{"id":384,"date":"2023-07-03T18:53:21","date_gmt":"2023-07-03T18:53:21","guid":{"rendered":"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/"},"modified":"2023-07-03T18:53:21","modified_gmt":"2023-07-03T18:53:21","slug":"derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/","title":{"rendered":"Derivata di una funzione logaritmica"},"content":{"rendered":"<p>Qui troverai come risolvere la derivata di una funzione logaritmica in qualsiasi base (formula). Inoltre, potrai esercitarti con esercizi passo passo sulle derivate delle funzioni logaritmiche.<\/p>\n<p> <strong>La formula per dividere una funzione logaritmica varia a seconda che il logaritmo sia naturale (in base e) o un&#8217;altra base<\/strong> . Quindi vedremo prima le due formule separatamente con un esempio per ogni caso, e poi faremo un riassunto delle due regole. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-un-logaritmo-natural-o-neperiano\"><\/span> Derivato di un naturale o logaritmo naturale<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La derivata di un logaritmo naturale (o logaritmo naturale) \u00e8 il quoziente della derivata dell&#8217;argomento del logaritmo diviso per la funzione dell&#8217;argomento.<\/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-491d95a8a33d226da4fc5d62a8e70f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"335\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Logicamente, se la funzione all&#8217;interno del logaritmo \u00e8 la funzione identit\u00e0, al numeratore della derivata rimane un 1:<\/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-1519db0b90e430fab54b04113c435118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"331\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Guarda il seguente esempio in cui viene risolta la derivata del logaritmo naturale di 3x:<\/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-420e2f2cb107eb22019157bcb76c5645_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{3}{3x}=\\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"384\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Ricorda che il logaritmo naturale \u00e8 un logaritmo la cui base \u00e8 il numero e (numero di Eulero). <\/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-d5698ad2315473c75950453c15326f81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\ln(x)=\\log_e(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-un-logaritmo-en-base-a\"><\/span> Derivata di un logaritmo basato su<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La derivata di un logaritmo rispetto a qualsiasi base \u00e8 uguale a 1 diviso per il prodotto di x volte il logaritmo naturale della base del logaritmo originale.<\/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-8b37542882d2bccf84707a3341af5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x\\cdot\\ln(a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"397\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Quindi se applichiamo la regola della catena, la regola della derivata logaritmica \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-2fac5d7501d02d27d74a95272a64e756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\ln(a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"397\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Ad esempio, la derivata del logaritmo in base 2 di x al quadrato \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-e934db01ce50b0ef6f597d5952637cfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_2(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2x}{x^2\\cdot\\ln(2)}=\\cfrac{2}{x\\ln(2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"488\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-una-funcion-logaritmica\"><\/span> Formula per la derivata di una funzione logaritmica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Considerando la definizione di derivata logaritmica e le sue due possibili varianti, ecco un riassunto delle due formule per facilitarne la memorizzazione. <\/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\/fonction-logarithmique-derivee.webp\" alt=\"derivata di una funzione logaritmica\" class=\"wp-image-1842\" width=\"395\" height=\"279\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-derivadas-de-funciones-logaritmicas\"><\/span> Risolti problemi di derivate di funzioni logaritmiche<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Derivare la seguente funzione logaritmica: <\/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-8118283c3444e6c13b9aefdf0d8a11aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log(3x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" 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 soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In questo caso \u00e8 necessario risolvere la derivata di un logaritmo in base decimale, dobbiamo quindi applicare la seguente formula:<\/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-2fac5d7501d02d27d74a95272a64e756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\ln(a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"397\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La derivata del logaritmo in base 10 \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-f340fe2c5f62e4f0c4499aca10845cf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log(3x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{6x}{3x^2\\cdot \\ln(10)}=\\cfrac{2}{x \\ln(10)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"516\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ricorda che se un logaritmo non ha base significa che la sua base \u00e8 10.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 2<\/h3>\n<p> Deriva il seguente logaritmo naturale (o naturale): <\/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-a44fc37965c07092cdfa5cb2679a8b8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln\\left(x^3+4x^2\\right)^5\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"165\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La funzione in questo problema \u00e8 un logaritmo naturale, quindi dobbiamo utilizzare la seguente regola per derivare la funzione logaritmica:<\/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-491d95a8a33d226da4fc5d62a8e70f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"335\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La derivata del logaritmo naturale \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-fc06150c0093afdd84076e69171b7d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=\\cfrac{5\\left(x^3+4x^2\\right)^4\\cdot (3x^2+8x)}{\\left(x^3+4x^2\\right)^5}\\\\[2ex] &amp;=\\cfrac{5\\cdot (3x^2+8x)}{x^3+4x^2}\\\\[2ex] &amp;=\\cfrac{15x^2+40x}{x^3+4x^2}\\\\[2ex] &amp;=\\cfrac{15x+40}{x^2+4x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"245\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 3<\/h3>\n<p> Ricavare il seguente logaritmo: <\/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-5682f4fd6180c07879cfe9fb6a4b2583_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_7(x^5+7x^2-3x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"239\" 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 soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In questo esercizio dobbiamo ricavare un logaritmo in base 7, quindi utilizzeremo la seguente formula:<\/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-2fac5d7501d02d27d74a95272a64e756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\ln(a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"397\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E la derivata del logaritmo \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-182fbf09950c4930013d2f863888bdd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{5x^4+14x-3}{(x^5+7x^2-3x+1)\\cdot \\ln(7)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"267\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 4<\/h3>\n<p> Trova la derivata della seguente funzione logaritmica con una 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-218d7543ba82562bbf91b5f4e0ca3f1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\log_4\\left(\\frac{5x}{8x^2-1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"176\" 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>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per risolvere la derivata logaritmica possiamo prima semplificare la funzione applicando le propriet\u00e0 dei logaritmi:<\/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-2d0cedfe8d0bd4de138099938b10e39f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_4(5x)-\\log_4(8x^2-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora dobbiamo usare due volte la formula della derivata logaritmica, ma entrambe le derivate sono pi\u00f9 facili da calcolare.<\/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-2fac5d7501d02d27d74a95272a64e756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u\\cdot \\ln(a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"397\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In sintesi la derivata della funzione \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-85874bff9f3259727a78b50aece1f1e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=\\cfrac{5}{5x\\cdot \\ln(4)}-\\cfrac{16x}{(8x^2-1)\\cdot \\ln(4)}\\\\[2ex]&amp;=\\cfrac{1}{x\\ln(4)}-\\cfrac{16x}{(8x^2-1)\\ln(4)}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"111\" width=\"279\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 5<\/h3>\n<p> Calcola la derivata della seguente funzione logaritmica con una radice: <\/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-9518b9623edc80e9f5de230edb5e573c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln\\left(\\sqrt[4]{\\text{cos}(9x)}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"170\" style=\"vertical-align: -11px;\"><\/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 soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per prima cosa semplificheremo la funzione utilizzando le propriet\u00e0 dei logaritmi: <\/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-95cc0f73a05b0cde647035b17d0fed60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln\\left(\\text{cos}(9x)\\right)^{\\frac{1}{4}}\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"154\" 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-fac1c4306bdc844dc069a28c995e5dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{1}{4}\\ln\\left(\\text{cos}(9x)\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"161\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E una volta eliminato il radicale dalla funzione, usiamo la regola per la derivata del naturale o logaritmo naturale:<\/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-491d95a8a33d226da4fc5d62a8e70f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{u}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"335\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertanto, la derivata della funzione logaritmica composta \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-440b871bd7321bb0121db9a588adde6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)&amp;=\\cfrac{1}{4}\\cdot \\cfrac{-\\text{sen}(9x)\\cdot 9}{\\text{cos}(9x)}=\\cfrac{-9\\text{sen}(9x)}{4\\text{cos}(9x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"304\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Qui troverai come risolvere la derivata di una funzione logaritmica in qualsiasi base (formula). Inoltre, potrai esercitarti con esercizi passo passo sulle derivate delle funzioni logaritmiche. La formula per dividere una funzione logaritmica varia a seconda che il logaritmo sia naturale (in base e) o un&#8217;altra base . Quindi vedremo prima le due formule separatamente &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\"> <span class=\"screen-reader-text\">Derivata di una funzione logaritmica<\/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-384","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>Derivato di una funzione logaritmica - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivato di una funzione logaritmica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Qui troverai come risolvere la derivata di una funzione logaritmica in qualsiasi base (formula). Inoltre, potrai esercitarti con esercizi passo passo sulle derivate delle funzioni logaritmiche. La formula per dividere una funzione logaritmica varia a seconda che il logaritmo sia naturale (in base e) o un&#8217;altra base . Quindi vedremo prima le due formule separatamente &hellip; Derivata di una funzione logaritmica Leggi altro &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-03T18:53:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-491d95a8a33d226da4fc5d62a8e70f61_l3.png\" \/>\n<meta name=\"author\" content=\"Squadra di Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Scritto da\" \/>\n\t<meta name=\"twitter:data1\" content=\"Squadra di Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tempo di lettura stimato\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minuti\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\"},\"author\":{\"name\":\"Squadra di Mathority\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8\"},\"headline\":\"Derivata di una funzione logaritmica\",\"datePublished\":\"2023-07-03T18:53:21+00:00\",\"dateModified\":\"2023-07-03T18:53:21+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\"},\"wordCount\":456,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/it\/#organization\"},\"articleSection\":[\"Derivati\"],\"inLanguage\":\"it-IT\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\",\"url\":\"https:\/\/mathority.org\/it\/derivata-di-una-funzione-logaritmica-logaritmo-naturale-neperiano\/\",\"name\":\"Derivato di una funzione logaritmica - 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