{"id":80,"date":"2023-09-17T11:00:46","date_gmt":"2023-09-17T11:00:46","guid":{"rendered":"https:\/\/mathority.org\/nl\/afgeleide-van-een-delingsquotient\/"},"modified":"2023-09-17T11:00:46","modified_gmt":"2023-09-17T11:00:46","slug":"afgeleide-van-een-delingsquotient","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/afgeleide-van-een-delingsquotient\/","title":{"rendered":"Afgeleide van een quoti\u00ebnt (of deling)"},"content":{"rendered":"<p>In dit artikel leggen we uit hoe je uit twee functies een quoti\u00ebnt (of deling) kunt afleiden. Je vindt voorbeelden van afgeleiden van quoti\u00ebnten van functies en daarnaast kun je oefenen met stapsgewijze oefeningen over afgeleiden van delingen. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-un-cociente\"><\/span> Formule voor de afgeleide van een quoti\u00ebnt<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> De <strong>afgeleide van een co\u00ebffici\u00ebnt (of deling) van de functies<\/strong> is identiek aan de afgeleide van de tellerfunctie door de noemerfunctie kleiner dan de tellerfunctie door de afgeleide van de noemerfunctie gedeeld door het kwadraat van de hoge noemerfunctie. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-quotient-de-division-derivee.webp\" alt=\"formule voor de afgeleide van een deling of quoti\u00ebnt\" class=\"wp-image-2194\" width=\"326\" height=\"304\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Zoals je kunt zien, hebben we na differentiatie nog steeds een breuk als we de regel voor de afgeleide van een quoti\u00ebnt (of deling) toepassen. Maar bovendien hebben we in de teller twee vermenigvuldigingen en een aftrekking, en de noemer wordt verheven tot de macht twee. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-derivadas-de-cocientes\"><\/span> Voorbeelden van afgeleiden van quoti\u00ebnten<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> We hebben zojuist gezien wat de formule is voor de afgeleide van een quoti\u00ebnt van twee functies. Vervolgens zullen we verschillende voorbeelden van afgeleiden van dit soort bewerkingen oplossen. Houd er rekening mee dat als u niet begrijpt hoe een functioneel quoti\u00ebnt wordt afgeleid, u ons dit kunt vragen in het opmerkingengedeelte.<\/p>\n<h3 class=\"wp-block-heading\"> voorbeeld 1<\/h3>\n<p> In dit voorbeeld zullen we een potenti\u00eble functie afleiden gedeeld door een trigonometrische 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-9d260d4cdca9f28e43607a9c1e7b3404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{3x^2+4x}{\\text{sen}(2x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"128\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> De formule voor de afgeleide van een deling van twee verschillende functies is als volgt:<\/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-fc09ff88e92ee46b5c98d6fc81a5d5a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=\\cfrac{f(x)}{g(x)}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=\\cfrac{f'(x)\\cdot g(x)-f(x)\\cdot g'(x)}{\\bigl(g(x)\\bigr)^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"139\" width=\"255\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> We moeten dus eerst de afgeleide van elke functie afzonderlijk berekenen: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-21\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-24719cd47158514d54e16f4994a1c2b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (3x^2+4x)=6x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"180\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-093fb274a2a393453833ed572dc1bc62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ \\text{sen}(2x)=2\\text{cos}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"171\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> De afgeleide van de gehele functie 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-65ce4673f3ad5a4c09a9b2e7c611821d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\cfrac{3x^2+4x}{\\text{sen}(2x)}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=\\cfrac{(6x+4)\\cdot\\text{sen}(2x)-(3x^2+4x)\\cdot 2\\text{cos}(2x)}{\\text{sen}^2(2x)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Voorbeeld 2<\/h3>\n<p> In dit geval vinden we de afgeleide van een constante gedeeld door een 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-e7d5ddfdf95f11b94783ca40437e371a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{10}{x^2+3x-9}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"150\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Zoals we hierboven hebben gezien, is de regel voor de afgeleide van een deling van twee verschillende functies als volgt:<\/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-fc09ff88e92ee46b5c98d6fc81a5d5a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=\\cfrac{f(x)}{g(x)}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=\\cfrac{f'(x)\\cdot