{"id":33373,"date":"2024-11-30T13:00:12","date_gmt":"2024-11-30T13:00:12","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33373"},"modified":"2025-07-21T02:52:37","modified_gmt":"2025-07-21T02:52:37","slug":"%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d1%8b%d0%b5-%d0%bc%d0%bd%d0%be%d0%b3%d0%be%d1%87%d0%bb%d0%b5%d0%bd%d0%be%d0%b2-%d1%82%d1%80%d0%b8%d0%b3%d0%be%d0%bd%d0%be%d0%bc%d0%b5%d1%82","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d1%8b%d0%b5-%d0%bc%d0%bd%d0%be%d0%b3%d0%be%d1%87%d0%bb%d0%b5%d0%bd%d0%be%d0%b2-%d1%82%d1%80%d0%b8%d0%b3%d0%be%d0%bd%d0%be%d0%bc%d0%b5%d1%82\/","title":{"rendered":"\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432, \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<style><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><br \/>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432, \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0446\u0435\u043d\u0442\u0440\u0430\u043b\u044c\u043d\u044b\u043c \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f, \u0441 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u043c\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u043c\u0438 \u0432 \u043d\u0430\u0443\u043a\u0435, \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u0438\u0438 \u0438 \u044d\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0435. \u0414\u0430\u043d\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0441\u043e\u0431\u043e\u0439 \u043f\u043e\u0448\u0430\u0433\u043e\u0432\u043e\u0435 \u0440\u0443\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u043f\u043e \u043e\u0441\u0432\u043e\u0435\u043d\u0438\u044e \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u2014 \u043e\u0442 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432 \u0434\u043e \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u041f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432 \u0438 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u044b\u0445 \u043f\u0440\u0438\u043c\u0435\u0440\u043e\u0432 \u0441\u0442\u0440\u0435\u043c\u0438\u043c\u0441\u044f \u043f\u043e\u043d\u044f\u0442\u044c \u043a\u0430\u043a \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u043f\u0440\u0430\u0432\u0438\u043b, \u0442\u0430\u043a \u0438 \u0438\u0445 \u043e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f<\/strong><\/p>\n<ol>\n<li><strong>\u041f\u043e\u043d\u044f\u0442\u044c<\/strong> \u043e\u0431\u0449\u0435\u0435 \u043f\u043e\u043d\u044f\u0442\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0438 \u0435\u0451 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430.<\/li>\n<li><strong>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c<\/strong> \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0431\u0430\u0437\u043e\u0432\u044b\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445.<\/li>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0442\u044c<\/strong> \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438\u0434\u0435\u043d\u0442\u0438\u0447\u043d\u043e\u0441\u0442\u0438.<\/li>\n<li><strong>\u0412\u044b\u0432\u0435\u0441\u0442\u0438<\/strong> \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u043d\u0430\u0447\u0438\u043d\u0430\u044f \u0441 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u0441\u0438\u043d\u0443\u0441\u0430 \u0438 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430.<\/li>\n<li><strong>\u0412\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c<\/strong> \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043f\u0440\u0430\u0432\u0438\u043b.<\/li>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0442\u044c<\/strong> \u0441\u0442\u0440\u043e\u0433\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432.<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>\u0421\u041e\u0414\u0415\u0420\u0416\u0410\u041d\u0418\u0415<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/a><br \/>\n<a href=\"#2\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" title=\"YouTube video player\" data-src=\"https:\/\/www.youtube.com\/embed\/LudVXLRnmLw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>\u0414\u043e \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0435\u0433\u043e \u043c\u043e\u043c\u0435\u043d\u0442\u0430 \u043c\u044b \u043b\u0438\u0448\u044c \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043b\u0438, \u0447\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f, \u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0435\u0451 \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043d\u043e \u043f\u043e\u043a\u0430 \u043d\u0438\u0447\u0435\u0433\u043e \u043d\u0435 \u0441\u043a\u0430\u0437\u0430\u043b\u0438 \u043e \u043c\u0435\u0442\u043e\u0434\u0430\u0445 \u0435\u0451 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f. \u0417\u0434\u0435\u0441\u044c \u043c\u044b \u0440\u0435\u0448\u0438\u043c \u044d\u0442\u0443 \u0437\u0430\u0434\u0430\u0447\u0443, \u043f\u043e\u043a\u0430\u0437\u0430\u0432 \u043a\u0430\u0436\u0434\u0443\u044e \u0438\u0437 \u0442\u0435\u0445\u043d\u0438\u043a \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0441\u043f\u043e\u0441\u043e\u0431\u044b \u0438\u0445 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0441\u043b\u0443\u0447\u0430\u044f.