{"id":384,"date":"2023-07-03T15:44:25","date_gmt":"2023-07-03T15:44:25","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/"},"modified":"2023-07-03T15:44:25","modified_gmt":"2023-07-03T15:44:25","slug":"derivada-seno-hiperbolica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/","title":{"rendered":"Derivada do seno hiperb\u00f3lico"},"content":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 como derivar o seno hiperb\u00f3lico (f\u00f3rmula). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos resolvidos da derivada do seno hiperb\u00f3lico. E por fim, provamos a f\u00f3rmula da derivada deste tipo de fun\u00e7\u00e3o trigonom\u00e9trica. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-del-seno-hiperbolico\"><\/span> F\u00f3rmula derivada do seno hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada do seno hiperb\u00f3lico de x \u00e9 o cosseno hiperb\u00f3lico 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-5824c2bfe983f8a9b725fab69a97ecca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"394\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do seno hiperb\u00f3lico de uma fun\u00e7\u00e3o<\/strong> \u00e9 igual ao produto do cosseno hiperb\u00f3lico da fun\u00e7\u00e3o e a derivada dessa 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-8d49318fdfae22c716e856e18e7440db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"422\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Na verdade, as duas f\u00f3rmulas acima s\u00e3o iguais, a \u00fanica diferen\u00e7a \u00e9 que na segunda f\u00f3rmula aplicamos a regra da cadeia. E como a derivada de x \u00e9 1, isso n\u00e3o altera a fun\u00e7\u00e3o. <\/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-du-sinus-hyperbolique.webp\" alt=\"derivada do seno hiperb\u00f3lico\" class=\"wp-image-2021\" width=\"423\" height=\"286\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver, a f\u00f3rmula da derivada senoidal hiperb\u00f3lica \u00e9 muito semelhante \u00e0 <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-sinusal\/\">f\u00f3rmula da derivada senoidal<\/a><\/span> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-seno-hiperbolico\"><\/span> Exemplos da derivada seno hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Uma vez que j\u00e1 vimos o que \u00e9 a f\u00f3rmula da derivada do seno hiperb\u00f3lico, passamos agora a resolver v\u00e1rios exemplos da derivada do seno hiperb\u00f3lico. Ent\u00e3o, certamente voc\u00ea n\u00e3o tem d\u00favidas sobre como isso \u00e9 feito. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-del-seno-hiperbolico-de-2x\"><\/span> Exemplo 1: Derivada do seno hiperb\u00f3lico 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-11e5d630f508c66d1be884d9a6454d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, no argumento do seno hiperb\u00f3lico, temos uma fun\u00e7\u00e3o diferente de x, portanto, devemos utilizar a f\u00f3rmula da derivada do seno hiperb\u00f3lico com a regra da cadeia para encontrar a 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-8d49318fdfae22c716e856e18e7440db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"422\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A derivada de 2x \u00e9 2, ent\u00e3o a derivada do seno hiperb\u00f3lico de 2x ser\u00e1 o cosseno hiperb\u00f3lico de 2x vezes 2. <\/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-4e763d6973e52217ac14f6f6b3344737_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(2x)\\cdot 2=2\\text{cosh}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"533\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-del-seno-hiperbolico-de-x-al-cuadrado\"><\/span> Exemplo 2: Derivada do seno hiperb\u00f3lico 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-432b20159f3f131533249e4ee44702e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula para a derivada da fun\u00e7\u00e3o seno hiperb\u00f3lica \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-8d49318fdfae22c716e856e18e7440db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"422\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por outro lado, a derivada da fun\u00e7\u00e3o quadr\u00e1tica x <sup>2<\/sup> \u00e9 2x. A derivada de toda a fun\u00e7\u00e3o \u00e9, portanto: <\/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-56db45ee5a51f7ed624f7e88ce4cf906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cosh}(x^2)\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"443\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-formula-de-la-derivada-del-seno-hiperbolico\"><\/span> Prova da f\u00f3rmula da derivada do seno hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Por fim, demonstraremos a f\u00f3rmula da derivada do seno hiperb\u00f3lico. Para isso, partiremos da defini\u00e7\u00e3o matem\u00e1tica do seno