g(x)-f(x)\\cdot g'(x)}{\\bigl(g(x)\\bigr)^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"139\" width=\"255\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> We berekenen dus afzonderlijk de afgeleide van de teller en de noemer: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-24\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-191c03d69e261059308133b99f87bf1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 10=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c63da81dbf763b0a24cf929aef024c51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (x^2+3x-9)=2x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"201\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> En tenslotte vinden we de afgeleide van de deling van gehele getallen:<\/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-8f8bdea77dc91b1aff40695511593e86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\cfrac{10}{x^2+3x-9}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=\\cfrac{0\\cdot (x^2+3x-9)-10\\cdot (2x+3)}{\\left(x^2+3x-9\\right)^2}=\\cfrac{-20x+30}{\\left(x^2+3x-9\\right)^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"441\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> In feite kunnen we een formule afleiden om direct te differenti\u00ebren als we een constante in de teller hebben gedeeld door een functie, omdat de afgeleide van de constante altijd 0 is. Daarom zal de volgende formule altijd waar 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-1f0bd615634f5205f91674f96f5c2514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{c}z(x)=\\cfrac{k}{f(x)} \\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=\\cfrac{-k\\cdot f'(x)}{\\bigl(f(x)\\bigr)^2}\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Voorbeeld 3<\/h3>\n<p> In deze oefening zullen we een quoti\u00ebnt van twee polynomen afleiden:<\/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-59b390fee61ab3c2cbb4dc2230386658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3+4x^2}{5x^2-8}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Om de afgeleide op te lossen, moeten we de regel toepassen voor de afgeleide van een quoti\u00ebnt van twee verschillende functies, die als volgt luidt:<\/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-fc09ff88e92ee46b5c98d6fc81a5d5a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=\\cfrac{f(x)}{g(x)}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=\\cfrac{f'(x)\\cdot g(x)-f(x)\\cdot g'(x)}{\\bigl(g(x)\\bigr)^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"139\" width=\"255\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Laten we nu de afgeleide van het tellerpolynoom en het noemerpolynoom vinden: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-27\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-39e82139a124bf031f31b84007bcb923_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (x^3+4x^2)=3x^2+8x\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"196\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73db77f7639fc8b4cc305fbbb2b1cf2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (5x^2-8)=10x\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"148\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> De afgeleide van de verdeling van polyniemen 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-065ad49556f264b4cfb505522ad7566b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\cfrac{x^3+4x^2}{5x^2-8}\\\\[2.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=\\cfrac{(3x^2+8x)\\cdot (5x^2-8)-(x^3+4x^2)\\cdot 10x}{\\left(5x^2-8\\right)^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"137\" width=\"373\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> En ten slotte voeren we de bewerkingen uit en vereenvoudigen we de breuk zoveel mogelijk: <\/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-af3f7cb513883d1fa5dadca23701c19d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=\\cfrac{(3x^2+8x)\\cdot (5x^2-8)-(x^3+4x^2)\\cdot 10x}{\\left(5x^2-8\\right)^2}\\\\[2ex]&amp;=\\cfrac{15x^4-24x^2+40x^3-64x-10x^4-40x^3}{25x^4+64-80x^2}\\\\[2ex]&amp;=\\cfrac{5x^4-24x^2-64x}{25x^4-80x^2+64}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"178\" width=\"379\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-de-un-cociente\"><\/span> Opgeloste oefeningen over de afgeleide van een quoti\u00ebnt<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Leid de volgende functieverdelingen af: <\/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-e6244d0e6cfcb8c4b82806d40cab93fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) }f(x)=\\cfrac{9x^2+5x}{6x^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"154\" 