{"id":493,"date":"2025-02-13T19:45:27","date_gmt":"2025-02-13T19:45:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/calculus-of-the-hyperbolic-functions-learn-it-1\/"},"modified":"2025-02-13T19:45:27","modified_gmt":"2025-02-13T19:45:27","slug":"calculus-of-the-hyperbolic-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/calculus-of-the-hyperbolic-functions-learn-it-1\/","title":{"raw":"Calculus of the Hyperbolic Functions: Learn It 1","rendered":"Calculus of the Hyperbolic Functions: Learn It 1"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Differentiate and integrate hyperbolic functions and their inverse forms<\/li>\n\t<li>Understand the practical situations where the catenary curve appears<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives and Integrals of the Hyperbolic Functions<\/h2>\n<p id=\"fs-id1167793637684\">The hyperbolic sine ([latex]sinh[\/latex]) and hyperbolic cosine ([latex]cosh[\/latex]) functions are defined as:<\/p>\n<div id=\"fs-id1167793288534\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sinh}x=\\dfrac{{e}^{x}-{e}^{\\text{\u2212}x}}{2}\\text{&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;}\\text{cosh}x=\\dfrac{{e}^{x}+{e}^{\\text{\u2212}x}}{2}[\/latex]<\/div>\n<p id=\"fs-id1167793233791\">The other hyperbolic functions are then defined in terms of [latex]\\text{sinh}x[\/latex] and [latex]\\text{cosh}x.[\/latex] The graphs of these functions provide insights into their behaviors.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"958\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213342\/CNX_Calc_Figure_06_09_001.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled \u201cc\u201d is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled \u201cf\u201d and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" width=\"958\" height=\"749\"> Figure 1. Graphs of the hyperbolic functions.[\/caption]\n\n<p id=\"fs-id1167793925306\">It is straightforward to develop differentiation formulas for hyperbolic functions. For instance:<\/p>\n<div id=\"fs-id1167794049237\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill \\frac{d}{dx}(\\text{sinh}x)&amp; =\\frac{d}{dx}(\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2})\\hfill \\\\ &amp; =\\frac{1}{2}\\left[\\frac{d}{dx}({e}^{x})-\\frac{d}{dx}({e}^{\\text{\u2212}x})\\right]\\hfill \\\\ &amp; =\\frac{1}{2}\\left[{e}^{x}+{e}^{\\text{\u2212}x}\\right]=\\text{cosh}x.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793278441\">Similarly, [latex](\\frac{d}{dx})\\text{cosh}x=\\text{sinh}x.[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>derivatives of the hyperbolic functions<\/h3>\n<table class=\"center\">\n<caption>Derivatives of the Hyperbolic Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th style=\"text-align: center;\">[latex]f(x)[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{sinh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{cosh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{cosh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{sinh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{tanh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{\\text{sech}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{coth}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{sech}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}\\text{sech}x\\text{tanh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{csch}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}\\text{csch}x\\text{coth}x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<p id=\"fs-id1167794139383\">The derivatives of hyperbolic functions share similarities with those of trigonometric functions. For example:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{d}{dx}) \\sin x= \\cos x[\/latex] and [latex](\\frac{d}{dx})\\text{sinh}x=\\text{cosh}x.[\/latex]<\/p>\n<p>However, the derivatives of the cosine functions differ in sign:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{d}{dx}) \\cos x=\\text{\u2212} \\sin x,[\/latex] but [latex](\\frac{d}{dx})\\text{cosh}x=\\text{sinh}x.