{"id":398,"date":"2023-07-03T01:50:55","date_gmt":"2023-07-03T01:50:55","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/"},"modified":"2023-07-03T01:50:55","modified_gmt":"2023-07-03T01:50:55","slug":"derivado-de-larcosecante-hiperbolico","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/","title":{"rendered":"Derivada do arco cossecante hiperb\u00f3lico"},"content":{"rendered":"<p>Neste artigo explicamos como derivar o arco secante hiperb\u00f3lico de uma fun\u00e7\u00e3o. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos pr\u00e1ticos da derivada do arcossecante hiperb\u00f3lico. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-arcocosecante-hiperbolica\"><\/span> F\u00f3rmula para a derivada do arco cossecante hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada do arco cossecante hiperb\u00f3lico de x \u00e9 igual a menos 1 dividido pelo produto de x vezes a raiz de um mais 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-0367e01bffbaea95b002c0261e6597d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-1}{x\\sqrt{1+x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do arco cossecante hiperb\u00f3lico de uma fun\u00e7\u00e3o<\/strong> \u00e9 menos a derivada da referida fun\u00e7\u00e3o dividida pelo produto da fun\u00e7\u00e3o vezes a raiz de um mais a fun\u00e7\u00e3o quadrada.<\/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-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Em resumo, a f\u00f3rmula para calcular a derivada da fun\u00e7\u00e3o arcossecante hiperb\u00f3lica \u00e9: <\/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-larccosecante-hyperbolique.webp\" alt=\"derivada do arco cossecante hiperb\u00f3lico\" class=\"wp-image-2797\" width=\"365\" height=\"266\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Embora tenhamos colocado duas f\u00f3rmulas, isso n\u00e3o significa que sejam diferentes. Se voc\u00ea olhar de perto, a segunda f\u00f3rmula \u00e9 igual \u00e0 primeira, mas aplica a regra da cadeia. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-arcocosecante-hiperbolica\"><\/span> Exemplos da derivada do arco cossecante hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dada a f\u00f3rmula para a derivada da cossecante do arco hiperb\u00f3lico, derivaremos ent\u00e3o duas dessas fun\u00e7\u00f5es para que voc\u00ea possa ver como isso \u00e9 feito.<\/p>\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-24a3e6958bf4fd28f73062bb78280dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste exerc\u00edcio devemos utilizar a f\u00f3rmula da derivada do arco cossecante hiperb\u00f3lico com a regra da cadeia, pois no argumento existe uma fun\u00e7\u00e3o diferente 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-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o, para encontrar a derivada, precisamos substituir u por 3x e u&#8217; por sua derivada, que \u00e9 3:<\/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-aa1b0c4d4233e274d3996620bacdb869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-3}{3x\\sqrt{1+(3x)^2}}=\\cfrac{-3}{3x\\sqrt{1+9x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"582\" style=\"vertical-align: -16px;\"><\/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-a99a903e923a1fd4dc9442e903f244ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(x^5-2x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso temos uma fun\u00e7\u00e3o polinomial no argumento da arcossecante hiperb\u00f3lica, ent\u00e3o tamb\u00e9m precisamos usar a 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-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o colocamos a derivada da fun\u00e7\u00e3o argumento no numerador da fra\u00e7\u00e3o, e no denominador trocamos u pela fun\u00e7\u00e3o polinomial: <\/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-6257584425bd348ba75c7680d8ff6f70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f(x)=\\text{arccsch}(x^5-2x^3) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}f'(x)&amp;=\\cfrac{-(5x^4-6x^2)}{(x^5-2x^3)\\sqrt{1+(x^5-2x^3)^2}}\\\\[1.5ex] &amp;=\\cfrac{-5x^4+6x^2}{(x^5-2x^3)\\sqrt{1+(x^5-2x^3)^2}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"634\" style=\"vertical-align: 0px;\"><\/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\/derivado-de-arcosina-hiperbolico\/\">Derivada do arco cosseno hiperb\u00f3lico<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-cossecante-hiperbolica\/\">Derivada da cossecante hiperb\u00f3lica<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-cosseno-hiperbolico\/\">Derivada do cosseno hiperb\u00f3lico<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivado-de-arcossecante\/\">Derivada do arco cossecante<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/deriva-da-cossecante\/\">derivado da cosecante<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivado-de-larcosina-1\/\">Derivada do arco cosseno<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/deriva-do-cosseno\/\">derivada do cosseno<\/a><\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar o arco secante hiperb\u00f3lico de uma fun\u00e7\u00e3o. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos pr\u00e1ticos da derivada do arcossecante hiperb\u00f3lico. F\u00f3rmula para a derivada do arco cossecante hiperb\u00f3lico A derivada do arco cossecante hiperb\u00f3lico de x \u00e9 igual a menos 1 dividido pelo produto de x vezes a raiz de um mais x &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\"> <span class=\"screen-reader-text\">Derivada do arco cossecante hiperb\u00f3lico<\/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-398","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 do arco cossecante hiperb\u00f3lico - 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\/derivado-de-larcosecante-hiperbolico\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada do arco cossecante hiperb\u00f3lico - Mathority\" \/>\n<meta property=\"og:description\" content=\"Neste artigo explicamos como derivar o arco secante hiperb\u00f3lico de uma fun\u00e7\u00e3o. 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F\u00f3rmula para a derivada do arco cossecante hiperb\u00f3lico A derivada do arco cossecante hiperb\u00f3lico de x \u00e9 igual a menos 1 dividido pelo produto de x vezes a raiz de um mais x &hellip; Derivada do arco cossecante hiperb\u00f3lico Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-03T01:50:55+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0367e01bffbaea95b002c0261e6597d0_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=\"2 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Derivada do arco cossecante hiperb\u00f3lico\",\"datePublished\":\"2023-07-03T01:50:55+00:00\",\"dateModified\":\"2023-07-03T01:50:55+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\"},\"wordCount\":315,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\",\"url\":\"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/\",\"name\":\"Derivada do arco cossecante hiperb\u00f3lico - 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Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos pr\u00e1ticos da derivada do arcossecante hiperb\u00f3lico. F\u00f3rmula para a derivada do arco cossecante hiperb\u00f3lico A derivada do arco cossecante hiperb\u00f3lico de x \u00e9 igual a menos 1 dividido pelo produto de x vezes a raiz de um mais x &hellip; Derivada do arco cossecante hiperb\u00f3lico Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/","article_published_time":"2023-07-03T01:50:55+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0367e01bffbaea95b002c0261e6597d0_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"2 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Derivada do arco cossecante hiperb\u00f3lico","datePublished":"2023-07-03T01:50:55+00:00","dateModified":"2023-07-03T01:50:55+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/"},"wordCount":315,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Derivados"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/","url":"https:\/\/mathority.org\/pt\/derivado-de-larcosecante-hiperbolico\/","name":"Derivada do arco cossecante hiperb\u00f3lico - 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