{"id":386,"date":"2023-07-03T13:35:39","date_gmt":"2023-07-03T13:35:39","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/"},"modified":"2023-07-03T13:35:39","modified_gmt":"2023-07-03T13:35:39","slug":"derivada-da-tangente-hiperbolica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/","title":{"rendered":"Derivada da tangente hiperb\u00f3lica"},"content":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a derivada da tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos resolvidos de derivadas de tangentes hiperb\u00f3licas. E, finalmente, mostramos a f\u00f3rmula da derivada da tangente hiperb\u00f3lica. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-tangente-hiperbolica\"><\/span> F\u00f3rmula para a derivada da tangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada da tangente hiperb\u00f3lica de x \u00e9 igual a 1 dividido pelo quadrado do cosseno hiperb\u00f3lico de x.<\/strong> A derivada da tangente de x tamb\u00e9m \u00e9 equivalente ao quadrado da secante hiperb\u00f3lica de x e 1 menos o quadrado da tangente hiperb\u00f3lica de x.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a7c392afdb3bbf504e167e15fb2fee6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{tanh}(x)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=\\cfrac{1}{\\text{cosh}^2(x)}=\\text{sech}^2(x)=1-\\text{tanh}^2(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por outro lado, se no argumento da fun\u00e7\u00e3o tivermos uma fun\u00e7\u00e3o diferente de x, devemos aplicar a regra da cadeia. E ent\u00e3o as tr\u00eas f\u00f3rmulas para a derivada da tangente hiperb\u00f3lica s\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b5ee01c6675067b20f71ea8ac4efcfe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{tanh}(u)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=\\cfrac{u'}{\\text{cosh}^2(u)}=\\text{sech}^2(u)\\cdot u'=\\left(1-\\text{tanh}^2(u)\\right)\\cdot u'\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"409\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Isto n\u00e3o significa que sempre que derivamos a tangente hiperb\u00f3lica tenhamos de utilizar todas as tr\u00eas f\u00f3rmulas, mas sim que podemos utilizar qualquer uma delas para deriv\u00e1-la. Ent\u00e3o, dependendo da fun\u00e7\u00e3o do argumento da tangente hiperb\u00f3lica, ser\u00e1 melhor usar uma f\u00f3rmula ou outra. Abaixo est\u00e3o v\u00e1rios exemplos nos quais voc\u00ea pode ver como a tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o \u00e9 derivada. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-de-la-tangente-hyperbolique.webp\" alt=\"derivada da tangente hiperb\u00f3lica\" class=\"wp-image-2072\" width=\"420\" height=\"366\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A derivada da tangente hiperb\u00f3lica \u00e9 quase id\u00eantica \u00e0 derivada da tangente, mas possui um pequeno detalhe que as torna totalmente diferentes. Voc\u00ea pode ver qual \u00e9 a diferen\u00e7a no seguinte link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente\/\">f\u00f3rmula da derivada tangente<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-tangente-hiperbolica\"><\/span> Exemplos de derivada da tangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de ver qual \u00e9 a f\u00f3rmula da derivada da tangente hiperb\u00f3lica, aqui est\u00e3o v\u00e1rios exemplos resolvidos de derivadas deste tipo de fun\u00e7\u00f5es trigonom\u00e9tricas para que voc\u00ea entenda perfeitamente como derivar a tangente hiperb\u00f3lica. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-la-tangente-hiperbolica-de-2x\"><\/span> Exemplo 1: Derivada da tangente hiperb\u00f3lica de 2x<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79ac7ea68ec9b155da28c4fbcaa0ee15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para derivar a tangente hiperb\u00f3lica neste exemplo, usaremos a f\u00f3rmula do cosseno hiperb\u00f3lico, embora voc\u00ea possa usar a que preferir.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7db9abcf1f642aef23b75d912cde9280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{\\text{cosh}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"407\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Sabemos que a derivada de 2x \u00e9 2, ent\u00e3o a derivada de toda a fun\u00e7\u00e3o \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ece3b50dd3b4705574fda3f7cda6e66a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(2x)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2}{\\text{cosh}^2(2x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"425\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-la-tangente-hiperbolica-de-x-al-cuadrado\"><\/span> Exemplo 2: Derivada da tangente hiperb\u00f3lica de x ao quadrado<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ff96c9470dca3d696a7002d26563a63e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A regra para a derivada da tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7db9abcf1f642aef23b75d912cde9280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{\\text{cosh}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"407\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Por um lado, diferenciamos a fun\u00e7\u00e3o do argumento x <sup>2<\/sup> , que d\u00e1 2x, e ent\u00e3o resolvemos a derivada de toda a fun\u00e7\u00e3o usando a f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d16a26c06c4b3843facf3f0a980a61d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(x^2)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2x}{\\text{cosh}^2(x^2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"421\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-la-tangente-hiperbolica-al-cubo\"><\/span> Exemplo 3: Derivada da tangente hiperb\u00f3lica ao cubo<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e766da71a69fd53df69de8305a592844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}^3(7x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, devemos derivar a tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o que, al\u00e9m disso, \u00e9 elevada a uma pot\u00eancia. Portanto, precisamos usar a f\u00f3rmula da derivada de uma fun\u00e7\u00e3o potencial, a regra da derivada da tangente hiperb\u00f3lica e a regra da