{"id":391,"date":"2023-07-03T05:06:38","date_gmt":"2023-07-03T05:06:38","guid":{"rendered":"https:\/\/mathority.org\/pt\/deriva-de-arco-secante\/"},"modified":"2023-07-03T05:06:38","modified_gmt":"2023-07-03T05:06:38","slug":"deriva-de-arco-secante","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/deriva-de-arco-secante\/","title":{"rendered":"Derivado de arco secante"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 qual \u00e9 a derivada do arco secante (f\u00f3rmula). Voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para derivadas do arco secante de uma fun\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-del-arcosecante\"><\/span> F\u00f3rmula derivada do arco secante<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada do arco secante de x \u00e9 um sobre o produto de x vezes a raiz de x ao quadrado menos 1.<\/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-c711ae2ad7e63c11dadabe5b1fa34460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x\\cdot \\sqrt{x^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"439\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do arco secante de uma fun\u00e7\u00e3o<\/strong> \u00e9 igual ao quociente da derivada dessa fun\u00e7\u00e3o dividida pela fun\u00e7\u00e3o vezes a raiz dessa fun\u00e7\u00e3o ao quadrado menos um.<\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-8101c1f6c91528fc3a8c0a3c35509c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\sqrt{u^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"439\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Obviamente, a segunda f\u00f3rmula \u00e9 semelhante \u00e0 primeira f\u00f3rmula, a \u00fanica diferen\u00e7a entre as duas \u00e9 que a regra da cadeia \u00e9 aplicada na segunda 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\/derive-darcosecant.webp\" alt=\"derivada de arco secante\" class=\"wp-image-2718\" width=\"411\" height=\"292\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Embora possa parecer estranho por serem fun\u00e7\u00f5es inversas, a derivada do arco secante nada tem a ver com a derivada da secante. Voc\u00ea pode ver a f\u00f3rmula da derivada da secante clicando aqui:<\/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-secante\/\">derivada da secante<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-arcosecante\"><\/span> Exemplos de derivada de arco secante <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\" id=\"block-46cfc7df-b680-41c2-ad53-bd8a19834b32\"> Exemplo 1<\/h3>\n<p> Neste exemplo, veremos quanto \u00e9 a derivada do arco secante da fun\u00e7\u00e3o linear 7x.<\/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-9191ef645841e7a1f497896df1a8b14e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(7x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-2a112ce1-0dbe-43d5-95b3-4d8506c1a246\"> Para encontrar a derivada do arco secante voc\u00ea deve aplicar sua f\u00f3rmula correspondente, que \u00e9 a seguinte: <\/p>\n<p class=\"has-text-align-center\" id=\"block-f73c7b5e-fb98-43c1-a3bb-52c9144eee8a\"><meta charset=\"utf-8\"><\/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-8101c1f6c91528fc3a8c0a3c35509c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\sqrt{u^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"439\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p id=\"block-a4fe1876-f662-49c1-8d09-6a6c4b5528dd\"> A derivada da fun\u00e7\u00e3o 7x \u00e9 7, ent\u00e3o a derivada do arco secante da fun\u00e7\u00e3o 7x \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-095e6d51d8dc9dd49fa9cefb1c115ea3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(7x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{7}{7x\\cdot \\sqrt{(7x)^2-1}}=\\cfrac{1}{x\\cdot \\sqrt{49x^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"582\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\" id=\"block-1446420a-0d61-44d3-9e31-8c5935a432a7\"> Exemplo 2<\/h3>\n<p> Neste segundo exemplo, derivaremos o arco secante de uma fun\u00e7\u00e3o potencial.<\/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-109aa2df6a4b0a2d174e98c663b78dca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(x^4-5x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-0514a2ea-d85a-4b25-a7db-9c27533e7436\"> Como existe um termo diferente de x no argumento da fun\u00e7\u00e3o arco-secante, precisamos aplicar a regra da derivada do arco-secante com a regra da cadeia para derivar a fun\u00e7\u00e3o inteira. <\/p>\n<p class=\"has-text-align-center\" id=\"block-86fe6911-58a5-4f15-be8f-9e340c0a1401\"><meta charset=\"utf-8\"><\/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-8101c1f6c91528fc3a8c0a3c35509c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\sqrt{u^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"439\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p id=\"block-6abf3c5a-c400-48c6-8375-c05fcb255b20\"> Assim, no numerador escrevemos a derivada do argumento da fun\u00e7\u00e3o, e no denominador reescrevemos a fun\u00e7\u00e3o potencial e multiplicamos pela raiz quadrada da fun\u00e7\u00e3o do argumento elevada \u00e0 pot\u00eancia de 2 menos 1: <\/p>\n<p class=\"has-text-align-center\" id=\"block-99708268-e595-48a1-8a30-244af13d52f2\"><meta charset=\"utf-8\"><\/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-13a238ac3f7cb034fec6d5ae6b5b3826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arcsec}(x^4-5x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{4x^3-10x}{(x^4-5x^2)\\cdot \\sqrt{\\left(x^4-5x^2\\right)^2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"582\" style=\"vertical-align: -30px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 qual \u00e9 a derivada do arco secante (f\u00f3rmula). Voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para derivadas do arco secante de uma fun\u00e7\u00e3o. F\u00f3rmula derivada do arco secante A derivada do arco secante de x \u00e9 um sobre o produto de x vezes a raiz de x ao quadrado menos 1. Portanto, a derivada &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/deriva-de-arco-secante\/\"> <span class=\"screen-reader-text\">Derivado de arco secante<\/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-391","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 de arcosecante - 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\/deriva-de-arco-secante\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada de arcosecante - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 qual \u00e9 a derivada do arco secante (f\u00f3rmula). 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Voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para derivadas do arco secante de uma fun\u00e7\u00e3o. F\u00f3rmula derivada do arco secante A derivada do arco secante de x \u00e9 um sobre o produto de x vezes a raiz de x ao quadrado menos 1. 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