{"id":34,"date":"2023-09-17T11:02:40","date_gmt":"2023-09-17T11:02:40","guid":{"rendered":"https:\/\/mathority.org\/it\/derivata-della-tangente\/"},"modified":"2023-09-17T11:02:40","modified_gmt":"2023-09-17T11:02:40","slug":"derivata-della-tangente","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/derivata-della-tangente\/","title":{"rendered":"Derivata della tangente"},"content":{"rendered":"<p>Qui scoprirai come viene derivata la funzione tangente. Inoltre, potrai vedere esempi di derivata della tangente e anche esercitarti con esercizi risolti passo dopo passo. Infine, dimostriamo anche la formula della derivata tangente e ti mostriamo la formula della derivata 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 \u00e8 la derivata della tangente?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La derivata della tangente di x \u00e8 uguale a 1 sul quadrato del coseno di x.<\/strong> Anche la derivata della tangente di x \u00e8 equivalente al quadrato della secante di x e 1 pi\u00f9 il quadrato della tangente di 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> Tutte le espressioni sono equivalenti, quindi la funzione tangente ha tre possibili formule per derivarla.<\/p>\n<p> Quando invece nell&#8217;argomento tangente abbiamo una funzione diversa da x (chiamiamola u), dobbiamo applicare la regola della catena. La derivata della tangente di u \u00e8 quindi:<\/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> In breve, la regola della derivata tangente pu\u00f2 essere riassunta come segue: <\/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=\"derivata 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> Esempi di derivata tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Data la formula della derivata tangente, in questa sezione risolveremo diversi esempi di questo tipo di derivate trigonometriche in modo che tu possa capire come derivare la funzione tangente. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-la-tangente-de-2x\"><\/span> Esempio 1: Derivata della tangente di 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> Per calcolare la derivata della tangente puoi utilizzare una delle tre formule che abbiamo visto sopra. In questo caso utilizzeremo la formula del coseno:<\/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> La funzione 2x \u00e8 lineare, quindi la sua derivata \u00e8 2. Quindi la derivata della tangente di 2x \u00e8 2 fratto il quadrato del coseno di 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> Esempio 2: Derivata della tangente di x al quadrato<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> In questo esempio, la funzione argomento tangente non \u00e8 una x, ma una funzione con una derivata. Ci\u00f2 significa che dobbiamo applicare la regola della catena per ricavarlo.<\/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> La derivata di x al quadrato \u00e8 2x, quindi la derivata della tangente di x <sup>2<\/sup> \u00e8: <\/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> Esempio 3: Derivata della tangente al 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> In questo problema abbiamo una funzione composta, quindi dovremo utilizzare anche la regola della catena per differenziare la 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> Inoltre la tangente viene elevata alla potenza di 3, il che significa che prima di applicare la formula per la derivata della tangente \u00e8 necessario utilizzare la formula per la derivata di una potenza: <\/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> Derivata della tangente inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Come ogni funzione inversa, anche la funzione tangente ha una funzione inversa, la funzione arcotangente. Anche se la formula per ricavarla non \u00e8 simile alla formula della tangente, ve la mostriamo perch\u00e9 pu\u00f2 essere utile in alcuni casi.<\/p>\n<p> La <strong>derivata dell&#8217;arcotangente<\/strong> di una funzione \u00e8 il quoziente della derivata della funzione diviso per uno pi\u00f9 detta funzione al quadrato<\/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> Ad esempio, la derivata della tangente inversa di 3x \u00e8: <\/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> Esercizi risolti sulla derivata della tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Calcolare la derivata delle seguenti funzioni tangenti: <\/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>Vedi la soluzione<\/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> Dimostrazione della derivata della tangente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Affinch\u00e9 tu possa verificare che non si tratta di un&#8217;espressione inventata, in questa sezione dimostreremo la formula per la derivata della tangente utilizzando la definizione matematica di tangente.<\/p>\n<p> Per fare ci\u00f2 partiremo dall\u2019identit\u00e0 trigonometrica che collega i tre rapporti trigonometrici:<\/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 usiamo la <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivata-di-un-quoziente-di-divisione\/\">formula per la derivata di una divisione<\/a><\/span> , la derivata sarebbe: <\/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> Ma, utilizzando l&#8217;identit\u00e0 trigonometrica fondamentale, sappiamo che il quadrato del seno pi\u00f9 il quadrato del coseno \u00e8 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 cos\u00ec siamo gi\u00e0 arrivati alla prima formula per la derivata della tangente. Inoltre, la secante \u00e8 l&#8217;inverso moltiplicativo del coseno, quindi si deriva anche la seconda espressione:<\/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> Infine, la terza regola della derivata tangente pu\u00f2 essere dimostrata trasformando la frazione del passaggio precedente in una somma di frazioni: <\/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>Qui scoprirai come viene derivata la funzione tangente. Inoltre, potrai vedere esempi di derivata della tangente e anche esercitarti con esercizi risolti passo dopo passo. Infine, dimostriamo anche la formula della derivata tangente e ti mostriamo la formula della derivata tangente inversa. Qual \u00e8 la derivata della tangente? La derivata della tangente di x \u00e8 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/derivata-della-tangente\/\"> <span class=\"screen-reader-text\">Derivata della tangente<\/span> Leggi altro &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":[6],"tags":[],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-derivati"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Derivata della tangente (formula ed esercizi risolti)<\/title>\n<meta name=\"description\" content=\"Spieghiamo come derivare la funzione tangente (formula). 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