{"id":393,"date":"2023-07-03T04:11:00","date_gmt":"2023-07-03T04:11:00","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-de-arco-tangente\/"},"modified":"2023-07-03T04:11:00","modified_gmt":"2023-07-03T04:11:00","slug":"derivada-de-arco-tangente","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-de-arco-tangente\/","title":{"rendered":"Derivada da arcotangente"},"content":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada do arco tangente e explicaremos como derivar o arco tangente de uma fun\u00e7\u00e3o com exemplos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-del-arcocotangente\"><\/span> f\u00f3rmula derivada arcotangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong><strong>A derivada do arco tangente de x \u00e9 menos um dividido por um mais x ao quadrado.<\/strong><\/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-50e6d74bd307289a24eb7f891974643f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{1}{1+x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"424\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>derivada do arco tangente de uma fun\u00e7\u00e3o<\/strong> \u00e9 igual a menos a derivada dessa fun\u00e7\u00e3o dividida por 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-e4a94f9fe1b97f77b028dfe2296debcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"424\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Observe que a primeira e a segunda f\u00f3rmulas s\u00e3o iguais, a \u00fanica diferen\u00e7a \u00e9 que a regra da cadeia \u00e9 aplicada \u00e0 segunda express\u00e3o. Na verdade, se voc\u00ea substituir u por x, obter\u00e1 a primeira f\u00f3rmula, pois a derivada da fun\u00e7\u00e3o x \u00e9 1. <\/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-larctangente.webp\" alt=\"derivado do arco tangente\" class=\"wp-image-2747\" width=\"390\" height=\"281\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Embora a arcotangente seja a fun\u00e7\u00e3o inversa da cotangente, suas derivadas s\u00e3o bem diferentes. Na verdade, a cotangente de uma fun\u00e7\u00e3o tem tr\u00eas maneiras de ser derivada, voc\u00ea pode ver todas 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-cotangente\/\">f\u00f3rmula da derivada da cotangente<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-arcocotangente\"><\/span> Exemplos de derivada do arco tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de ver qual \u00e9 a f\u00f3rmula da derivada do arco tangente, aqui est\u00e3o dois exerc\u00edcios resolvidos deste tipo de derivada trigonom\u00e9trica. Lembre-se tamb\u00e9m que caso tenha alguma d\u00favida, voc\u00ea pode deixar sua pergunta abaixo 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 tangente 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-f643c789a6794a49d13eb08ba6a91fc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"145\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-2a112ce1-0dbe-43d5-95b3-4d8506c1a246\"> No argumento da arcotangente temos uma fun\u00e7\u00e3o diferente de x, ent\u00e3o precisamos aplicar a f\u00f3rmula da derivada da arcotangente com a 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-e4a94f9fe1b97f77b028dfe2296debcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"424\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p id=\"block-a4fe1876-f662-49c1-8d09-6a6c4b5528dd\"> A derivada de x elevado a dois \u00e9 2x, ent\u00e3o no numerador devemos colocar 2x e no denominador a fun\u00e7\u00e3o do argumento ao quadrado: <\/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-eac3a555cca0fee987537eb7171a9145_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{2x}{1+(\\left(x^2\\right)^2}=-\\cfrac{2x}{1+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"548\" style=\"vertical-align: -20px;\"><\/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 tangente de uma fun\u00e7\u00e3o polinomial de terceiro grau.<\/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-a62d6933c54de6f0c63f01a6edcaf0b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(x^3-9x+2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-0514a2ea-d85a-4b25-a7db-9c27533e7436\"> Usamos a regra da derivada arcotangente 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-e4a94f9fe1b97f77b028dfe2296debcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"424\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p id=\"block-6abf3c5a-c400-48c6-8375-c05fcb255b20\"> Portanto, a derivada do arco tangente da fun\u00e7\u00e3o \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-634cbaeb640fd0886691ef88d12afc52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccotg}(x^3-9x+2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{3x^2-9}{1+(x^3-9x+2)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"596\" style=\"vertical-align: -17px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada do arco tangente e explicaremos como derivar o arco tangente de uma fun\u00e7\u00e3o com exemplos. f\u00f3rmula derivada arcotangente A derivada do arco tangente de x \u00e9 menos um dividido por um mais x ao quadrado. Portanto, a derivada do arco tangente de uma fun\u00e7\u00e3o \u00e9 igual a menos &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-de-arco-tangente\/\"> <span class=\"screen-reader-text\">Derivada da arcotangente<\/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-393","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 da arcotangente - 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-de-arco-tangente\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada da arcotangente - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada do arco tangente e explicaremos como derivar o arco tangente de uma fun\u00e7\u00e3o com exemplos. f\u00f3rmula derivada arcotangente A derivada do arco tangente de x \u00e9 menos um dividido por um mais x ao quadrado. 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