{"id":31,"date":"2023-09-17T11:02:40","date_gmt":"2023-09-17T11:02:40","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-da-tangente\/"},"modified":"2023-09-17T11:02:40","modified_gmt":"2023-09-17T11:02:40","slug":"derivada-da-tangente","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-da-tangente\/","title":{"rendered":"Derivada da tangente"},"content":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 como a fun\u00e7\u00e3o tangente \u00e9 derivada. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de derivada da tangente e at\u00e9 praticar com exerc\u00edcios resolvidos passo a passo. Finalmente, tamb\u00e9m demonstramos a f\u00f3rmula da derivada tangente e mostramos a f\u00f3rmula da derivada tangente inversa. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-derivada-de-la-tangente\"><\/span> Qual \u00e9 a derivada da tangente?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada da tangente de x \u00e9 igual a 1 sobre o quadrado do cosseno de x.<\/strong> A derivada da tangente de x tamb\u00e9m \u00e9 equivalente ao quadrado da secante de x, e 1 mais o quadrado da tangente de x.<\/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-dfb81626a982a908c4e517b1ecb748e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{tan}(x)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=\\cfrac{1}{\\text{cos}^2(x)}=\\text{sec}^2(x)=1+\\text{tan}^2(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Todas as express\u00f5es s\u00e3o equivalentes, portanto a fun\u00e7\u00e3o tangente possui tr\u00eas f\u00f3rmulas poss\u00edveis para deriv\u00e1-la.<\/p>\n<p> Por outro lado, quando no argumento da tangente temos uma fun\u00e7\u00e3o diferente de x (vamos cham\u00e1-la de u), devemos aplicar a regra da cadeia. A derivada da tangente de voc\u00ea \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-1ad272ab857ecf57ebc79e68a4370fc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{tan}(u)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=\\cfrac{u'}{\\text{cos}^2(u)}=\\text{sec}^2(u)\\cdot u'=\\left(1+\\text{tan}^2(u)\\right)\\cdot u'\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Resumindo, a regra da derivada tangente pode ser resumida da seguinte forma: <\/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-de-la-tangente.webp\" alt=\"derivada tangente\" class=\"wp-image-1929\" width=\"418\" height=\"365\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-tangente\"><\/span> Exemplos de derivada tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dada a f\u00f3rmula da derivada tangente, nesta se\u00e7\u00e3o resolveremos v\u00e1rios exemplos deste tipo de derivadas trigonom\u00e9tricas para que voc\u00ea entenda como derivar a fun\u00e7\u00e3o tangente. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-la-tangente-de-2x\"><\/span> Exemplo 1: Derivada da tangente 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-f238988096540344626a3079f65a0753_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para calcular a derivada da tangente, voc\u00ea pode usar uma das tr\u00eas f\u00f3rmulas que vimos acima. Neste caso, usaremos a f\u00f3rmula do cosseno:<\/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-47c1f81edd8b591f33ab986d4de73a34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{\\text{cos}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A fun\u00e7\u00e3o 2x \u00e9 linear, ent\u00e3o sua derivada \u00e9 2. Portanto, a derivada da tangente de 2x \u00e9 2 sobre o quadrado do cosseno de 2x: <\/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-e18c22b2cabb93a6081363bc618840b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(2x)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2}{\\text{cos}^2(2x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"405\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-la-tangente-de-x-al-cuadrado\"><\/span> Exemplo 2: Derivada da tangente 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-9c6defebe72239c5288ece20976d9a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste exemplo, a fun\u00e7\u00e3o de argumento tangente n\u00e3o \u00e9 um x, mas uma fun\u00e7\u00e3o com derivada. O que significa que precisamos de aplicar a regra da cadeia para deriv\u00e1-lo.<\/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-47c1f81edd8b591f33ab986d4de73a34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{\\text{cos}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A derivada de x ao quadrado \u00e9 2x, ent\u00e3o a derivada da tangente de x <sup>2<\/sup> \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-111ca482f4c688c676c10b2ed80d6567_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2x}{\\text{cos}^2(x^2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"403\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-la-tangente-al-cubo\"><\/span> Exemplo 3: Derivada da tangente ao cubo<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-a70568c32830f1f20ab7a5885bf999ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}^3(9x^2-4x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"172\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste problema temos uma fun\u00e7\u00e3o composta, portanto tamb\u00e9m precisaremos utilizar a regra da cadeia para derivar a tangente.