{"id":392,"date":"2023-07-03T09:49:11","date_gmt":"2023-07-03T09:49:11","guid":{"rendered":"https:\/\/mathority.org\/ru\/%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%b0%d1%8f-%d0%b3%d0%b8%d0%bf%d0%b5%d1%80%d0%b1%d0%be%d0%bb%d0%b8%d1%87%d0%b5%d1%81%d0%ba%d0%be%d0%b3%d0%be-%d0%b0%d1%80%d0%ba%d1%82%d0%b0%d0%bd\/"},"modified":"2023-07-03T09:49:11","modified_gmt":"2023-07-03T09:49:11","slug":"%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%b0%d1%8f-%d0%b3%d0%b8%d0%bf%d0%b5%d1%80%d0%b1%d0%be%d0%bb%d0%b8%d1%87%d0%b5%d1%81%d0%ba%d0%be%d0%b3%d0%be-%d0%b0%d1%80%d0%ba%d1%82%d0%b0%d0%bd","status":"publish","type":"post","link":"https:\/\/mathority.org\/ru\/%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%b0%d1%8f-%d0%b3%d0%b8%d0%bf%d0%b5%d1%80%d0%b1%d0%be%d0%bb%d0%b8%d1%87%d0%b5%d1%81%d0%ba%d0%be%d0%b3%d0%be-%d0%b0%d1%80%d0%ba%d1%82%d0%b0%d0%bd\/","title":{"rendered":"\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430"},"content":{"rendered":"<p>\u0417\u0434\u0435\u0441\u044c \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0432\u044b\u0432\u0435\u0441\u0442\u0438 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0412\u044b \u0442\u0430\u043a\u0436\u0435 \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u0443\u0432\u0438\u0434\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u044d\u0442\u043e\u0433\u043e \u0442\u0438\u043f\u0430 \u0438, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043c\u044b \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u0430\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 x \u0440\u0430\u0432\u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0435 \u043c\u0438\u043d\u0443\u0441 x \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0435.<\/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-33ae63a662489900a94430ce0dac1b60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{1-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, <strong>\u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/strong> \u0440\u0430\u0432\u043d\u0430 \u0447\u0430\u0441\u0442\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0434\u0435\u043b\u0435\u043d\u043d\u043e\u043c\u0443 \u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0443 \u043c\u0438\u043d\u0443\u0441 \u0443\u043a\u0430\u0437\u0430\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0435.<\/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-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> \u041f\u043e \u0441\u0443\u0442\u0438, \u043e\u0431\u0435 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b, \u043d\u043e \u0432\u043e \u0432\u0442\u043e\u0440\u043e\u0439 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0437\u0430\u043c\u0435\u043d\u0430 x \u043d\u0430 u \u0434\u0430\u0435\u0442 \u043d\u0430\u043c \u0432 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u043f\u0435\u0440\u0432\u0443\u044e \u0444\u043e\u0440\u043c\u0443\u043b\u0443, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f x \u0440\u0430\u0432\u043d\u0430 1. <\/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-de-larctangente-hyperbolique.webp\" alt=\"\u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430\" class=\"wp-image-2343\" width=\"393\" height=\"298\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> \u0422\u043e\u0447\u043d\u043e \u0442\u0430\u043a \u0436\u0435, \u043a\u0430\u043a \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439 \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430, \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430. \u041d\u0435\u0441\u043c\u043e\u0442\u0440\u044f \u043d\u0430 \u044d\u0442\u043e, \u0438\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043e\u0447\u0435\u043d\u044c \u0440\u0430\u0437\u043d\u044b\u0435, \u0432\u044b \u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u044d\u0442\u043e\u0439 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0437\u0434\u0435\u0441\u044c:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>\u0421\u043c.:<\/strong> <span style=\"text-decoration: underline;\"> \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span> \u041f\u0440\u0438\u043c\u0435\u0440\u044b \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u041f\u0440\u0438\u043c\u0435\u0440 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-ad1cd9320973ca2c5d2b83434086f629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u041f\u043e \u043b\u043e\u0433\u0438\u043a\u0435 \u0432\u0435\u0449\u0435\u0439 \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430:<\/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-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> \u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f 2\u0445 \u0440\u0430\u0432\u043d\u0430 2, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432 \u0447\u0438\u0441\u043b\u0438\u0442\u0435\u043b\u0435 \u0434\u0440\u043e\u0431\u0438 \u043f\u043e\u0441\u0442\u0430\u0432\u044c\u0442\u0435 \u0434\u0432\u043e\u0439\u043a\u0443, \u0430 \u0432 \u0437\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u0435 &#8212; \u0435\u0434\u0438\u043d\u0438\u0446\u0443 \u043c\u0438\u043d\u0443\u0441 2\u0445 \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0435:<\/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-9820d63e99b4b29c41d6fd14a3426815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2}{1-(2x)^2}}=\\cfrac{2}{1- 4x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"528\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> \u041f\u0440\u0438\u043c\u0435\u0440 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-3e4e12eb6cf782403fe0de4f37bc025f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(e^{3x})\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0427\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430.<\/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-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0441\u043e\u0441\u0442\u0430\u0432\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043d\u0430\u043c \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u0442\u0440\u0435\u0431\u0443\u0435\u0442\u0441\u044f \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u0442\u044c \u0446\u0435\u043f\u043d\u043e\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u043e: <\/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-0eb0da6a9477e040476051a829238c84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(e^{3x}) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{3\\cdot e^{3x}}{1-\\left(e^{3x}\\right)^2}=\\cfrac{3e^{3x}}{1-3^{6x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"534\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u0412 \u044d\u0442\u043e\u043c \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u043c\u044b \u043f\u0440\u043e\u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u0443\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430.