cadeia: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a7e6b023f634820c69181cfa8d0a16b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3\\text{tanh}^2(7x^2)\\cdot \\cfrac{14x}{\\text{cosh}^2(7x^2)}}=\\cfrac{42x\\cdot \\text{tanh}^2(7x^2)}{\\text{cosh}^2(7x^2)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"402\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-derivada-de-la-tangente\"><\/span> Prova da derivada da tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nesta se\u00e7\u00e3o, demonstraremos a f\u00f3rmula da derivada da tangente hiperb\u00f3lica. E, para isso, partiremos da identidade trigonom\u00e9trica que conecta as tr\u00eas raz\u00f5es trigonom\u00e9tricas hiperb\u00f3licas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12f286528bc0635705aadbe510b6ceb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x)=\\cfrac{\\text{senh}(x)}{\\text{cosh}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"144\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Nota:<\/strong> Para entender a prova, voc\u00ea precisa saber qual \u00e9 a <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/\">derivada do seno hiperb\u00f3lico<\/a><\/span> e qual \u00e9 a <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-cosseno-hiperbolico\/\">derivada do cosseno hiperb\u00f3lico<\/a><\/span> . Portanto, recomendamos que voc\u00ea visite as p\u00e1ginas vinculadas antes de continuar.<\/p>\n<p> Agora, vamos aplicar a f\u00f3rmula da derivada de um quociente: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b01359155f318f95df8e21e428d2026_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\left(\\text{tanh}(x)\\right)'=\\left(\\frac{\\text{senh}(x)}{\\text{cosh}(x)}\\right)'\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"193\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6f85187679a1b95d64d3afdb78efd4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=\\cfrac{\\text{cosh}(x)\\cdot \\text{cosh}(x)-\\text{senh}(x)\\text{senh}(x) }{\\text{cosh}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"356\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48dd21086a84d52131322f0aa9086a4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=\\cfrac{\\text{cosh}^2(x)-\\text{senh}^2(x)}{\\text{cosh}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"243\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Reduzimos a express\u00e3o do numerador da fra\u00e7\u00e3o usando a seguinte f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4317a445a90e4d139b47db7cf4a49a1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}^2(x)-\\text{senh}^2(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7acd0e926ab13e13a82d0bbed6f20fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=\\cfrac{1}{\\text{cosh}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"155\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver, a igualdade anterior corresponde \u00e0 primeira f\u00f3rmula da derivada da tangente hiperb\u00f3lica. Da mesma forma, a secante hiperb\u00f3lica \u00e9 o inverso multiplicativo do cosseno hiperb\u00f3lico, ent\u00e3o a segunda f\u00f3rmula tamb\u00e9m \u00e9 derivada:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-efd858fd9bbcc28bbba771ddfe60479d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=\\text{sech}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"150\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Finalmente, podemos chegar \u00e0 terceira regra da derivada da tangente hiperb\u00f3lica convertendo a fra\u00e7\u00e3o do passo anterior numa subtra\u00e7\u00e3o de fra\u00e7\u00f5es: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-258c135ebd5bf9f28981900d19ca20e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{\\text{cosh}^2(x)-\\text{senh}^2(x)}{\\text{cosh}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"234\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8316a3ad4867e7135dfae9a7f49506e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=\\cfrac{\\text{cosh}^2(x)}{\\text{cosh}^2(x)}-\\cfrac{\\text{senh}^2(x)}{\\text{cosh}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"245\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acc11824e13677fc21ae1f0e9dd24733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}'(x)=1-\\text{tanh}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a derivada da tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos resolvidos de derivadas de tangentes hiperb\u00f3licas. E, finalmente, mostramos a f\u00f3rmula da derivada da tangente hiperb\u00f3lica. F\u00f3rmula para a derivada da tangente hiperb\u00f3lica A derivada da tangente hiperb\u00f3lica de x \u00e9 igual a 1 dividido &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\"> <span class=\"screen-reader-text\">Derivada da tangente hiperb\u00f3lica<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[11],"tags":[],"class_list":["post-386","post","type-post","status-publish","format-standard","hentry","category-derivados"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Derivada da tangente hiperb\u00f3lica - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada da tangente hiperb\u00f3lica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a derivada da tangente hiperb\u00f3lica de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos resolvidos de derivadas de tangentes hiperb\u00f3licas. E, finalmente, mostramos a f\u00f3rmula da derivada da tangente hiperb\u00f3lica. F\u00f3rmula para a derivada da tangente hiperb\u00f3lica A derivada da tangente hiperb\u00f3lica de x \u00e9 igual a 1 dividido &hellip; Derivada da tangente hiperb\u00f3lica Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-03T13:35:39+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a7c392afdb3bbf504e167e15fb2fee6_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Derivada da tangente hiperb\u00f3lica\",\"datePublished\":\"2023-07-03T13:35:39+00:00\",\"dateModified\":\"2023-07-03T13:35:39+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\"},\"wordCount\":612,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\",\"url\":\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\",\"name\":\"Derivada da tangente hiperb\u00f3lica - 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