{"id":36,"date":"2023-09-17T11:02:01","date_gmt":"2023-09-17T11:02:01","guid":{"rendered":"https:\/\/mathority.org\/it\/derivata-di-una-somma\/"},"modified":"2023-09-17T11:02:01","modified_gmt":"2023-09-17T11:02:01","slug":"derivata-di-una-somma","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/derivata-di-una-somma\/","title":{"rendered":"Derivato da una somma"},"content":{"rendered":"<p>Qui spieghiamo come derivare una somma di funzioni (formula). Inoltre, potrai vedere esempi di derivate di somme e potrai anche esercitarti con esercizi risolti sulla derivata di una somma. E infine troverai la dimostrazione della formula per la derivata di una somma. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-una-suma\"><\/span> Formula per la derivata di una somma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La derivata di una somma di due funzioni \u00e8 uguale alla somma delle derivate di ciascuna funzione separatamente.<\/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-cec257ea65c3d37da6460f8f1ab6bb1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)+g(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=f'(x)+g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"463\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In altre parole, derivare due funzioni separatamente e poi sommarle equivale a sommare prima le funzioni e poi ricavare la derivata. <\/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-dune-somme.webp\" alt=\"derivato da una somma\" class=\"wp-image-2035\" width=\"243\" height=\"243\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Tieni presente che la regola della derivata dell&#8217;addizione si applica anche alla sottrazione, quindi se una funzione ha davanti un segno negativo invece di un segno positivo, dobbiamo usare la stessa formula anche per differenziarla.<\/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-6a93098fe108be6019d44ba2d0f82108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)\\pm g(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=f'(x)\\pm g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"463\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Inoltre, l&#8217;addizione \u00e8 un&#8217;operazione che possiede la propriet\u00e0 associativa, ovvero che il numero di addizioni coinvolte nell&#8217;addizione \u00e8 indifferente, poich\u00e9 la derivata dell&#8217;intera funzione continuer\u00e0 ad essere la somma della derivata di ciascuna 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-05ceda7dfbdbac4960b012f62d17c8a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=f(x)\\pm g(x) \\pm h(x)\\pm \\dots\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex]z'(x)=f'(x)\\pm g'(x)\\pm h'(x)\\pm \\dots\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-una-suma\"><\/span> Esempi di derivata di una somma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Una volta che abbiamo visto qual \u00e8 la formula per la derivata di una somma, vedremo diversi esempi di derivate di questo tipo di operazioni per comprendere appieno come si derivano le somme di funzioni. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-una-suma-de-funciones-potenciales\"><\/span> Esempio 1: Derivata di una somma di funzioni potenziali<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-efca27bad0818b86e6e42ee15a31ed6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x^2+5x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> La derivata della somma di due funzioni \u00e8 uguale alla derivata di ciascuna funzione separatamente. Pertanto, calcoliamo prima la derivata di ciascuna funzione separatamente: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-69\">\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-5e625836da38ce78d816c887fd842fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx} \\ 3x^2=6x\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"95\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\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-f575779fe1f9c29e907e67f622cb5411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 5x=5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Pertanto, la derivata dell&#8217;intera funzione sar\u00e0 la somma delle due derivate calcolate: <\/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-8a8dcfca37df757df3fd79292ead67b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=6x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-una-suma-de-funciones-diferentes\"><\/span> Esempio 2: Derivata di una somma di funzioni diverse<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-0c94ae34a2f2fe6741b9233bde69c420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}(x)+\\ln(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"166\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Per differenziare la somma delle funzioni, \u00e8 necessario differenziare le due funzioni separatamente e poi sommarle. Ricaviamo quindi le funzioni: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-72\">\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-e1fd0d9e783b091d26fe14a6f9523801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx} \\ \\text{sen}(x)=\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"144\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\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-630f2b7757bfc7fbc7fe540313c526b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ \\ln (x)=\\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"103\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> E poi aggiungiamo le due derivate trovate: <\/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-d74c893fd6ae38b019a934d75b39b921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\text{cos}(x)+\\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"145\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-una-suma-al-cuadrado\"><\/span> Esempio 3: derivata di una somma quadrata<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-fc55182666ca330acaf33748f94e1b06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(3x^4+7x^2+1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"187\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> In questo caso abbiamo una funzione composta, poich\u00e9 abbiamo una somma di funzioni elevate a una potenza. Pertanto, dobbiamo applicare la regola della catena per derivare l&#8217;intera 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-3d5f04a22b460b05368b60eba4a74c4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2\\left(3x^4+7x^2+1\\right)\\cdot (12x^3+14x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"303\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Vedi:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-potenziale-di-potenza\/\">ricavare una potenza<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-derivadas-de-sumas-de-funciones\"><\/span> Esercizi risolti sulle derivate di somme di funzioni<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Derivare le seguenti somme 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-ae19a6909252c6385e64b78c67688db4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=6x^3+9x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"159\" 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-d7239b728b0ae9ef7f14e34910f71c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=x^4+10x^3+5x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" 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-d421d72da5a4af38dda14af4151fba4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f(x)=3x^2-4x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"182\" 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-2df5e88a0c25ffd19b52aa2c950a1c1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\text{cos}(x)+e^{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"177\" 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-3f844607b5234ee8282802ef2ad19dac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=\\left(x^3+4x^2+6x\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"213\" 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-202ed150e3c26b6b56de412d7f8a5884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f(x)=\\log_3(8x^2+2x)-x^7+e^{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"277\" style=\"vertical-align: -5px;\"><\/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-c5e6d4e0f405a797739dba71003c60ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=18x^2+18x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"174\" 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-1069a5e43f43a569cfb3db5f8d9b7331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=4x^3+30x^2+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"202\" 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-93b4f2f1938846ce64f102e4964cbbef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f'(x)=6x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" 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-190b6cc92e8c9a9119d0255247b7e14e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=-\\text{sen}(x)+3e^{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"205\" 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-fa0eabdbceafaf3e9b7535da9d1d520a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=3\\left(x^3+4x^2+6x\\right)^2\\cdot (3x^2+8x+6)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"354\" 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-40fc02db80aa8a889f70ca9f25d295c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f'(x)=\\cfrac{16x+2}{(8x^2+2x)\\ln(3)}-7x^6+2x\\cdot e^{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"334\" style=\"vertical-align: -17px;\"><\/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-formula-de-la-derivada-de-una-suma\"><\/span> Dimostrazione della formula per la derivata di una somma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> In quest&#8217;ultima sezione dimostreremo la formula per la derivata di una somma di funzioni. E, per fare ci\u00f2, ricorriamo alla definizione matematica della derivata, 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-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 quindi z la somma 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-4d2d91569fa3b33d834dead8ca02ff3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)+g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" 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-17f0dc216847872551637f676d1ff0b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{z(x+h)-z(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"215\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Sostituiamo ora z alla somma delle funzioni nell&#8217;espressione limite:<\/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-feab6b99cc6e3bfad015b988f16c5241_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{\\bigl[f(x+h)+g(x+h)\\bigr]-\\bigl[f(x)+g(x)\\bigr]}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"389\" 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-73d3c0e1781a7ec65721ab6aef3d840a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"359\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Trasformiamo la frazione per avere la somma di due frazioni, ciascuna corrispondente a ciascuna funzione di addizione:<\/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-e7b00aae0064e00a5e8a2f57e54fa339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\left[\\frac{f(x+h)-f(x)}{h}+\\frac{g(x+h)-g(x)}{h}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"379\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Grazie alle propriet\u00e0 dei limiti possiamo separare l&#8217;espressione precedente in due limiti, poich\u00e9 il limite di una somma equivale alla somma 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-6c8091e8c717504ae86efff94b5d8e3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}+\\lim_{h \\to 0}\\frac{g(x+h)-g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"394\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> E, come abbiamo visto sopra nella definizione di derivata, ogni limite corrisponde alla derivata di una funzione. Si ottiene quindi la seguente uguaglianza:<\/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-cea8c29e93381736297a8f6aa46c83fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=f'(x)+g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"158\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Qui spieghiamo come derivare una somma di funzioni (formula). Inoltre, potrai vedere esempi di derivate di somme e potrai anche esercitarti con esercizi risolti sulla derivata di una somma. E infine troverai la dimostrazione della formula per la derivata di una somma. Formula per la derivata di una somma La derivata di una somma di &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/derivata-di-una-somma\/\"> <span class=\"screen-reader-text\">Derivato da una somma<\/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-36","post","type-post","status-publish","format-standard","hentry","category-derivati"],"yoast_head":"<!-- This site is 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