{"id":392,"date":"2023-07-03T04:31:43","date_gmt":"2023-07-03T04:31:43","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivado-de-arcossecante\/"},"modified":"2023-07-03T04:31:43","modified_gmt":"2023-07-03T04:31:43","slug":"derivado-de-arcossecante","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivado-de-arcossecante\/","title":{"rendered":"Derivada do arco cossecante"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 qual \u00e9 a f\u00f3rmula da derivada do arcossecante. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exerc\u00edcios resolvidos para derivadas do arco cossecante de uma fun\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-del-arcocosecante\"><\/span> F\u00f3rmula derivada de arcossecante<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada do arco-secante de x \u00e9 menos 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-f0d7defc746af3a5f6f4f0c6b9032eb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(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=\"469\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do arco-secante de uma fun\u00e7\u00e3o<\/strong> \u00e9 igual a menos o 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-319f0b4d00e6c138ce50d898a9948363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(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=\"469\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Na verdade, as duas f\u00f3rmulas anteriores s\u00e3o iguais, mas na segunda express\u00e3o \u00e9 aplicada a regra da cadeia. Na verdade, se voc\u00ea substituir a fun\u00e7\u00e3o identidade x em u, obter\u00e1 a derivada do arcossecante de x, j\u00e1 que a derivada de x \u00e9 um. <\/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\/formule-derivee-de-larc-cosecant.webp\" alt=\"f\u00f3rmula derivada de arcossecante\" class=\"wp-image-2734\" width=\"463\" height=\"308\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea bem sabe, a arcossecante \u00e9 a fun\u00e7\u00e3o trigonom\u00e9trica inversa da cossecante, por\u00e9m suas derivadas s\u00e3o bem diferentes. Voc\u00ea pode ver a f\u00f3rmula para este outro tipo de fun\u00e7\u00e3o trigonom\u00e9trica no seguinte link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/deriva-da-cossecante\/\">derivada da cossecante<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-arcocosecante\"><\/span> Exemplos de derivada do arco cossecante<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Vendo o que \u00e9 a regra da derivada arcossecante, resolveremos a seguir dois exemplos deste tipo de derivada. Mas se voc\u00ea ainda tiver alguma d\u00favida sobre como derivar o arco cossecante, pode nos perguntar nos coment\u00e1rios.<\/p>\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 cossecante da fun\u00e7\u00e3o quadr\u00e1tica x <sup>2<\/sup> .<\/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-fca6065258c2a26d98bf6b5fb0eb4b54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-2a112ce1-0dbe-43d5-95b3-4d8506c1a246\"> Para calcular a derivada do arco secante de x ao quadrado, aplicamos a f\u00f3rmula que vimos acima: <\/p>\n<p class=\"has-text-align-center\" id=\"block-f73c7b5e-fb98-43c1-a3bb-52c9144eee8a\"><meta charset=\"utf-8\"><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-319f0b4d00e6c138ce50d898a9948363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(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=\"469\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p id=\"block-a4fe1876-f662-49c1-8d09-6a6c4b5528dd\"> A derivada de x elevado a dois \u00e9 2x, ent\u00e3o a derivada da fun\u00e7\u00e3o composta \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-eb4dcb62cafb03a267428465ac221f2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{2x}{x^2\\cdot \\sqrt{\\left(x^2\\right)^2-1}}=-\\cfrac{2x}{x^2\\cdot \\sqrt{x^4-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" 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-8ef686c080851299fb2bf1960a042424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(2x^3+9x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"202\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-0514a2ea-d85a-4b25-a7db-9c27533e7436\"> Precisamos de utilizar a regra da derivada do arco secante para determinar 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-319f0b4d00e6c138ce50d898a9948363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(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=\"469\" 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 ao quadrado menos 1:<\/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-ead8a1afc8e88458fea7abdab11ee836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccosec}(2x^3+9x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{6x^2+9}{(2x^3+9x)\\cdot \\sqrt{\\left(2x^3+9x\\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 f\u00f3rmula da derivada do arcossecante. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exerc\u00edcios resolvidos para derivadas do arco cossecante de uma fun\u00e7\u00e3o. F\u00f3rmula derivada de arcossecante A derivada do arco-secante de x \u00e9 menos um sobre o produto de x vezes a raiz de x ao quadrado menos 1. Portanto, &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivado-de-arcossecante\/\"> <span class=\"screen-reader-text\">Derivada do arco cossecante<\/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-392","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 - 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-arcossecante\/\" \/>\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 - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 qual \u00e9 a f\u00f3rmula da derivada do arcossecante. 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Al\u00e9m disso, voc\u00ea poder\u00e1 ver exerc\u00edcios resolvidos para derivadas do arco cossecante de uma fun\u00e7\u00e3o. F\u00f3rmula derivada de arcossecante A derivada do arco-secante de x \u00e9 menos um sobre o produto de x vezes a raiz de x ao quadrado menos 1. 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