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-e94071c6dc40cd4a7280be617cdddd3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) }f(x)=\\cfrac{19}{2x^2-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"143\" 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-214e8c32a9ffb1c37f164935c3ad6bfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) }f(x)=\\cfrac{8x^3-4x^2+3x}{e^{4x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"202\" 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-a83ff4c06137279870296a80b12b0cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) }f(x)=\\cfrac{\\text{cos}(x^2)}{\\text{sen}(6x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"144\" 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-4bed0ff9464adb5897528d5b47ed477c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) }f(x)=\\cfrac{\\ln(x^3+4)}{\\left(4x^2-3x\\right)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"174\" 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-74ddf8caa494c9a60dcbfc9d57c90d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) }f(x)=\\cfrac{\\sqrt{x^2+4x}}{5^{x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"158\" style=\"vertical-align: -13px;\"><\/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-26b0af84dd46ca29727eee97380b4ca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{A) }f'(x)&amp;=\\cfrac{(18x+5)\\cdot 6x^3-(9x^2+5x)\\cdot 18x^2}{\\left(6x^3\\right)^2}\\\\[1.5ex]&amp;=\\cfrac{108x^4+30x^3-162x^4-90x^3}{36x^6}\\\\[1.5ex]&amp;=\\cfrac{-54x^4-60x^3}{36x^6}\\\\[1.5ex]&amp;=\\cfrac{-9x-10}{6x^3}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"225\" width=\"354\" style=\"vertical-align: 0px;\"><\/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-97bee45dee6ebba49cd8a9822ef70308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) }f'(x)=\\cfrac{-19\\cdot 4x}{\\left(2x^2-2\\right)^2}=\\cfrac{-76x}{\\left(2x^2-2\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"273\" 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-11f9c8fda61edb1ce51bd33e022a0a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{C) }f'(x)&amp;=\\cfrac{(24x^2-8x+3)e^{4x}-(8x^3-4x^2+3x)\\cdot 4e^{4x}}{\\left(e^{4x}\\right)^2}\\\\[1.5ex]&amp;=\\cfrac{e^{4x}(24x^2-8x+3-32x^3+16x^2-12x)}{e^{8x}}\\\\[1.5ex]&amp;=\\cfrac{-32x^3+40x^2-20x+3}{e^{4x}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"431\" style=\"vertical-align: 0px;\"><\/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-da9da045ccfc03ecc1d9d44e1ea9caee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) }f'(x)=\\cfrac{-2x\\text{sen}(x^2)\\cdot\\text{sen}(6x)-\\text{cos}(x^2)\\text{cos}(6x)\\cdot 6}{\\text{sen}^2(6x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"414\" 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-ec87daa1a463bacd5a42a1b16e826449_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{E) }f'(x)&amp;=\\cfrac{\\cfrac{3x^2}{x^3+4}\\cdot\\left(4x^2-3x\\right)^3-\\ln(x^3+4)\\cdot 3\\left(4x^2-3x\\right)^2\\cdot (8x-3) }{\\left(\\left(4x^2-3x\\right)^3\\right)^2}\\\\[1.5ex]&amp;=\\cfrac{\\cfrac{3x^2}{x^3+4}\\cdot\\left(4x^2-3x\\right)^3-\\ln(x^3+4)\\cdot 3\\left(4x^2-3x\\right)^2\\cdot (8x-3) }{\\left(4x^2-3x\\right)^6}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"170\" width=\"535\" style=\"vertical-align: 0px;\"><\/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-bef2b22482e39cea7e82047c0d9911b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{F) }f'(x)&amp;=\\cfrac{\\cfrac{2x+4}{2\\sqrt{x^2+4x}}\\cdot 5^{x^2} - \\sqrt{x^2+4x}\\cdot 5^{x^2}\\cdot \\ln(5) \\cdot 2x }{\\left(5^{x^2}\\right)^2}\\\\[1.5ex]&amp;=\\cfrac{\\cfrac{2x+4}{2\\sqrt{x^2+4x}}\\cdot 5^{x^2} - \\sqrt{x^2+4x}\\cdot 5^{x^2}\\cdot \\ln(5) \\cdot 2x }{5^{2x^2}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"155\" width=\"424\" 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-de-un-cociente\"><\/span> Demonstratie van de afgeleide van een quoti\u00ebnt<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ten slotte zullen we de formule demonstreren voor de afgeleide van een deling. Om dit te doen, zullen we de algemene definitie van een derivaat gebruiken, namelijk:<\/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> Laat <em>z<\/em> een deling zijn van twee verschillende functies:<\/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-83357f61a7cd6587a3fd5e5348b056fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=\\cfrac{f(x)}{g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"94\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Dan zal de afgeleide van de functie <em>z,<\/em> waarbij de wiskundige definitie wordt toegepast, 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-db6545eb9e109966a362acf510f101a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\cfrac{f(x+h)}{g(x+h)}-\\cfrac{f(x)}{g(x)}}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"223\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> We