{"id":91,"date":"2023-09-17T10:54:18","date_gmt":"2023-09-17T10:54:18","guid":{"rendered":"https:\/\/mathority.org\/nl\/afgeleide-van-de-cotangens\/"},"modified":"2023-09-17T10:54:18","modified_gmt":"2023-09-17T10:54:18","slug":"afgeleide-van-de-cotangens","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/afgeleide-van-de-cotangens\/","title":{"rendered":"Afgeleide van de cotangens"},"content":{"rendered":"<p>In dit artikel zullen we zien hoe we de cotangens van een functie kunnen afleiden. Je vindt voorbeelden van de afgeleide van de cotangens en zelfs oefeningen die stap voor stap worden opgelost. Ten slotte bewijzen we de formule voor de afgeleide van de cotangens. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-cotangente\"><\/span> Formule voor de afgeleide van de cotangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>De afgeleide van de cotangens van x is gelijk aan negatief \u00e9\u00e9n over het kwadraat van de sinus van x.<\/strong> De afgeleide van de cotangens van x is ook gelijk aan minus het kwadraat van de cotangens van x, en minus de som van \u00e9\u00e9n plus het kwadraat van de cotangens 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-0a3653f5c765d773ebc789107bf1a825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cotg}(x)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=-\\cfrac{1}{\\text{sen}^2(x)}=-\\text{cosec}^2(x)=-\\left(1+\\text{cotg}^2(x)\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"393\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Als het argument cotangens een andere functie is dan x, zijn de formules voor de afgeleide van de cotangens van een functie hetzelfde als de vorige, maar worden de uitdrukkingen vermenigvuldigd met de afgeleide van de functie van het argument.<\/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-38ea1d1edeaf5664c56a946b5a87577d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cotg}(u)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\color{black}\\\\ f'(x)=-\\cfrac{u'}{\\text{sen}^2(u)}=-u' \\cdot \\text{cosec}^2(u)=-u' \\cdot \\left(1+\\text{cotg}^2(u)\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"445\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dit betekent dat er drie verschillende formules zijn om de afgeleide van de cotangens te vinden. Maar logischerwijs is het niet nodig om alle drie de formules te gebruiken, maar u kunt deze afleiden met de formule die u verkiest. <\/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-la-cotangente.webp\" alt=\"afgeleid van de cotangens\" class=\"wp-image-2685\" width=\"428\" height=\"361\" 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-cotangente\"><\/span> Voorbeelden van afgeleide van de cotangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nu we de formule voor de afgeleide van de cotangens van een functie hebben gezien, zullen we in deze sectie verschillende voorbeelden van dit soort trigonometrische afgeleiden oplossen. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-la-cotangente-de-2x\"><\/span> Voorbeeld 1: Afgeleide van de cotangens van 2x<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> In dit voorbeeld zullen we zien wat de afgeleide is van de cotangens van de functie 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-b95db136ea1e222c9f810d724216b083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Zoals we hebben gezien, kun je om de afgeleide van de cotangens te berekenen een van de drie bovenstaande formules gebruiken. In dit geval gebruiken we de sinuso\u00efdale formule:<\/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-b071b560415fc193171a74fd0b4b84cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{\\text{sen}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"409\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Omdat 2x een term van de eerste graad is, is de afgeleide ervan 2. Dus de afgeleide van de cotangens van 2x is negatief twee gedeeld door het kwadraat van de sinus van 2x: <\/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-4151decbf7dd792fd0ea6aa6ca4b55f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(2x)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{2}{\\text{sen}^2(2x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"426\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-la-cotangente-de-x-al-cuadrado\"><\/span> Voorbeeld 2: Afgeleide van de cotangens van x kwadraat<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> In het tweede voorbeeld gaan we bepalen wat de afgeleide van de cotangens van x kwadraat 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-ec3733080f720a5115d2d6719d33d7e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In dit voorbeeld is de functie van het cotangens-argument geen x, dus moeten we de kettingregel toepassen om de cotangens te differenti\u00ebren.