[\/latex]<\/p>\n<p>Using the differentiation formulas, we can derive the integral formulas for hyperbolic functions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>&nbsp;integral formulas for hyperbolic functions<\/h3>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccccccc}\\hfill \\displaystyle\\int \\text{sinh}udu&amp; =\\hfill &amp; \\text{cosh}u+C\\hfill &amp; &amp; &amp; \\hfill \\displaystyle\\int {\\text{csch}}^{2}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{coth}u+C\\hfill \\\\ \\hfill \\displaystyle\\int \\text{cosh}udu&amp; =\\hfill &amp; \\text{sinh}u+C\\hfill &amp; &amp; &amp; \\hfill \\displaystyle\\int \\text{sech}u\\text{tanh}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{sech}u+C\\hfill \\\\ \\hfill \\displaystyle\\int {\\text{sech}}^{2}udu&amp; =\\hfill &amp; \\text{tanh}u+C\\hfill &amp; &amp; &amp; \\hfill \\displaystyle\\int \\text{csch}u\\text{coth}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{csch}u+C\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793278457\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167794060382\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}[\/latex]<\/li>\n<\/ol>\n\n[reveal-answer q=\"fs-id1167794098810\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794098810\"]\n\n<p id=\"fs-id1167794098810\">Using the formulas in the table on derivatives&nbsp; of the hyperbolic functions and the chain rule, we get<\/p>\n<ol id=\"fs-id1167793500488\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))=\\text{cosh}({x}^{2})\u00b72x[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}=2\\text{cosh}x\\text{sinh}x[\/latex]<\/li>\n<\/ol>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793965336\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167794050806\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\displaystyle\\int x\\text{cosh}({x}^{2})dx[\/latex]<\/li>\n\t<li>[latex]\\displaystyle\\int \\text{tanh}xdx[\/latex]<\/li>\n<\/ol>\n\n[reveal-answer q=\"fs-id1167793254882\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793254882\"]\n\n<p id=\"fs-id1167793254882\">We can use [latex]u[\/latex]-substitution in both cases.<\/p>\n<ol id=\"fs-id1167794042760\" style=\"list-style-type: lower-alpha;\">\n\t<li>Let [latex]u={x}^{2}.[\/latex] Then, [latex]du=2xdx[\/latex] and<br>\n<div id=\"fs-id1167794337482\" class=\"equation unnumbered\">[latex]\\displaystyle\\int x\\text{cosh}({x}^{2})dx=\\displaystyle\\int \\frac{1}{2}\\text{cosh}udu=\\frac{1}{2}\\text{sinh}u+C=\\frac{1}{2}\\text{sinh}({x}^{2})+C.[\/latex]<\/div>\n<\/li>\n\t<li>Let [latex]u=\\text{cosh}x.[\/latex] Then, [latex]du=\\text{sinh}xdx[\/latex] and<br>\n<div id=\"fs-id1167794043265\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\text{tanh}xdx=\\displaystyle\\int \\frac{\\text{sinh}x}{\\text{cosh}x}dx=\\displaystyle\\int \\frac{1}{u}du=\\text{ln}|u|+C=\\text{ln}|\\text{cosh}x|+C.[\/latex]<\/div>\n<p>Note that [latex]\\text{cosh}x&gt;0[\/latex] for all [latex]x,[\/latex] so we can eliminate the absolute value signs and obtain<\/p>\n<div id=\"fs-id1167793358384\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\text{tanh}xdx=\\text{ln}(\\text{cosh}x)+C.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]223494[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Differentiate and integrate hyperbolic functions and their inverse forms<\/li>\n<li>Understand the practical situations where the catenary curve appears<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives and Integrals of the Hyperbolic Functions<\/h2>\n<p id=\"fs-id1167793637684\">The hyperbolic sine ([latex]sinh[\/latex]) and hyperbolic cosine ([latex]cosh[\/latex]) functions are defined as:<\/p>\n<div id=\"fs-id1167793288534\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sinh}x=\\dfrac{{e}^{x}-{e}^{\\text{\u2212}x}}{2}\\text{&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;}\\text{cosh}x=\\dfrac{{e}^{x}+{e}^{\\text{\u2212}x}}{2}[\/latex]<\/div>\n<p id=\"fs-id1167793233791\">The other hyperbolic functions are then defined in terms of [latex]\\text{sinh}x[\/latex] and [latex]\\text{cosh}x.[\/latex] The graphs of these functions provide insights into their behaviors.<\/p>\n<figure style=\"width: 958px\" 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\/11213342\/CNX_Calc_Figure_06_09_001.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled \u201cc\u201d is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled \u201cf\u201d and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" width=\"958\" height=\"749\" \/><figcaption class=\"wp-caption-text\">Figure 1. Graphs of the hyperbolic functions.<\/figcaption><\/figure>\n<p id=\"fs-id1167793925306\">It is straightforward to develop differentiation formulas for hyperbolic functions. For instance:<\/p>\n<div id=\"fs-id1167794049237\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill \\frac{d}{dx}(\\text{sinh}x)& =\\frac{d}{dx}(\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2})\\hfill \\\\ & =\\frac{1}{2}\\left[\\frac{d}{dx}({e}^{x})-\\frac{d}{dx}({e}^{\\text{\u2212}x})\\right]\\hfill \\\\ & =\\frac{1}{2}\\left[{e}^{x}+{e}^{\\text{\u2212}x}\\right]=\\text{cosh}x.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793278441\">Similarly, [latex](\\frac{d}{dx})\\text{cosh}x=\\text{sinh}x.