{"id":394,"date":"2023-07-03T03:43:52","date_gmt":"2023-07-03T03:43:52","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-da-secante-hiperbolica\/"},"modified":"2023-07-03T03:43:52","modified_gmt":"2023-07-03T03:43:52","slug":"derivada-da-secante-hiperbolica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-da-secante-hiperbolica\/","title":{"rendered":"Derivada secante hiperb\u00f3lica"},"content":{"rendered":"<p>Neste artigo explicamos como derivar a secante hiperb\u00f3lica de uma fun\u00e7\u00e3o. Voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada da secante hiperb\u00f3lica e v\u00e1rios exemplos trabalhados deste tipo de derivada. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-secante-hiperbolica\"><\/span> F\u00f3rmula para a derivada da secante hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada da secante hiperb\u00f3lica de x \u00e9 igual a menos o produto da secante hiperb\u00f3lica de x vezes a tangente hiperb\u00f3lica de x.<\/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-c07ba93179ede436aa585653d7c4e07f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(x)\\cdot \\text{tanh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"476\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada da secante hiperb\u00f3lica de uma fun\u00e7\u00e3o<\/strong> \u00e9 menos o produto da secante hiperb\u00f3lica da fun\u00e7\u00e3o vezes a tangente hiperb\u00f3lica da fun\u00e7\u00e3o vezes a derivada da referida 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-f51426f6d6f9cb5df2135bf16c720ce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(u)\\cdot \\text{tanh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"504\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Resumindo, a f\u00f3rmula para a derivada da fun\u00e7\u00e3o secante 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-la-secante-hyperbolique.webp\" alt=\"derivado da secante hiperb\u00f3lica\" class=\"wp-image-2756\" width=\"473\" height=\"297\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Observe que ambas as express\u00f5es pertencem, na verdade, a uma \u00fanica f\u00f3rmula. A \u00fanica diferen\u00e7a \u00e9 que na segunda f\u00f3rmula \u00e9 aplicada a regra da cadeia. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-secante-hiperbolica\"><\/span> Exemplos de derivada da secante hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agora que conhecemos a f\u00f3rmula da derivada da secante hiperb\u00f3lica, veremos v\u00e1rios exerc\u00edcios resolvidos deste tipo de derivada trigonom\u00e9trica.<\/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-1077c4d8341190071e3d52fc9b7dd587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste exemplo, temos uma fun\u00e7\u00e3o diferente de x no argumento da secante hiperb\u00f3lica, portanto, para deriv\u00e1-la, precisamos usar a f\u00f3rmula da 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-f51426f6d6f9cb5df2135bf16c720ce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(u)\\cdot \\text{tanh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"504\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como a fun\u00e7\u00e3o 2x \u00e9 linear, sua derivada \u00e9 2. Portanto, para encontrar a derivada, simplesmente substitu\u00edmos u por 2x e u&#8217; por 2 na 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-947df587e6457c9a82023c6ea76e3d1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(2x)\\cdot \\text{tanh}(2x)\\cdot 2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"525\" style=\"vertical-align: -5px;\"><\/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-f759fd3a41187cc0764c458c21481eb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A fun\u00e7\u00e3o deste exerc\u00edcio \u00e9 composta, pois a secante hiperb\u00f3lica tem outra fun\u00e7\u00e3o no seu argumento. Devemos, portanto, usar a f\u00f3rmula da secante hiperb\u00f3lica com a regra da cadeia para fazer sua deriva\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-f51426f6d6f9cb5df2135bf16c720ce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(u)\\cdot \\text{tanh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"504\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A derivada de x elevada a 2 d\u00e1 2x, ent\u00e3o a derivada da secante hiperb\u00f3lica de x ao quadrado \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-2afe9c3dc5fc592bf5714f34f8016ef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sech}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sech}(x^2)\\cdot \\text{tanh}(x^2)\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"532\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar a secante hiperb\u00f3lica de uma fun\u00e7\u00e3o. Voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada da secante hiperb\u00f3lica e v\u00e1rios exemplos trabalhados deste tipo de derivada. F\u00f3rmula para a derivada da secante hiperb\u00f3lica A derivada da secante hiperb\u00f3lica de x \u00e9 igual a menos o produto da secante hiperb\u00f3lica de x vezes a &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-da-secante-hiperbolica\/\"> <span class=\"screen-reader-text\">Derivada secante 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-394","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 secante 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-secante-hiperbolica\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada secante hiperb\u00f3lica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Neste artigo explicamos como derivar a secante hiperb\u00f3lica de uma fun\u00e7\u00e3o. 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