<\/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-b071b560415fc193171a74fd0b4b84cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(u)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{\\text{sen}^2(u)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"409\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> De afgeleide van x kwadraat is 2x, dus de afgeleide van de cotangens van x <sup>2<\/sup> is: <\/p>\n<p class=\"has-text-align-center\"><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-6aef87160d0da1da0b32e1e200f31b7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}(x^2)\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{2x}{\\text{sen}^2(x^2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"424\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-la-cotangente-al-cubo\"><\/span> Voorbeeld 3: Afgeleide van de cotangens in blokjes<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Ten slotte zullen we ontdekken hoeveel de afgeleide is van de derde cotangens van een polynoomfunctie:<\/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-9be8c3c10c50e3ef6bb701a11d3f46c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cotg}^3(x^5-6x^2+10)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In dit geval hebben we een samenstelling van functies, dus moeten we de kettingregel gebruiken met de formule voor de afgeleide van een macht om de afgeleide van de cotangens te vinden: <\/p>\n<p class=\"has-text-align-center\"><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-ed0c6f314584b0f00021e3833de2b223_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=-3\\cdot\\text{cotg}^2(x^5-6x^2+10)\\cdot\\frac{5x^4-12x}{\\text{sen}^2(x^5-6x^2+10)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"424\" 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-cotangente\"><\/span> Opgeloste oefeningen over de afgeleide van de cotangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Bereken de afgeleide van de volgende cotangensfuncties: <\/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-e1e391853b53d81ed500fd590799cbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=\\text{cotg}(5x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" 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-8581122adcd9d79db9862a27b57af0a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=\\text{cotg}(2x^4+10x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"236\" 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-e7814ccc95124c55f86b2f7036178254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f(x)=\\text{cotg}^5\\left(\\frac{x}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"160\" 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-1dcf8b02c8e9d9712210a78d116e2fc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\text{cotg}\\left(e^{x^2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"162\" style=\"vertical-align: -11px;\"><\/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-f358f9329696ee71ac2c1abfd5f9668e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=\\text{cotg}\\bigl(\\ln(x^2)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"176\" 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-e26ae47cb88a301a3353d868fc9b74f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f(x)=\\text{cotg}\\left(\\sqrt{8x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"166\" 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-b002740b34952198a8284265a444cbed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=-\\cfrac{5}{\\text{sen}^2(5x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"171\" 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-d9b86f549fab91f93d0fc4e437e1c4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=-\\cfrac{8x+10}{\\text{sen}^2(2x^4+10x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"257\" 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-eb1435fb7eb0c7bcff84741362417548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f'(x)=5\\cdot \\text{cotg}^4\\left(\\frac{x}{2}\\right)\\cdot \\left(-\\frac{1}{\\text{sen}^2\\left(\\frac{x}{2}\\right)}\\right)\\cdot \\frac{1}{2}=-\\frac{5\\cdot \\text{cotg}^4\\left(\\frac{x}{2}\\right)}{2\\cdot \\text{sen}^2\\left(\\frac{x}{2}\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"468\" style=\"vertical-align: -23px;\"><\/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-4fb5fc0cea94ec46660b5bb16d9daaa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=-\\cfrac{2x\\cdot e^{x^2}}{\\text{sen}^2(e^{x^2})}\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"175\" style=\"vertical-align: -18px;\"><\/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-efc6c63eb1a5cfc3f89deb4cfc3b4586_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=-\\cfrac{\\cfrac{2x}{x^2}}{\\text{sen}^2\\bigl(\\ln(x^2)\\bigr)}=-\\cfrac{2}{x\\cdot\\text{sen}^2\\bigl(\\ln(x^2)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"356\" 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-70d6c7ac291ee53490646ae841842eef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f'(x)=-\\cfrac{\\frac{8}{2\\sqrt{8x}}}{\\text{sen}^2\\left(\\sqrt{8x}\\right)}=-\\cfrac{4}{\\sqrt{8x}\\cdot \\text{sen}^2\\left(\\sqrt{8x}\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"360\" 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-cotangente\"><\/span> Bewijs van de afgeleide van de cotangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> In dit laatste deel zullen we de formule demonstreren voor de afgeleide van de cotangens. Om dit te doen, zullen we uitgaan van de wiskundige definitie van de cotangensfunctie, die gelijk is aan de cosinus gedeeld door de sinus:<\/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-dd85e2f7ac86aa67c6bd2f82fedfa926_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}(x)=\\cfrac{\\text{cos}(x)}{\\text{sen}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"131\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Nu differenti\u00ebren we de functie door de regel toe te passen voor de afgeleide van een quoti\u00ebnt; <\/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-0eb90358967efc64de6d45b8eabe5e37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\bigl(\\text{cotg}(x)\\bigr)'=\\left(\\frac{\\text{cos}(x)}{\\text{sen}(x)}\\right)'\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"181\" 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-7685acfe693a3c5e8dd2e543ad8ec7c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=\\cfrac{-\\text{sen}(x)\\cdot \\text{sen}(x)-\\text{cos}(x)\\cdot \\text{cos}(x) }{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"349\" 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-9988024649ed4fdb4b1a9bb913aa813f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=\\cfrac{-\\text{sen}^2(x)-\\text{cos}^2(x)}{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"243\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> We nemen de gemeenschappelijke factor in de noemer en verwijderen het negatieve teken uit de breuk:<\/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-b23352ad101cdc7e1adeb6316c6d66c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=\\cfrac{-\\bigl(\\text{sen}^2(x)+\\text{cos}^2(x)\\bigr)}{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"259\" 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-df4c94284c00f257f7be601a98bf9475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=-\\cfrac{\\text{sen}^2(x)+\\text{cos}^2(x)}{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"234\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Aan de andere kant weten we dat het kwadraat van de sinus plus het kwadraat van de cosinus gelijk is aan \u00e9\u00e9n dankzij de fundamentele trigonometrische identiteit.<\/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-dd14ec817754afcafd5d862afc0703b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=-\\cfrac{1}{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"157\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> En zo verkregen we de eerste formule voor de afgeleide van de cotangens. Op dezelfde manier is de cosecans de multiplicatieve inverse van de sinus, dus de tweede regel van de afgeleide van de cotangens is ook bewezen:<\/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-41aafad77ef896612fe6851ee97d914a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=-\\text{sec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ten slotte kan de derde formule voor de afgeleide van deze trigonometrische functie 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-df4c94284c00f257f7be601a98bf9475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cotg}'(x)=-\\cfrac{\\text{sen}^2(x)+\\text{cos}^2(x)}{\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"234\" 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-3780e44f1c85235473d47f418c7bc889_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cotg}'(x)=-\\left(\\frac{\\text{sen}^2(x)}{\\text{sen}^2(x)}+\\frac{\\text{cos}^2(x)}{\\text{sen}^2(x)}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"266\" 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-e7a8b05807749ddd4b8979253dcb2f45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}'(x)=-\\bigl(1+\\text{cotg}^2(x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"200\" style=\"vertical-align: -7px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel zullen we zien hoe we de cotangens van een functie kunnen afleiden. Je vindt voorbeelden van de afgeleide van de cotangens en zelfs oefeningen die stap voor stap worden opgelost. Ten slotte bewijzen we de formule voor de afgeleide van de cotangens. Formule voor de afgeleide van de cotangens De afgeleide van &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/afgeleide-van-de-cotangens\/\"> <span class=\"screen-reader-text\">Afgeleide van de cotangens<\/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-91","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 cotangens (formule en voorbeelden)<\/title>\n<meta name=\"description\" content=\"Hoe de cotangens van een functie af te leiden (formule). 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