<\/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-47c1f81edd8b591f33ab986d4de73a34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{\\text{cos}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, a tangente \u00e9 elevada \u00e0 pot\u00eancia de 3, o que significa que antes de aplicar a f\u00f3rmula da derivada da tangente deve-se usar a f\u00f3rmula da derivada de uma pot\u00eancia: <\/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-424a7372a1d97a5c17a86d6253666164_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=3\\text{tan}^2(9x^2-4x)\\cdot \\cfrac{18x-4}{\\text{cos}^2(9x^2-4x)} \\\\[2ex]&amp;=\\cfrac{3\\text{tan}^2(9x^2-4x)\\cdot(18x-4)}{\\text{cos}^2(9x^2-4x)}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"314\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-la-tangente-inversa\"><\/span> Derivada da tangente inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como qualquer fun\u00e7\u00e3o inversa, a fun\u00e7\u00e3o tangente tamb\u00e9m possui uma inversa, a fun\u00e7\u00e3o arcotangente. Embora a f\u00f3rmula para deriv\u00e1-la n\u00e3o seja semelhante \u00e0 f\u00f3rmula da tangente, mostramos-lhe porque pode ser \u00fatil em alguns casos.<\/p>\n<p> A <strong>derivada da tangente inversa<\/strong> de uma fun\u00e7\u00e3o \u00e9 o quociente da derivada da fun\u00e7\u00e3o dividida por um mais a referida fun\u00e7\u00e3o 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-d26f5f19ebcdab218e6d1924e18845f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}^{-1}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"398\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Por exemplo, a derivada da tangente inversa de 3x \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-bdabf1792179bdd9281695a65dcd0912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tan}^{-1}(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{3}{1+(3x)^2}=\\cfrac{3}{1+9x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"513\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-de-la-tangente\"><\/span> Exerc\u00edcios resolvidos sobre a derivada da tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Calcule a derivada das seguintes fun\u00e7\u00f5es tangentes: <\/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-e0c187638a259878b3cf6382751c2718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=\\text{tan}(3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/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-573c097d9cddb7837803e4aceaec362a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=\\text{tan}(x^3-10x^2+8)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"228\" style=\"vertical-align: -5px;\"><\/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-4e140710c7f1fea51f3fe280f30fdb1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f(x)=\\text{tan}^2\\left(\\frac{x}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"153\" style=\"vertical-align: -12px;\"><\/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-49b86302e59ffb338f425f4e5a97be89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\text{tan}\\left(e^{2x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"151\" style=\"vertical-align: -7px;\"><\/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-4af042cb47d433a0eeff44d9c5349873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=\\text{tan}\\bigl(\\ln(4x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"171\" style=\"vertical-align: -7px;\"><\/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-21464f892729c58a42f796e0d35f6a89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f(x)=\\text{tan}\\left(\\sqrt{3x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"159\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-240652ca6b9fbabd52d65974bf3e4793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=\\cfrac{3}{\\text{cos}^2(3x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"156\" style=\"vertical-align: -17px;\"><\/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-5166bb8a8d3af33cc82165b63e2b6a52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=\\cfrac{3x^2-20x}{\\text{cos}^2(x^3-10x^2+8)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"241\" style=\"vertical-align: -17px;\"><\/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-94a4f0132583e89119dae1b25be65adf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f'(x)=2\\text{tan}\\left(\\frac{x}{2}\\right)\\cdot \\frac{1}{\\text{cos}^2\\left(\\frac{x}{2}\\right)}\\cdot \\frac{1}{2}=\\frac{\\text{tan}\\left(\\frac{x}{2}\\right)}{\\text{cos}^2\\left(\\frac{x}{2}\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"354\" style=\"vertical-align: -20px;\"><\/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-2d4f13da08be6a975b3e8710f5aee58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=\\cfrac{2e^{2x}}{\\text{cos}^2(e^{2x})}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"160\" style=\"vertical-align: -17px;\"><\/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-cd7569c56712ac42fa2fe9300d9e4896_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=\\cfrac{\\frac{4}{4x}}{\\text{cos}^2\\bigl(\\ln(4x)\\bigr)}=\\cfrac{1}{x\\cdot\\text{cos}^2\\bigl(\\ln(4x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"329\" style=\"vertical-align: -20px;\"><\/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-9a51deebd15244b70a6917a9ea2a456a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f'(x)=\\cfrac{\\frac{3}{2\\sqrt{3x}}}{\\text{cos}^2\\left(\\sqrt{3x}\\right)}=\\cfrac{3}{2\\sqrt{3x}\\cdot \\text{cos}^2\\left(\\sqrt{3x}\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"339\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-derivada-de-la-tangente\"><\/span> Prova da derivada da tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea possa verificar que esta n\u00e3o \u00e9 uma express\u00e3o inventada, nesta se\u00e7\u00e3o demonstraremos a f\u00f3rmula da derivada da tangente utilizando a defini\u00e7\u00e3o matem\u00e1tica de tangente.<\/p>\n<p> Para fazer isso, partiremos da identidade trigonom\u00e9trica que conecta as tr\u00eas raz\u00f5es trigonom\u00e9tricas:<\/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-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Se usarmos a <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-um-quociente-de-divisao\/\">f\u00f3rmula da derivada de uma divis\u00e3o<\/a><\/span> , a derivada seria: <\/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-334dc33e2ef413b8d99dd7de50cebc74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\left(\\text{tan}(x)\\right)'=\\left(\\frac{\\text{sen}(x)}{\\text{cos}(x)}\\right)'\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"173\" style=\"vertical-align: -17px;\"><\/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-2b05dcbadd57bacdab9a7d4eda718e3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{\\text{cos}(x)\\cdot \\text{cos}(x)+\\text{sen}(x)\\text{sen}(x) }{\\text{cos}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"308\" style=\"vertical-align: -17px;\"><\/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-cf6fae22356a5ba2fe4f327843c0da81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{\\text{cos}^2(x)+\\text{sen}^2(x)}{\\text{cos}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"214\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Mas, utilizando a identidade trigonom\u00e9trica fundamental, sabemos que o quadrado do seno mais o quadrado do cosseno \u00e9 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-92d80771f891319379b2e756c5524aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(x)+\\text{cos}^2(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/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-c737664b7a2ec3456d700d4939c15806_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{1}{\\text{cos}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"136\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E assim j\u00e1 chegamos \u00e0 primeira f\u00f3rmula para a derivada da tangente. Al\u00e9m disso, a secante \u00e9 o inverso multiplicativo do cosseno, ent\u00e3o a segunda express\u00e3o tamb\u00e9m \u00e9 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-41f558939bb7b23e97112acb0630c4bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\text{sec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"131\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por fim, a terceira regra da derivada tangente pode ser comprovada transformando a fra\u00e7\u00e3o da etapa anterior em uma soma de fra\u00e7\u00f5es: <\/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-cf6fae22356a5ba2fe4f327843c0da81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{\\text{cos}^2(x)+\\text{sen}^2(x)}{\\text{cos}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"214\" style=\"vertical-align: -17px;\"><\/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-187bc0e3bc1c35a7dfd18197b94aa845_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=\\cfrac{\\text{cos}^2(x)}{\\text{cos}^2(x)}+\\cfrac{\\text{sen}^2(x)}{\\text{cos}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"216\" style=\"vertical-align: -17px;\"><\/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-257a8cde825ce1b73cf5849d6a387507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=1+\\text{tan}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 como a fun\u00e7\u00e3o tangente \u00e9 derivada. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de derivada da tangente e at\u00e9 praticar com exerc\u00edcios resolvidos passo a passo. Finalmente, tamb\u00e9m demonstramos a f\u00f3rmula da derivada tangente e mostramos a f\u00f3rmula da derivada tangente inversa. Qual \u00e9 a derivada da tangente? A derivada da tangente de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-da-tangente\/\"> <span class=\"screen-reader-text\">Derivada da tangente<\/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-31","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>\u25b7 Derivada da tangente (f\u00f3rmula e exerc\u00edcios resolvidos)<\/title>\n<meta name=\"description\" content=\"Explicamos como derivar a fun\u00e7\u00e3o tangente (f\u00f3rmula). 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