<\/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-17261fa2031302bfad1883eb39b7116d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\text{arctanh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u0435 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u043f\u043e-\u0434\u0440\u0443\u0433\u043e\u043c\u0443:<\/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-27ba9a49fdc790b3131113b5ae592e2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\text{tanh}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0422\u0435\u043f\u0435\u0440\u044c \u043f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c \u043e\u0431\u0435 \u0447\u0430\u0441\u0442\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f:<\/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-4f8879b6d2f8df36bd6f3e5c817f21cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1=\\cfrac{1}{\\text{cosh}^2(y)}\\cdot y'\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> \u041c\u044b \u043e\u0447\u0438\u0449\u0430\u0435\u043c \u0432\u0430\u0441:<\/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-3bb09e461662267f0cbab52cf6e0bcac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\text{cosh}^2(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0421 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u043c\u044b \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u0440\u0430\u0437\u043d\u0438\u0446\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u0438 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0441\u0438\u043d\u0443\u0441\u0430 \u0434\u0430\u0435\u0442 1. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u044c \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u0434\u0440\u043e\u0431\u044c:<\/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-e726904c011eb3ab9ff264426988d029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}^2(y)-\\text{senh}^2(y)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"183\" 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-1820ba4560d8d0109af605b6e2757c93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\text{cosh}^2(y)}{1}=\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)-\\text{senh}^2(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"281\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> \u0412\u0441\u0435 \u0447\u043b\u0435\u043d\u044b \u0434\u0440\u043e\u0431\u0438 \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u043c \u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430:<\/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-9cc80abe73f130f4ec1c39cbf5d7e8ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)}}{\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)}-\\cfrac{\\text{senh}^2(y)}{\\text{cosh}^2(y)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"195\" style=\"vertical-align: -46px;\"><\/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-b150bbe95858decf7312b869b95d24b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-\\cfrac{\\text{senh}^2(y)}{\\text{cosh}^2(y)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"138\" style=\"vertical-align: -46px;\"><\/p>\n<\/p>\n<p> \u0427\u0430\u0441\u0442\u043d\u043e\u0435 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0441\u0438\u043d\u0443\u0441\u0430 \u043c\u0435\u0436\u0434\u0443 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043e\u043c \u0440\u0430\u0432\u043d\u043e \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u043c\u0443 \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0443, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e:<\/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-12f286528bc0635705aadbe510b6ceb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x)=\\cfrac{\\text{senh}(x)}{\\text{cosh}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" 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-725424805ce03fcabd470e9448c91f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-\\text{tanh}^2(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"136\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> \u041d\u043e, \u043a\u0430\u043a \u043c\u044b \u0432\u0438\u0434\u0435\u043b\u0438 \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430, \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u0435\u043d \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 x, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435, \u043f\u043e\u043b\u0443\u0447\u0438\u0432 \u0442\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430: <\/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-b90ecc88a8cbc7f110840727da48e632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"88\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"articulos-relacionados\"><\/span> \u041f\u043e\u0445\u043e\u0436\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> <span style=\"text-decoration: underline;\"> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"> \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0430\u0440\u043a\u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"> \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 <\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u0417\u0434\u0435\u0441\u044c \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0432\u044b\u0432\u0435\u0441\u0442\u0438 \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0412\u044b \u0442\u0430\u043a\u0436\u0435 \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u0443\u0432\u0438\u0434\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u044d\u0442\u043e\u0433\u043e \u0442\u0438\u043f\u0430 \u0438, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u043c\u044b 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\u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0435. \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0440\u0430\u0432\u043d\u0430 \u0447\u0430\u0441\u0442\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0434\u0435\u043b\u0435\u043d\u043d\u043e\u043c\u0443 \u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0443 \u043c\u0438\u043d\u0443\u0441 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/ru\/%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%b0%d1%8f-%d0%b3%d0%b8%d0%bf%d0%b5%d1%80%d0%b1%d0%be%d0%bb%d0%b8%d1%87%d0%b5%d1%81%d0%ba%d0%be%d0%b3%d0%be-%d0%b0%d1%80%d0%ba%d1%82%d0%b0%d0%bd\/\"> <span class=\"screen-reader-text\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430<\/span> \u0427\u0438\u0442\u0430\u0442\u044c \u0434\u0430\u043b\u0435\u0435 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\u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 x \u0440\u0430\u0432\u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0435 \u043c\u0438\u043d\u0443\u0441 x \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0435. \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0433\u0438\u043f\u0435\u0440\u0431\u043e\u043b\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0440\u0430\u0432\u043d\u0430 \u0447\u0430\u0441\u0442\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0434\u0435\u043b\u0435\u043d\u043d\u043e\u043c\u0443 \u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0443 \u043c\u0438\u043d\u0443\u0441 &hellip; 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