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1688"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1688"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}