{"id":399,"date":"2023-07-03T01:21:27","date_gmt":"2023-07-03T01:21:27","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/"},"modified":"2023-07-03T01:21:27","modified_gmt":"2023-07-03T01:21:27","slug":"derivada-do-arco-tangente-hiperbolico-2","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/","title":{"rendered":"Derivada da arcotangente hiperb\u00f3lica"},"content":{"rendered":"<p>Neste artigo explicamos como derivar o arco tangente hiperb\u00f3lico de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos resolvidos da derivada da arcotangente hiperb\u00f3lica. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-arcocotangente-hiperbolica\"><\/span> F\u00f3rmula para a derivada da arcotangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada do arco tangente hiperb\u00f3lico de x \u00e9 um sobre um menos x ao quadrado.<\/strong><\/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-42f0b83c10ae509a0cda1aa4abf25b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{1-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"411\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do arco tangente hiperb\u00f3lico de uma fun\u00e7\u00e3o<\/strong> \u00e9 igual ao quociente da derivada dessa fun\u00e7\u00e3o dividido por um menos essa fun\u00e7\u00e3o ao quadrado.<\/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-ccadba79123a5f5e81d2b777d5cb9ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"411\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Observe que a segunda f\u00f3rmula \u00e9 semelhante \u00e0 primeira, mas aplica a regra da cadeia, portanto, elas podem ser consideradas a mesma f\u00f3rmula. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-de-larccotangente-hyperbolique.webp\" alt=\"derivado do arco tangente hiperb\u00f3lico\" class=\"wp-image-2832\" width=\"391\" height=\"282\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Voc\u00ea pode ver em alguns livros de matem\u00e1tica que a derivada deste tipo de fun\u00e7\u00e3o trigonom\u00e9trica inversa \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-7c1e05f806ff6ead0b09d1af192c3345_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{-1}{x^2-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"114\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Por\u00e9m, se voc\u00ea olhar com aten\u00e7\u00e3o, s\u00e3o a mesma f\u00f3rmula, a \u00fanica diferen\u00e7a \u00e9 que o numerador e o denominador da fra\u00e7\u00e3o foram multiplicados por -1. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-arcocotangente-hiperbolica\"><\/span><meta charset=\"utf-8\"> Exemplos de derivada da arcotangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/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-5885f3abf1d025a4716d0233fdd5efda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(5x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> No argumento arcotangente hiperb\u00f3lico temos uma fun\u00e7\u00e3o diferente de x, ent\u00e3o precisamos usar a f\u00f3rmula da regra da cadeia para deriv\u00e1-la:<\/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-ccadba79123a5f5e81d2b777d5cb9ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"411\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> A derivada de 5x \u00e9 5, ent\u00e3o coloque 5 no numerador da fra\u00e7\u00e3o e coloque menos 5x ao quadrado no denominador:<\/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-36c2652240dffeccc7b2094f68c570c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(5x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{5}{1-(5x)^2}}=\\cfrac{5}{1- 25x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"535\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/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-4f6bb636493edeb7a8c2b9ee5e7890a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(e^{3x})\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para resolver a derivada desta fun\u00e7\u00e3o, precisamos aplicar a f\u00f3rmula da derivada da arcotangente hiperb\u00f3lica, que \u00e9 a seguinte:<\/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-ccadba79123a5f5e81d2b777d5cb9ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"411\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Neste caso temos uma fun\u00e7\u00e3o composta, pois existe uma fun\u00e7\u00e3o exponencial no argumento da fun\u00e7\u00e3o trigonom\u00e9trica. Portanto, precisamos usar a regra da cadeia para encontrar a derivada de toda a fun\u00e7\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-32563dc460ac67e1fbb529d3485e8297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccoth}(e^{3x}) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{3\\cdot e^{3x}}{1-\\left(e^{3x}\\right)^2}=\\cfrac{3e^{3x}}{1-3^{6x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"532\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"articulos-relacionados\"><\/span> itens similares<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico\/\">Derivada do arco tangente hiperb\u00f3lico<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente-hiperbolica\/\">Derivada da tangente hiperb\u00f3lica<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-cotangente-hiperbolica\/\">Derivada da cotangente hiperb\u00f3lica<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-arco-tangente\/\">Derivada da arcotangente<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-1\/\">Derivada de arco tangente<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-cotangente\/\">derivado da cotangente<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente\/\">derivada da tangente<\/a><\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar o arco tangente hiperb\u00f3lico de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos resolvidos da derivada da arcotangente hiperb\u00f3lica. F\u00f3rmula para a derivada da arcotangente hiperb\u00f3lica A derivada do arco tangente hiperb\u00f3lico de x \u00e9 um sobre um menos x ao quadrado. Portanto, a derivada do arco tangente hiperb\u00f3lico de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/\"> <span class=\"screen-reader-text\">Derivada da arcotangente 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-399","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 arcotangente 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-do-arco-tangente-hiperbolico-2\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada da arcotangente hiperb\u00f3lica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Neste artigo explicamos como derivar o arco tangente hiperb\u00f3lico de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos resolvidos da derivada da arcotangente hiperb\u00f3lica. F\u00f3rmula para a derivada da arcotangente hiperb\u00f3lica A derivada do arco tangente hiperb\u00f3lico de x \u00e9 um sobre um menos x ao quadrado. 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Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/","og_locale":"pt_BR","og_type":"article","og_title":"Derivada da arcotangente hiperb\u00f3lica - Mathority","og_description":"Neste artigo explicamos como derivar o arco tangente hiperb\u00f3lico de uma fun\u00e7\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos resolvidos da derivada da arcotangente hiperb\u00f3lica. F\u00f3rmula para a derivada da arcotangente hiperb\u00f3lica A derivada do arco tangente hiperb\u00f3lico de x \u00e9 um sobre um menos x ao quadrado. Portanto, a derivada do arco tangente hiperb\u00f3lico de &hellip; Derivada da arcotangente hiperb\u00f3lica Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/","article_published_time":"2023-07-03T01:21:27+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42f0b83c10ae509a0cda1aa4abf25b92_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Derivada da arcotangente hiperb\u00f3lica","datePublished":"2023-07-03T01:21:27+00:00","dateModified":"2023-07-03T01:21:27+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/"},"wordCount":304,"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-do-arco-tangente-hiperbolico-2\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/","url":"https:\/\/mathority.org\/pt\/derivada-do-arco-tangente-hiperbolico-2\/","name":"Derivada da arcotangente hiperb\u00f3lica - 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