[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>derivatives of the hyperbolic functions<\/h3>\n<table class=\"center\">\n<caption>Derivatives of the Hyperbolic Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th style=\"text-align: center;\">[latex]f(x)[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{sinh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{cosh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{cosh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{sinh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{tanh}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{\\text{sech}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{coth}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{sech}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}\\text{sech}x\\text{tanh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]\\text{csch}x[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\text{\u2212}\\text{csch}x\\text{coth}x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<p id=\"fs-id1167794139383\">The derivatives of hyperbolic functions share similarities with those of trigonometric functions. For example:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{d}{dx}) \\sin x= \\cos x[\/latex] and [latex](\\frac{d}{dx})\\text{sinh}x=\\text{cosh}x.[\/latex]<\/p>\n<p>However, the derivatives of the cosine functions differ in sign:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{d}{dx}) \\cos x=\\text{\u2212} \\sin x,[\/latex] but [latex](\\frac{d}{dx})\\text{cosh}x=\\text{sinh}x.[\/latex]<\/p>\n<p>Using the differentiation formulas, we can derive the integral formulas for hyperbolic functions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>&nbsp;integral formulas for hyperbolic functions<\/h3>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccccccc}\\hfill \\displaystyle\\int \\text{sinh}udu& =\\hfill & \\text{cosh}u+C\\hfill & & & \\hfill \\displaystyle\\int {\\text{csch}}^{2}udu& =\\hfill & \\text{\u2212}\\text{coth}u+C\\hfill \\\\ \\hfill \\displaystyle\\int \\text{cosh}udu& =\\hfill & \\text{sinh}u+C\\hfill & & & \\hfill \\displaystyle\\int \\text{sech}u\\text{tanh}udu& =\\hfill & \\text{\u2212}\\text{sech}u+C\\hfill \\\\ \\hfill \\displaystyle\\int {\\text{sech}}^{2}udu& =\\hfill & \\text{tanh}u+C\\hfill & & & \\hfill \\displaystyle\\int \\text{csch}u\\text{coth}udu& =\\hfill & \\text{\u2212}\\text{csch}u+C\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793278457\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167794060382\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794098810\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794098810\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794098810\">Using the formulas in the table on derivatives&nbsp; of the hyperbolic functions and the chain rule, we get<\/p>\n<ol id=\"fs-id1167793500488\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))=\\text{cosh}({x}^{2})\u00b72x[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}=2\\text{cosh}x\\text{sinh}x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793965336\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167794050806\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle\\int x\\text{cosh}({x}^{2})dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\text{tanh}xdx[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793254882\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793254882\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793254882\">We can use [latex]u[\/latex]-substitution in both cases.<\/p>\n<ol id=\"fs-id1167794042760\" style=\"list-style-type: lower-alpha;\">\n<li>Let [latex]u={x}^{2}.[\/latex] Then, [latex]du=2xdx[\/latex] and\n<div id=\"fs-id1167794337482\" class=\"equation unnumbered\">[latex]\\displaystyle\\int x\\text{cosh}({x}^{2})dx=\\displaystyle\\int \\frac{1}{2}\\text{cosh}udu=\\frac{1}{2}\\text{sinh}u+C=\\frac{1}{2}\\text{sinh}({x}^{2})+C.[\/latex]<\/div>\n<\/li>\n<li>Let [latex]u=\\text{cosh}x.[\/latex] Then, [latex]du=\\text{sinh}xdx[\/latex] and\n<div id=\"fs-id1167794043265\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\text{tanh}xdx=\\displaystyle\\int \\frac{\\text{sinh}x}{\\text{cosh}x}dx=\\displaystyle\\int \\frac{1}{u}du=\\text{ln}|u|+C=\\text{ln}|\\text{cosh}x|+C.[\/latex]<\/div>\n<p>Note that [latex]\\text{cosh}x>0[\/latex] for all [latex]x,[\/latex] so we can eliminate the absolute value signs and obtain<\/p>\n<div id=\"fs-id1167793358384\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\text{tanh}xdx=\\text{ln}(\\text{cosh}x)+C.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm223494\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=223494&theme=lumen&iframe_resize_id=ohm223494&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"2.9 Calculus of Hyperbolic Functions\",\"author\":\"Ryan 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