{"id":74,"date":"2023-09-17T11:02:59","date_gmt":"2023-09-17T11:02:59","guid":{"rendered":"https:\/\/mathority.org\/nl\/komt-voort-uit-de-cosinus\/"},"modified":"2023-09-17T11:02:59","modified_gmt":"2023-09-17T11:02:59","slug":"komt-voort-uit-de-cosinus","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/komt-voort-uit-de-cosinus\/","title":{"rendered":"Cosinus afgeleide"},"content":{"rendered":"<p>Hier ontdekt u hoe u de cosinusfunctie (formule) kunt afleiden. Je kunt voorbeelden zien van afgeleiden van cosinusfuncties en oefenen met stapsgewijze oefeningen. Daarnaast laten we u het bewijs van de formule zien, wat de tweede afgeleide van de cosinus is en zelfs de afgeleide van de inverse cosinus. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-derivada-del-coseno\"><\/span> Wat is de afgeleide van de cosinus?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>De afgeleide van de cosinusfunctie is de teken-gemodificeerde sinusfunctie. Met andere woorden: de afgeleide van de cosinus van x is gelijk aan minus de sinus van x.<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72551067d650b8d3797bc37497ec609d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"389\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Als er een functie in het cosinusargument aanwezig is, is de afgeleide van de cosinus het product van minus de sinus van die functie maal de afgeleide van de functie.<\/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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> De tweede formule is gelijk aan de eerste formule, maar met toepassing van de kettingregel. Samenvattend is de formule voor de afgeleide van de cosinus dus als volgt: <\/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-du-cosinus.webp\" alt=\"cosinus afgeleide\" class=\"wp-image-1902\" width=\"428\" height=\"292\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-coseno\"><\/span> Cosinus afgeleide voorbeelden<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nu we weten wat de cosinusformule is, zullen we verschillende voorbeelden van dit soort trigonometrische afgeleiden uitleggen, zodat je geen twijfels meer hebt over hoe je de cosinusfunctie kunt afleiden. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-del-coseno-de-2x\"><\/span> Voorbeeld 1: Afgeleide van de cosinus 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-87c696135df266b2d8498b353bf03c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In het cosinusargument hebben we geen enkele x, maar eerder een complexere functie. Daarom moeten we de volgende formule gebruiken om de cosinus 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Omdat de afgeleide van 2x 2 is, zal de afgeleide van de cosinus van 2x minus de sinus van 2x vermenigvuldigd met 2 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-ba75c906f1694fe3fbd16fa61e0d288e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(2x)\\cdot 2=-2\\text{sen}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"532\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-del-coseno-de-x-al-cuadrado\"><\/span> Voorbeeld 2: Afgeleide van de cosinus van x 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-89f1a1fc3f2d5e95aafbd2a37282f88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Net als in het vorige voorbeeld hebben we in het cosinusargument een andere functie dan x, dus zullen we de kettingregel gebruiken om de cosinus 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan is de afgeleide van x <sup>2<\/sup> 2x, daarom is de afgeleide van de cosinus van x verheven tot de macht 2: <\/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-a410f3316194c86b97a987b0ec7e9e6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(x^2)\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"437\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-del-coseno-al-cubo\"><\/span> Voorbeeld 3: Afgeleide van de gekubeerde cosinus<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-dfa2e76d23ef3aeb2ab3ff8e20e2aa07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^3(2x^6-5x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"178\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> De cosinusfunctie in dit voorbeeld bestaat uit een andere functie, dus we moeten de volgende formule toepassen om de afgeleide op te lossen:<\/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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Door de formule toe te passen komen we dus tot de afgeleide van de functie:<\/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-73284bcfb1d5647b2304e323e7fbaedf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=3\\text{cos}^2(2x^6-5x^3)\\cdot \\bigl(-\\text{sen}(2x^6-5x^3)\\bigr)\\cdot (12x^5-15x^2)\\\\[2ex]&amp;=-3\\text{cos}^2(2x^6-5x^3)\\cdot \\text{sen}(2x^6-5x^3)\\cdot (12x^5-15x^2)\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <u style=\"text-decoration-color:#ff951b;\">Om deze functie te differenti\u00ebren, moet je ook de formule voor de<\/u> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/afgeleide-van-een-machtspotentieelfunctie\/\">afgeleide van een potenti\u00eble functie<\/a><\/span> gebruiken. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"segunda-derivada-del-coseno\"><\/span> Tweede afgeleide van de cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Vervolgens zullen we zien dat de tweede afgeleide van de sinus eenvoudig kan worden berekend, dankzij de kenmerken van trigonometrische functies.