{"id":39,"date":"2023-09-17T11:00:46","date_gmt":"2023-09-17T11:00:46","guid":{"rendered":"https:\/\/mathority.org\/it\/derivata-di-un-quoziente-di-divisione\/"},"modified":"2023-09-17T11:00:46","modified_gmt":"2023-09-17T11:00:46","slug":"derivata-di-un-quoziente-di-divisione","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/derivata-di-un-quoziente-di-divisione\/","title":{"rendered":"Derivata di un quoziente (o divisione)"},"content":{"rendered":"<p>In questo articolo spieghiamo come ricavare un quoziente (o divisione) da due funzioni. Troverai esempi di derivate di quozienti di funzioni e, inoltre, potrai esercitarti con esercizi passo passo sulle derivate di divisioni. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-un-cociente\"><\/span> Formula per la derivata di un quoziente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> La <strong>derivata di un coefficiente (o divisione) delle funzioni<\/strong> \u00e8 identica alla derivata della funzione numeratore per la funzione denominatore minore della funzione numeratore per la derivata della funzione denominatore divisa per il quadrato della funzione denominatore alto. <\/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=\"formula per la derivata di una divisione o quoziente\" class=\"wp-image-2194\" width=\"326\" height=\"304\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Come puoi vedere, quando applichiamo la regola per la derivata di un quoziente (o di una divisione) abbiamo ancora una frazione dopo la differenziazione. Ma, in pi\u00f9, al numeratore abbiamo due moltiplicazioni e una sottrazione, e il denominatore \u00e8 elevato alla potenza di due. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-derivadas-de-cocientes\"><\/span> Esempi di derivate di quozienti<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Abbiamo appena visto qual \u00e8 la formula per la derivata di un quoziente di due funzioni, risolveremo poi alcuni esempi di derivate di questo tipo di operazioni. Ricorda, se non capisci come si ricava un quoziente funzionale, puoi chiedercelo nella sezione commenti.<\/p>\n<h3 class=\"wp-block-heading\"> Esempio 1<\/h3>\n<p> In questo esempio, deriveremo una funzione potenziale divisa per una funzione trigonometrica:<\/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> La formula per la derivata di una divisione di due funzioni diverse \u00e8 la seguente:<\/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> Quindi dobbiamo prima calcolare separatamente la derivata di ciascuna funzione: <\/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> La derivata dell&#8217;intera funzione \u00e8 quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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\"> Esempio 2<\/h3>\n<p> In questo caso troveremo la derivata di una costante divisa per una funzione:<\/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> Come abbiamo visto sopra, la regola per la derivata di una divisione di due funzioni diverse \u00e8 la seguente:<\/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> Quindi, calcoliamo separatamente la derivata del numeratore e del denominatore: <\/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> Infine troviamo la derivata della divisione intera:<\/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> Infatti, possiamo ricavare una formula per differenziare direttamente quando abbiamo una costante al numeratore divisa per una funzione, perch\u00e9 la derivata della costante \u00e8 sempre 0. Pertanto, la seguente formula sar\u00e0 sempre vera:<\/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\"> Esempio 3<\/h3>\n<p> In questo esercizio deriveremo il quoziente di due polinomi:<\/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> Per risolvere la derivata dobbiamo applicare la regola della derivata di un quoziente di due funzioni diverse, che \u00e8 la seguente:<\/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> Ora troviamo la derivata del polinomio al numeratore e del polinomio al denominatore: <\/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> La derivata della divisione dei polinonimi \u00e8 quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> E infine, eseguiamo le operazioni e semplifichiamo il pi\u00f9 possibile la frazione: <\/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> Esercizi risolti sulla derivata di un quoziente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Derivare le seguenti divisioni di funzioni: <\/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>Vedi la soluzione<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> Dimostrazione della derivata di un quoziente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Infine, dimostreremo la formula per la derivata di una divisione. Per fare ci\u00f2, utilizzeremo la definizione generale di derivata, che \u00e8:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> Sia <em>z<\/em> la divisione di due diverse funzioni:<\/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> Quindi, la derivata della funzione <em>z<\/em> applicando la definizione matematica sar\u00e0:<\/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> Risolviamo la sottrazione delle frazioni dal numeratore della frazione:<\/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> L&#8217;aggiunta di un termine di addizione e sottrazione a un&#8217;equazione non modifica l&#8217;equazione. Possiamo quindi passare allo step successivo:<\/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> Estraiamo il fattore comune:<\/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> Ora spostiamo il termine <em>h<\/em> dal denominatore al numeratore utilizzando le propriet\u00e0 delle frazioni:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> Trasformiamo l&#8217;equazione applicando le propriet\u00e0 dei limiti:<\/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> I limiti del numeratore corrispondono proprio alla definizione matematica della derivata di ciascuna funzione, quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> Risolviamo il limite del denominatore della frazione:<\/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> E cos\u00ec si dimostra la formula per la derivata di un quoziente di due funzioni:<\/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 questo articolo spieghiamo come ricavare un quoziente (o divisione) da due funzioni. Troverai esempi di derivate di quozienti di funzioni e, inoltre, potrai esercitarti con esercizi passo passo sulle derivate di divisioni. Formula per la derivata di un quoziente La derivata di un coefficiente (o divisione) delle funzioni \u00e8 identica alla derivata della funzione &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/derivata-di-un-quoziente-di-divisione\/\"> <span class=\"screen-reader-text\">Derivata di un quoziente (o divisione)<\/span> Leggi altro &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[6],"tags":[],"class_list":["post-39","post","type-post","status-publish","format-standard","hentry","category-derivati"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Derivata di un quoziente (divisione): formula ed esercizi svolti<\/title>\n<meta name=\"description\" content=\"Spieghiamo come derivare un quoziente (o divisione) da due funzioni. 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