{"id":585,"date":"2025-10-09T15:23:36","date_gmt":"2025-10-09T15:23:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=585"},"modified":"2025-11-20T15:42:14","modified_gmt":"2025-11-20T15:42:14","slug":"introduction-to-multivariable-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-multivariable-functions-background-youll-need-3\/","title":{"raw":"Introduction to Multivariable Functions: Background You'll Need 3","rendered":"Introduction to Multivariable Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Find the derivative of a function at a given point<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Derivative of a Function at a Point<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The limit we use to find the slope of a tangent line appears so frequently in calculus that it has a special name: the <strong>derivative<\/strong>. The process of finding a derivative is called <strong>differentiation<\/strong>.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">derivative<\/h3>\r\n<p id=\"fs-id1169738859079\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\r\n\r\n<div id=\"fs-id1169739179144\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n<p id=\"fs-id1169739032184\">provided this limit exists.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1169739032188\">Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\r\n\r\n<div id=\"fs-id1169739188551\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\nprovided this limit exists.\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739204510\">For [latex]f(x)=x^2[\/latex], estimate [latex]f^{\\prime}(3)[\/latex] using the first definition of the derivative.<\/p>\r\n[reveal-answer q=\"370671\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"370671\"]\r\n<p id=\"fs-id1169739033831\">Start by creating a table using values of [latex]x[\/latex] just below [latex]3[\/latex] and just above [latex]3[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1169739301095\" class=\"unnumbered\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column header is x and the second column is (x2 \u2212 9)\/(x \u2212 3). Under the first column are the values 2.9, 2.99, 2.999, 3.001, 3.01, and 3.1. Under the second column are the values 5.9, 5.99, 5.999, 6.001, 6.01, and 6.1.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{x^2-9}{x-3}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2.9[\/latex]<\/td>\r\n<td>[latex]5.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2.99[\/latex]<\/td>\r\n<td>[latex]5.99[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2.999[\/latex]<\/td>\r\n<td>[latex]5.999[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3.001[\/latex]<\/td>\r\n<td>[latex]6.001[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3.01[\/latex]<\/td>\r\n<td>[latex]6.01[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3.1[\/latex]<\/td>\r\n<td>[latex]6.1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<p id=\"fs-id1169739032830\">Look at the values in the table. As [latex]x[\/latex] gets closer to [latex]3[\/latex], the values of [latex]\\frac{x^2-9}{x-3}[\/latex] get closer to [latex]6[\/latex].<\/p>\r\nBased on the table, we can estimate that [latex]f^{\\prime}(3) \\approx 6[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739025974\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the second definition.<\/p>\r\n[reveal-answer q=\"fs-id1169739270484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739270484\"]\r\n<p id=\"fs-id1169739270484\">Using the second definition, we can substitute two values of the function into the equation.<\/p>\r\n\r\n<div id=\"fs-id1169736614162\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(2) &amp; =\\underset{h\\to 0}{\\lim}\\frac{f(2+h)-f(2)}{h} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{(3(2+h)^2-4(2+h)+1)-5}{h} &amp; &amp; &amp; \\begin{array}{l}\\text{Substitute} \\, f(2)=5 \\, \\text{and} \\\\ f(2+h)=3(2+h)^2-4(2+h)+1. \\end{array} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{3h^2+8h}{h} &amp; &amp; &amp; \\text{Simplify the numerator.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{h(3h+8)}{h} &amp; &amp; &amp; \\text{Factor the numerator.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}(3h+8) &amp; &amp; &amp; \\text{Cancel the common factor.} \\\\ &amp; =8 &amp; &amp; &amp; \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169738850760\">The results are the same whether we use the first or second definition.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]162456[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Find the derivative of a function at a given point<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Derivative of a Function at a Point<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The limit we use to find the slope of a tangent line appears so frequently in calculus that it has a special name: the <strong>derivative<\/strong>. The process of finding a derivative is called <strong>differentiation<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">derivative<\/h3>\n<p id=\"fs-id1169738859079\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\n<div id=\"fs-id1169739179144\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n<p id=\"fs-id1169739032184\">provided this limit exists.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739032188\">Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\n<div id=\"fs-id1169739188551\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n<p>provided this limit exists.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739204510\">For [latex]f(x)=x^2[\/latex], estimate [latex]f^{\\prime}(3)[\/latex] using the first definition of the derivative.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q370671\">Show Answer<\/button><\/p>\n<div id=\"q370671\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739033831\">Start by creating a table using values of [latex]x[\/latex] just below [latex]3[\/latex] and just above [latex]3[\/latex].<\/p>\n<table id=\"fs-id1169739301095\" class=\"unnumbered\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column header is x and the second column is (x2 \u2212 9)\/(x \u2212 3). Under the first column are the values 2.9, 2.99, 2.999, 3.001, 3.01, and 3.1. Under the second column are the values 5.9, 5.99, 5.999, 6.001, 6.01, and 6.1.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{x^2-9}{x-3}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2.9[\/latex]<\/td>\n<td>[latex]5.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2.99[\/latex]<\/td>\n<td>[latex]5.99[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2.999[\/latex]<\/td>\n<td>[latex]5.999[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3.001[\/latex]<\/td>\n<td>[latex]6.001[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3.01[\/latex]<\/td>\n<td>[latex]6.01[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3.1[\/latex]<\/td>\n<td>[latex]6.1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739032830\">Look at the values in the table. As [latex]x[\/latex] gets closer to [latex]3[\/latex], the values of [latex]\\frac{x^2-9}{x-3}[\/latex] get closer to [latex]6[\/latex].<\/p>\n<p>Based on the table, we can estimate that [latex]f^{\\prime}(3) \\approx 6[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739025974\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the second definition.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739270484\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739270484\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739270484\">Using the second definition, we can substitute two values of the function into the equation.<\/p>\n<div id=\"fs-id1169736614162\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(2) & =\\underset{h\\to 0}{\\lim}\\frac{f(2+h)-f(2)}{h} & & & \\text{Apply the definition.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{(3(2+h)^2-4(2+h)+1)-5}{h} & & & \\begin{array}{l}\\text{Substitute} \\, f(2)=5 \\, \\text{and} \\\\ f(2+h)=3(2+h)^2-4(2+h)+1. \\end{array} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{3h^2+8h}{h} & & & \\text{Simplify the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{h(3h+8)}{h} & & & \\text{Factor the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}(3h+8) & & & \\text{Cancel the common factor.} \\\\ & =8 & & & \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169738850760\">The results are the same whether we use the first or second definition.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm162456\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=162456&theme=lumen&iframe_resize_id=ohm162456&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-585","chapter","type-chapter","status-publish","hentry"],"part":426,"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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":2,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/585\/revisions"}],"predecessor-version":[{"id":2636,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/585\/revisions\/2636"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/585\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=585"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=585"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=585"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}