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <u style=\"text-decoration-color:#ff951b;\"><strong>Opmerking:<\/strong> Om het volgende te begrijpen, moet je weten<\/u> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/sinusderivaat\/\">wat de afgeleide van sinus is<\/a><\/span> .<\/p>\n<p> <strong>De tweede afgeleide van de cosinus van x is minus de cosinus van x.<\/strong> Dit lijkt misschien vreemd, maar wiskundig gezien is het zo. De afgeleide van de sinus is inderdaad de cosinus en daarom wordt, door de cosinus van x tweemaal te differenti\u00ebren, opnieuw de cosinus verkregen, maar met een aangepast teken.<\/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-d0d9dda8a4031c367120b1f950da4391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cos}(x)\\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f'(x)=-\\text{sen}(x)\\\\[2ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f''(x)=-\\text{cos}(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"157\" width=\"132\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Deze eigenschap verandert als het cosinusargument niet x is, omdat we in dit geval de term van de kettingregel verslepen: <\/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-83d86bd6508f06b0723153b3b9254c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cos}(u)\\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f'(x)=-\\text{sen}(u)\\cdot u' \\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f''(x)=-\\text{cos}(u)\\cdot u'^2 -\\text{sen}(u)\\cdot u'' \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"153\" width=\"263\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-del-coseno-inverso\"><\/span> Afgeleide van inverse cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Alle trigonometrische functies hebben een inverse functie en als zodanig kan de cosinusfunctie ook worden omgekeerd. Op dezelfde manier is de inverse cosinus differentieerbaar.<\/p>\n<p> De <strong>afgeleide van de inverse cosinus<\/strong> van een functie is minus de afgeleide van de functie gedeeld door de vierkantswortel van \u00e9\u00e9n minus het kwadraat van de functie.<\/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-307f91156ee9c404e9c1a1c0de56b102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^{-1}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{\\sqrt{1-u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"425\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Bedenk dat de inverse cosinus ook wel arccosinus wordt genoemd.<\/p>\n<p> De afgeleide van de inverse cosinus 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-7ce426dcd95d21e43b182ef593520c16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^{-1}(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{3}{\\sqrt{1-(3x)^2}}=-\\cfrac{3}{\\sqrt{1-9x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"571\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-del-coseno\"><\/span> Opgeloste oefeningen over de afgeleide van de cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Bereken de afgeleide van de volgende cosinusfuncties: <\/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-c1caebbb3b9acfa8cd25721299f9a22e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=\\text{cos}(4x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" 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-a89d6e415addae7423aa75362416686b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=\\text{cos}(2x^3-5x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"218\" 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-dad34285ad7ac07f34ef408c65cbb96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f(x)=9\\text{cos}\\left(\\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"152\" 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-752bff297c20cdf69d6fcb45290be935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\text{cos}^5(x^2+3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"187\" 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-939dee8bcafba3f55812c5a13f27a309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=\\text{cos}\\left(e^{5x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"148\" 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-f6c7c3c5786d010b99f4e65b692dfe1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } \\displaystyle f(x)=9\\text{cos}\\left(\\frac{e^x}{5x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"164\" style=\"vertical-align: -17px;\"><\/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-8a3e1b1b2fe486d1c432a075c0028b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=-\\text{sen}(4x)\\cdot 4 =-4\\text{sen}(4x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"285\" 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-8bc2f2e1676bb5f0d4ca231bd35b2b12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=-\\text{sen}(2x^3-5x+1)\\cdot (6x^2-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"321\" 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-cc5cf86d30b34d4cd1a794a4d2ee6a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f'(x)=-9\\text{sen}\\left(\\frac{x}{3}\\right)\\cdot \\frac{1}{3} =-3\\text{sen}\\left(\\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"306\" 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-f168f2e897b18c662f567a25ff09e881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=-5\\text{cos}^4(x^2+3x)\\cdot \\text{sen}(x^2+3x)\\cdot (2x+3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"401\" 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-fe49736d7a1ce1736679e8c25bc4a66b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=-\\text{sen}\\left(e^{5x}\\right)\\cdot e^{5x}\\cdot 5=-5\\text{sen}\\left(e^{5x}\\right)\\cdot e^{5x}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"377\" 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-9f4645fb77435daec6f696cffbd54884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{F) }\\displaystyle f'(x)&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{e^x\\cdot 5x-e^x\\cdot 5}{(5x)^2}\\\\[2ex]&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{5e^x(x-1)}{25x^2}\\\\[2ex]&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{e^x(x-1)}{5x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"172\" width=\"310\" style=\"vertical-align: 0px;\"><\/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-del-coseno\"><\/span> Bewijs van de cosinusafgeleide<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ten slotte zullen we wiskundig de formule demonstreren voor de afgeleide van de cosinus van x. Om dit te doen, zullen we de definitie van de afgeleide gebruiken, die overeenkomt met de volgende limiet:<\/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-dc1699622d128f888c1f20599aeccf60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"219\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> We gaan de cosinus bewijzen, dus de functie is cos(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-a00c11698e4b4f5caf0f227e18be8656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x+h)-\\text{cos}(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"245\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> We kunnen deze grens niet oplossen door vervanging, omdat we dan in onbepaaldheid terecht zouden komen. We kunnen de cosinus van een som echter op een andere manier uitdrukken door de volgende trigonometrische identiteit toe te passen:<\/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-5e06f1728cce31fb5650ba149b8e5b9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(a+b)=\\text{cos}(a)\\text{cos}(b)-\\text{sen}(a)\\text{sen}(b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"307\" 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-db64449e24b11a613417ebce4c7c7a85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x)\\text{cos}(h)-\\text{sen}(x)\\text{sen}(h)-\\text{cos}(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"380\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> De volgende stap is om de breuk in twee breuken te verdelen en de gemeenschappelijke factor van de cosinus te nemen:<\/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-c7c1cd89cf290b01d7d72fc8084f6529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\left[\\frac{\\text{cos}(x)\\bigl(\\text{cos}(h)-1\\bigr)}{h}-\\frac{\\text{sen}(x)\\text{sen}(h)}{h}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"380\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> De limiet van een aftrekking is gelijk aan de aftrekking van de limieten, dus:<\/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-739fc9a2280c7da1bf2ea830ee5ec88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x)\\bigl(\\text{cos}(h)-1\\bigr)}{h}-\\lim_{h \\to 0}\\frac{\\text{sen}(x)\\text{sen}(h)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"393\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> De cosinus van x en de sinus van x zijn niet afhankelijk van h, dus we kunnen ze buiten de grenzen halen:<\/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-bfbc83e5a84d91a0f6d98418a4f0041c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\text{cos}(x)\\lim_{h \\to 0}\\frac{\\text{cos}(h)-1}{h}-\\text{sen}(x)\\lim_{h \\to 0}\\frac{\\text{sen}(h)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"383\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Door de berekening van limieten met oneindig kleine equivalenten te gebruiken, concluderen we dat de eerste limiet 0 is en de tweede limiet 1. 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-f0c2ed1188b80356d05d6188fab5ca47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\text{cos}(x)\\cdot 0-\\text{sen}(x)\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"223\" 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-9f33ae6c9b18e01ba654772f22cab6d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> En we hebben de formule voor de afgeleide van de cosinusfunctie al bereikt, dus de gelijkheid is bewezen.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hier ontdekt u hoe u de cosinusfunctie (formule) kunt afleiden. Je kunt voorbeelden zien van afgeleiden van cosinusfuncties en oefenen met stapsgewijze oefeningen. Daarnaast laten we u het bewijs van de formule zien, wat de tweede afgeleide van de cosinus is en zelfs de afgeleide van de inverse cosinus. Wat is de afgeleide van de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/komt-voort-uit-de-cosinus\/\"> <span class=\"screen-reader-text\">Cosinus afgeleide<\/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-74","post","type-post","status-publish","format-standard","hentry","category-derivaten"],"yoast_head":"<!-- This site is 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