{"id":75,"date":"2023-09-17T11:02:40","date_gmt":"2023-09-17T11:02:40","guid":{"rendered":"https:\/\/mathority.org\/nl\/afgeleide-van-de-raaklijn\/"},"modified":"2023-09-17T11:02:40","modified_gmt":"2023-09-17T11:02:40","slug":"afgeleide-van-de-raaklijn","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/afgeleide-van-de-raaklijn\/","title":{"rendered":"Afgeleide van de raaklijn"},"content":{"rendered":"<p>Hier leert u hoe de tangensfunctie wordt afgeleid. Bovendien kunt u voorbeelden zien van de afgeleide van de raaklijn en zelfs oefenen met oefeningen die stap voor stap worden opgelost. Ten slotte demonstreren we ook de raaklijnafgeleide formule en laten we u de inverse raaklijnafgeleide formule zien. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-derivada-de-la-tangente\"><\/span> Wat is de afgeleide van de raaklijn?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>De afgeleide van de raaklijn van x is gelijk aan 1 over het kwadraat van de cosinus van x.<\/strong> De afgeleide van de raaklijn van x is ook gelijk aan het kwadraat van de secans van x, en 1 plus het kwadraat van de raaklijn van 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> Alle uitdrukkingen zijn gelijkwaardig, dus de raaklijnfunctie heeft drie mogelijke formules om deze af te leiden.<\/p>\n<p> Aan de andere kant, als we in het raaklijnargument een andere functie hebben dan x (laten we het u noemen), moeten we de kettingregel toepassen. De afgeleide van de tangens van u is daarom:<\/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> Kort gezegd kan de raaklijnafgeleide regel als volgt worden samengevat: <\/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=\"raaklijn afgeleide\" 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> Voorbeelden van raaklijnafgeleide<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Gegeven de formule voor de raaklijnafgeleide, zullen we in deze sectie verschillende voorbeelden van dit soort trigonometrische afgeleiden oplossen, zodat u begrijpt hoe u de raaklijnfunctie kunt afleiden. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-la-tangente-de-2x\"><\/span> Voorbeeld 1: Afgeleide van de tangens van 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> Om de afgeleide van de raaklijn te berekenen, kun je een van de drie formules gebruiken die we hierboven hebben gezien. In dit geval gebruiken we de cosinusformule:<\/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> De functie 2x is lineair, dus de afgeleide ervan is 2. Dus de afgeleide van de raaklijn van 2x is 2 over het kwadraat van de cosinus van 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> Voorbeeld 2: Afgeleide van de tangens van x in het kwadraat<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 dit voorbeeld is de tangens-argumentfunctie geen x, maar een functie met een afgeleide. Dat betekent dat we de kettingregel moeten toepassen om deze af te leiden.<\/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> De afgeleide van x kwadraat is 2x, dus de afgeleide van de raaklijn van x <sup>2<\/sup> is: <\/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> Voorbeeld 3: Afgeleide van de raaklijn aan de kubus<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 dit probleem hebben we een samengestelde functie, dus we zullen ook de kettingregel moeten gebruiken om de raaklijn te differenti\u00ebren.<\/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> Bovendien wordt de raaklijn verheven tot de macht 3, wat betekent dat voordat je de formule voor de afgeleide van de raaklijn toepast, je de formule voor de afgeleide van een macht moet gebruiken: <\/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> Afgeleide van de inverse tangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Zoals elke inverse functie heeft de tangensfunctie ook een inverse, de arctangensfunctie. Hoewel de formule om deze af te leiden niet vergelijkbaar is met de raaklijnformule, laten we deze u zien omdat deze in sommige gevallen nuttig kan zijn.<\/p>\n<p> De <strong>afgeleide van de inverse tangens<\/strong> van een functie is het quoti\u00ebnt van de afgeleide van de functie gedeeld door \u00e9\u00e9n plus de genoemde functie in het kwadraat<\/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> De afgeleide van de inverse tangens van 3x is bijvoorbeeld: <\/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> Opgeloste oefeningen over de afgeleide van de raaklijn<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Bereken de afgeleide van de volgende raaklijnfuncties: <\/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>Zie de oplossing<\/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> Bewijs van de afgeleide van de raaklijn<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Zodat u kunt verifi\u00ebren dat dit geen verzonnen uitdrukking is, zullen we in deze sectie de formule voor de afgeleide van de raaklijn demonstreren met behulp van de wiskundige definitie van raaklijn.<\/p>\n<p> Om dit te doen, gaan we uit van de trigonometrische identiteit die de drie trigonometrische verhoudingen met elkaar verbindt:<\/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> Als we de <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/afgeleide-van-een-delingsquotient\/\">formule voor de afgeleide van een deling<\/a><\/span> gebruiken, zou de afgeleide zijn: <\/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> Maar door gebruik te maken van de fundamentele trigonometrische identiteit weten we dat het kwadraat van de sinus plus het kwadraat van de cosinus 1 is:<\/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> En zo zijn we al aangekomen bij de eerste formule voor de afgeleide van de raaklijn. Bovendien is de secans de multiplicatieve inverse van de cosinus, dus wordt de tweede uitdrukking ook afgeleid:<\/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> Ten slotte kan de derde regel van de raaklijnafgeleide worden bewezen door de breuk uit de vorige stap om te zetten in een som van breuken: <\/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>Hier leert u hoe de tangensfunctie wordt afgeleid. Bovendien kunt u voorbeelden zien van de afgeleide van de raaklijn en zelfs oefenen met oefeningen die stap voor stap worden opgelost. Ten slotte demonstreren we ook de raaklijnafgeleide formule en laten we u de inverse raaklijnafgeleide formule zien. Wat is de afgeleide van de raaklijn? De &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/afgeleide-van-de-raaklijn\/\"> <span class=\"screen-reader-text\">Afgeleide van de raaklijn<\/span> Lees meer &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":[38],"tags":[],"class_list":["post-75","post","type-post","status-publish","format-standard","hentry","category-derivaten"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Afgeleide van de raaklijn (formule en opgeloste oefeningen)<\/title>\n<meta name=\"description\" content=\"We leggen uit hoe je de tangensfunctie (formule) kunt afleiden. 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