lossen het aftrekken van breuken op van de teller van 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-f9ec617a63f72bd4215ccb2b2998525e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\cfrac{f(x+h)\\cdot g(x)}{g(x+h)\\cdot g(x)}-\\cfrac{f(x)\\cdot g(x+h)}{g(x)\\cdot g(x+h)}}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"347\" style=\"vertical-align: -13px;\"><\/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-f8a9e6382fd7033298df2e7955ccd9fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x)-f(x)\\cdot g(x+h)}{h\\cdot g(x)\\cdot g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"343\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Het toevoegen van een optel- en aftrekkingsterm aan een vergelijking verandert de vergelijking niet. We kunnen daarom doorgaan naar de volgende stap:<\/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-8581512accedfadade2e1bbbeec84855_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x)\\color{orange}\\bm{-f(x)\\cdot g(x)}\\color{black}-f(x)\\cdot g(x+h)\\color{orange}\\bm{+f(x)\\cdot g(x)}\\color{black}}{h\\cdot g(x)\\cdot g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"720\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> We halen de gemeenschappelijke factor eruit:<\/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-b8f9b502b411bb77acebc63c00972053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{g(x)\\bigl[f(x+h)-f(x)\\bigr]-f(x)\\bigl[g(x+h)-g(x)\\bigr]}{h\\cdot g(x)\\cdot g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"457\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Laten we nu de term <em>h<\/em> van de noemer naar de teller verplaatsen met behulp van de eigenschappen 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-74338b745f98dd32abeea2df50b88ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{g(x)\\cdot \\cfrac{f(x+h)-f(x)\\cdot g(x)}{h}-f(x)\\cdot\\cfrac{g(x+h)-g(x)}{h}}{g(x)\\cdot g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"503\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> We transformeren de vergelijking door de eigenschappen van de limieten 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-a699ddaf78abfcfbd1aa3993b6a0b033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\frac{g(x)\\cdot \\displaystyle\\lim_{h \\to 0}\\cfrac{f(x+h)-f(x)\\cdot g(x)}{h}-f(x)\\cdot\\lim_{h \\to 0}\\cfrac{g(x+h)-g(x)}{h}}{g(x)\\cdot \\displaystyle\\lim_{h \\to 0}g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"73\" width=\"535\" style=\"vertical-align: -25px;\"><\/p>\n<\/p>\n<p> De grenzen van de teller komen precies overeen met de wiskundige definitie van de afgeleide van elke functie, 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-8735d5a5a43fa63c27443c2fe34a1530_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\frac{g(x)\\cdot f'(x)-f(x)\\cdot g'(x)}{g(x)\\cdot \\displaystyle\\lim_{h \\to 0}g(x+h)}\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"257\" style=\"vertical-align: -25px;\"><\/p>\n<\/p>\n<p> We lossen de limiet van de noemer van de breuk op:<\/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-1ef05117a36f586b6c8441b829bd4c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\frac{g(x)\\cdot f'(x)-f(x)\\cdot g'(x)}{g(x)\\cdot g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"257\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> En zo wordt de formule voor de afgeleide van een quoti\u00ebnt van twee functies gedemonstreerd:<\/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-85517a8cdcfda040b304fbdabe67a5fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\frac{f'(x)\\cdot g(x)-f(x)\\cdot g'(x)}{\\bigl(g(x)\\bigr)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"257\" style=\"vertical-align: -23px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel leggen we uit hoe je uit twee functies een quoti\u00ebnt (of deling) kunt afleiden. Je vindt voorbeelden van afgeleiden van quoti\u00ebnten van functies en daarnaast kun je oefenen met stapsgewijze oefeningen over afgeleiden van delingen. Formule voor de afgeleide van een quoti\u00ebnt De afgeleide van een co\u00ebffici\u00ebnt (of deling) van de functies &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/afgeleide-van-een-delingsquotient\/\"> <span class=\"screen-reader-text\">Afgeleide van een quoti\u00ebnt (of deling)<\/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-80","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>Afgeleide van een quoti\u00ebnt (deling): formule en opgeloste oefeningen<\/title>\n<meta name=\"description\" content=\"We leggen uit hoe je uit twee functies een quoti\u00ebnt (of deling) kunt afleiden. 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