hiperb\u00f3lico:<\/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-ef003e8f8367d5cacacb4d67fb8dbd01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{senh}(x)=\\cfrac{e^x-e^{-x}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"150\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Deduzimos agora os dois lados da igualdade:<\/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-39aa6ef096997dc19e3850036a95b0d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\bigl(\\text{senh}(x)\\bigr)'=\\left(\\frac{e^x-e^{-x}}{2}\\right)'\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"201\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Para derivar o lado direito da equa\u00e7\u00e3o, usaremos a f\u00f3rmula da derivada da divis\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-c11def15692e1374b5f75743bfc04b40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}'(x)=\\frac{(e^x+e^{-x})\\cdot 2}{2^2}=\\frac{e^x+e^{-x}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"284\" style=\"vertical-align: -12px;\"><\/p>\n<\/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-funcao-exponencial\/\">derivada da fun\u00e7\u00e3o exponencial com base e<\/a><\/span><\/p>\n<p> E cheg\u00e1mos precisamente \u00e0 express\u00e3o que define o cosseno hiperb\u00f3lico. Para que a derivada do seno hiperb\u00f3lico seja provada:<\/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-fb3756efaf241b9435c36856a5dad2b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}'(x)=\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"143\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 como derivar o seno hiperb\u00f3lico (f\u00f3rmula). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos resolvidos da derivada do seno hiperb\u00f3lico. E por fim, provamos a f\u00f3rmula da derivada deste tipo de fun\u00e7\u00e3o trigonom\u00e9trica. F\u00f3rmula derivada do seno hiperb\u00f3lico A derivada do seno hiperb\u00f3lico de x \u00e9 o cosseno hiperb\u00f3lico de x. Portanto, a derivada &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/\"> <span class=\"screen-reader-text\">Derivada do seno 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-384","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 seno hiperb\u00f3lico - Matoridade<\/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-seno-hiperbolica\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada do seno hiperb\u00f3lico - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea descobrir\u00e1 como derivar o seno hiperb\u00f3lico (f\u00f3rmula). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos resolvidos da derivada do seno hiperb\u00f3lico. E por fim, provamos a f\u00f3rmula da derivada deste tipo de fun\u00e7\u00e3o trigonom\u00e9trica. F\u00f3rmula derivada do seno hiperb\u00f3lico A derivada do seno hiperb\u00f3lico de x \u00e9 o cosseno hiperb\u00f3lico de x. 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Matoridade","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-seno-hiperbolica\/","og_locale":"pt_BR","og_type":"article","og_title":"Derivada do seno hiperb\u00f3lico - Matoridade","og_description":"Aqui voc\u00ea descobrir\u00e1 como derivar o seno hiperb\u00f3lico (f\u00f3rmula). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos resolvidos da derivada do seno hiperb\u00f3lico. E por fim, provamos a f\u00f3rmula da derivada deste tipo de fun\u00e7\u00e3o trigonom\u00e9trica. F\u00f3rmula derivada do seno hiperb\u00f3lico A derivada do seno hiperb\u00f3lico de x \u00e9 o cosseno hiperb\u00f3lico de x. Portanto, a derivada &hellip; Derivada do seno hiperb\u00f3lico Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/","article_published_time":"2023-07-03T15:44:25+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5824c2bfe983f8a9b725fab69a97ecca_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\/derivada-seno-hiperbolica\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Derivada do seno hiperb\u00f3lico","datePublished":"2023-07-03T15:44:25+00:00","dateModified":"2023-07-03T15:44:25+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/"},"wordCount":396,"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-seno-hiperbolica\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/","url":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/","name":"Derivada do seno hiperb\u00f3lico - Matoridade","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-07-03T15:44:25+00:00","dateModified":"2023-07-03T15:44:25+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/derivada-seno-hiperbolica\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Derivada do seno hiperb\u00f3lico"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/384","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=384"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/384\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=384"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=384"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}