{"id":401,"date":"2023-07-03T01:50:55","date_gmt":"2023-07-03T01:50:55","guid":{"rendered":"https:\/\/mathority.org\/it\/derivato-del-larccosecante-iperbolico\/"},"modified":"2023-07-03T01:50:55","modified_gmt":"2023-07-03T01:50:55","slug":"derivato-del-larccosecante-iperbolico","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/derivato-del-larccosecante-iperbolico\/","title":{"rendered":"Derivata dell&#39;arco cosecante iperbolico"},"content":{"rendered":"<p>In questo articolo spieghiamo come derivare l&#8217;arcosecante iperbolico di una funzione. Troverai anche esempi pratici della derivata dell&#8217;arcosecante iperbolico. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-arcocosecante-hiperbolica\"><\/span> Formula per la derivata dell&#8217;arco cosecante iperbolico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La derivata dell&#8217;arco cosecante iperbolico di x \u00e8 uguale a meno 1 diviso per il prodotto di x per la radice di uno pi\u00f9 x al quadrato.<\/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-0367e01bffbaea95b002c0261e6597d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-1}{x\\sqrt{1+x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Pertanto, la <strong>derivata dell&#8217;arco cosecante iperbolico di una funzione<\/strong> \u00e8 meno la derivata di detta funzione divisa per il prodotto della funzione per la radice di uno pi\u00f9 la funzione quadrata.<\/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-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> In sintesi, la formula per calcolare la derivata della funzione arcosecante iperbolica \u00e8: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-de-larccosecante-hyperbolique.webp\" alt=\"derivata dell'arco cosecante iperbolico\" class=\"wp-image-2797\" width=\"365\" height=\"266\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Anche se abbiamo messo due formule, ci\u00f2 non significa che siano diverse. Se guardi da vicino, la seconda formula \u00e8 come la prima ma applica la regola della catena. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-arcocosecante-hiperbolica\"><\/span> Esempi di derivata dell&#8217;arco cosecante iperbolico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Data la formula per la derivata dell&#8217;arco cosecante iperbolico, deriveremo quindi due di queste funzioni in modo da poter vedere come si fa.<\/p>\n<h3 class=\"wp-block-heading\"> Esempio 1<\/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-24a3e6958bf4fd28f73062bb78280dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In questo esercizio dobbiamo utilizzare la formula per la derivata dell&#8217;arco cosecante iperbolico con la regola della catena, perch\u00e9 nell&#8217;argomento \u00e8 presente una funzione diversa da x:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Quindi, per trovare la derivata, dobbiamo sostituire la u con 3x e la u&#8217; con la sua derivata, che \u00e8 3:<\/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-aa1b0c4d4233e274d3996620bacdb869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-3}{3x\\sqrt{1+(3x)^2}}=\\cfrac{-3}{3x\\sqrt{1+9x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"582\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Esempio 2<\/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-a99a903e923a1fd4dc9442e903f244ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(x^5-2x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In questo caso abbiamo una funzione polinomiale nell&#8217;argomento arcosecante iperbolico, quindi dobbiamo usare anche la regola della catena per derivarla:<\/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-c84a1cfa851e7f8db30d3953efcf1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arccsch}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{-u'}{u\\sqrt{1+u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"435\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Quindi inseriamo la derivata della funzione argomento al numeratore della frazione e al denominatore cambiamo la u con la funzione polinomiale: <\/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-6257584425bd348ba75c7680d8ff6f70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f(x)=\\text{arccsch}(x^5-2x^3) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}f'(x)&amp;=\\cfrac{-(5x^4-6x^2)}{(x^5-2x^3)\\sqrt{1+(x^5-2x^3)^2}}\\\\[1.5ex] &amp;=\\cfrac{-5x^4+6x^2}{(x^5-2x^3)\\sqrt{1+(x^5-2x^3)^2}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"634\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"articulos-relacionados\"><\/span> Articoli simili<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivato-dellarcocosina-iperbolica\/\">Derivata dell&#8217;arco coseno iperbolico<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivata-della-cosecante-iperbolica\/\">Derivata della cosecante iperbolica<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivata-del-coseno-iperbolico\/\">Derivata del coseno iperbolico<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivato-arcocosecante\/\">Derivata dell&#8217;arco cosecante<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/deriva-dalla-cosecante\/\">derivato dalla cosecante<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/derivato-della-larccosina\/\">Derivata dell&#8217;arcocoseno<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/deriva-dal-coseno\/\">derivata del coseno<\/a><\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>In questo articolo spieghiamo come derivare l&#8217;arcosecante iperbolico di una funzione. Troverai anche esempi pratici della derivata dell&#8217;arcosecante iperbolico. Formula per la derivata dell&#8217;arco cosecante iperbolico La derivata dell&#8217;arco cosecante iperbolico di x \u00e8 uguale a meno 1 diviso per il prodotto di x per la radice di uno pi\u00f9 x al quadrato. Pertanto, la &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/derivato-del-larccosecante-iperbolico\/\"> <span class=\"screen-reader-text\">Derivata dell&#39;arco cosecante iperbolico<\/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-401","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 dell&#039;arco cosecante iperbolico - 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\/derivato-del-larccosecante-iperbolico\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivata dell&#039;arco cosecante iperbolico - Mathority\" \/>\n<meta property=\"og:description\" content=\"In questo articolo spieghiamo come derivare l&#8217;arcosecante iperbolico di una funzione. Troverai anche esempi pratici della derivata dell&#8217;arcosecante iperbolico. Formula per la derivata dell&#8217;arco cosecante iperbolico La derivata dell&#8217;arco cosecante iperbolico di x \u00e8 uguale a meno 1 diviso per il prodotto di x per la radice di uno pi\u00f9 x al quadrato. 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Troverai anche esempi pratici della derivata dell&#8217;arcosecante iperbolico. Formula per la derivata dell&#8217;arco cosecante iperbolico La derivata dell&#8217;arco cosecante iperbolico di x \u00e8 uguale a meno 1 diviso per il prodotto di x per la radice di uno pi\u00f9 x al quadrato. 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