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h2>\n<h3>\u041f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=104s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>\u2014 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u0430\u044f \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430, \u0442\u043e\u0433\u0434\u0430 \u0438\u043c\u0435\u0435\u043c:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{d}{dx}c = 0<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/span> \u041d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u044d\u0442\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u0432\u0441\u0435\u0433\u043e \u0432 \u043e\u0434\u0438\u043d \u0448\u0430\u0433:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}c &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{c - c}{\\Delta x} \\quad \\text{; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 $f(x)=c$} \\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{0}{\\Delta x} = 0\n\n\\end{array}\n\n<\/span>\n<h3>\u0424\u0443\u043d\u043a\u0446\u0438\u044f \u0438\u0434\u0435\u043d\u0442\u0438\u0447\u043d\u043e\u0441\u0442\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> \u0442\u043e\u0433\u0434\u0430:<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx}{dx}=1<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/span> \u041f\u043e\u0447\u0442\u0438 \u043a\u0430\u043a \u0432 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435, \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u043e\u0434\u0438\u043d \u0448\u0430\u0433:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x &amp;= \\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{(x+\\Delta x) - x}{\\Delta x} \\quad \\text{; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0434\u043b\u044f $f(x) = x$}\\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\Delta x}{\\Delta x} = 1\n\n\\end{array}\n\n<\/span>\n<h3>\u041d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> \u0433\u0434\u0435<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u2014 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u043e\u0435 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u0442\u043e\u0433\u0434\u0430 \u0438\u043c\u0435\u0435\u043c:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx^n}{dx} =nx^{n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/span> \u0427\u0442\u043e\u0431\u044b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u044d\u0442\u0443 \u0442\u0435\u043e\u0440\u0435\u043c\u0443, \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">\u0431\u0438\u043d\u0438\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u0443\u044e \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u041d\u044c\u044e\u0442\u043e\u043d\u0430<\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}x^n = \\lim_{\\Delta x \\to 0} \\frac{(x+\\Delta x)^n -x^n}{\\Delta x} &amp;\\text{ ; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0434\u043b\u044f $f(x)= x^n$} \\\\ \\\\\n\n&amp; \\displaystyle \\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=0}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; \u0431\u0438\u043d\u043e\u043c \u041d\u044c\u044e\u0442\u043e\u043d\u0430, \u043f\u0440\u0438\u043c\u0435\u043d\u0451\u043d\u043d\u044b\u0439 \u043a (1)} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle x^n + \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; \u0432\u044b\u0434\u0435\u043b\u044f\u044f \u043f\u0435\u0440\u0432\u044b\u0439 \u0447\u043b\u0435\u043d \u0441\u0443\u043c\u043c\u044b} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right]}{\\Delta x} &amp; \\text{; \u0441\u043e\u043a\u0440\u0430\u0449\u0430\u044f \u043f\u043e\u0434\u043e\u0431\u043d\u044b\u0435 \u0447\u043b\u0435\u043d\u044b} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[ {{n}\\choose{1}} x^{n-1}(\\Delta x)^{0} + \\sum_{k=2}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\text{; \u0432\u044b\u0434\u0435\u043b\u044f\u044f \u043f\u0435\u0440\u0432\u044b\u0439 \u0447\u043b\u0435\u043d \u0441\u0443\u043c\u043c\u044b} \\\\ \\\\\n\n&amp; \\displaystyle \\color{blue} {\\displaystyle \\dfrac{d}{dx}x^n} = n x^{n-1} &amp; \\color{black}\n\n\\end{array}\n\n<\/span>\n<h3>\u0426\u0435\u043b\u044b\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/h3>\n<p>\u041f\u0440\u0438\u0432\u0435\u0434\u0451\u043d\u043d\u043e\u0435 \u0432\u044b\u0448\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043e\u0431\u043e\u0441\u043d\u043e\u0432\u044b\u0432\u0430\u0435\u0442 \u0441\u043b\u0443\u0447\u0430\u0439, \u043a\u043e\u0433\u0434\u0430 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c, \u043d\u043e \u043e\u043d\u043e \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u043e \u043d\u0430 \u043b\u044e\u0431\u044b\u0435 \u0446\u0435\u043b\u044b\u0435 \u0447\u0438\u0441\u043b\u0430. \u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span>, \u0442\u043e\u0433\u0434\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^a}{dx} = ax^{a-1}<\/span>\n<p>\u041c\u044b \u0443\u0436\u0435 \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u044d\u0442\u043e \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0434\u043b\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b; \u043d\u0430\u043c \u043e\u0441\u0442\u0430\u0451\u0442\u0441\u044f \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c, \u0447\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442, \u043a\u043e\u0433\u0434\u0430 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430. \u0414\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^{-n}}{dx} = {-n}x^{-n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/span> \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x^{-n} &amp;= \\dfrac{d}{dx} \\left( \\dfrac{1}{x^n}\\right) \\\\ \\\\\n\n&amp; &amp;= \\dfrac{0 \\cdot nx^{n-1} - nx^{n-1} \\cdot 1}{x^{2n}}\\\\ \\\\\n\n&amp; &amp;= -nx^{n-1-2n} \\\\ \\\\\n\n&amp; &amp;= -nx^{-n-1}\n\n\\end{array}\n\n<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h2>\n<h3>\u0422\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=667s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u041e\u043d\u0438 \u043e\u0445\u0432\u0430\u0442\u044b\u0432\u0430\u044e\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435<\/span><\/strong><\/a> \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sin(x) = \\cos(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sec(x) = \\sec(x)\\tan(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cos(x) = -\\sin(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\csc(x) = -\\csc(x)\\cot(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\tan(x) = \\sec^2(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cot(x) = -\\csc^2(x)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0427\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u043a\u0430\u0436\u0434\u043e\u0435 \u0438\u0437 \u044d\u0442\u0438\u0445 \u043f\u0440\u0430\u0432\u0438\u043b, \u043d\u0430\u0438\u043b\u0443\u0447\u0448\u0438\u0439 \u043f\u0443\u0442\u044c \u2014 \u043d\u0430\u0447\u0430\u0442\u044c \u0441 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u0441\u0438\u043d\u0443\u0441\u0430 \u0438 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430, \u0430 \u0437\u0430\u0442\u0435\u043c, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445, \u0432\u044b\u0432\u0435\u0441\u0442\u0438 \u043e\u0441\u0442\u0430\u043b\u044c\u043d\u044b\u0435 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435.<\/p>\n<h4>\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 \u0441\u0438\u043d\u0443\u0441\u0430<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\dfrac{d}{dx}\\sin(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x+\\Delta x) - \\sin(x)}{\\Delta x} &amp; \\text{; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0441\u0438\u043d\u0443\u0441\u0430} \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x)\\cos(\\Delta x) + \\sin(\\Delta x)\\cos(x) - \\sin(x)}{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\sin(x)\\left[\\cos(\\Delta x) -1\\right] + \\sin(\\Delta x)\\cos(x) }{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\sin(x)\\lim_{\\Delta x \\to 0} \\left[\\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\right] + \\cos(x) \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\right] &amp; \\\\ \\\\\n\n(2)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} = 1 &amp; \\text{; \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u00ab\u0431\u0443\u0442\u0435\u0440\u0431\u0440\u043e\u0434\u0430\u00bb (\u0441\u0430\u043d\u0434\u0432\u0438\u0447\u0430)}\\\\ \\\\\n\n(3)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\cdot \\dfrac{\\cos(\\Delta x) + 1}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos^2(\\Delta x) - 1}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{-\\sin^2(\\Delta x)}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\cdot \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- (1)\\cdot(0) = 0 \\\\ \\\\\n\n(4) &amp;\\color{blue}\\dfrac{d}{dx}\\sin(x) = \\cos(x) \\color{black} &amp; \\text{; \u0438\u0437 (1,2,3)}\n\n\\end{array}<\/span>\n<h4>\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 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}\\cos(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x + \\Delta x) - \\cos(x)}{\\Delta x} &amp; \\text{; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x)\\cos(\\Delta x) - \\sin(x)\\sin(\\Delta x) - \\cos(x)}{\\Delta x} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x) [ \\cos(\\Delta x) - 1] - \\sin(x)\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ [ \\cos(\\Delta x) - 1]}{\\Delta x} - \\sin(x) \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\cdot(0) - \\sin(x)\\cdot (1)\\\\ \\\\\n\n&amp;\\color{blue}\\dfrac{d}{dx}\\cos(x) = - \\sin(x) \\color{black}\n\n\\end{array}<\/span>\n<h4>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430, \u0441\u0435\u043a\u0430\u043d\u0441\u0430, \u043a\u043e\u0441\u0435\u043a\u0430\u043d\u0441\u0430 \u0438 \u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430<\/h4>\n<p>\u0418\u043c\u0435\u044f \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0434\u043b\u044f \u0441\u0438\u043d\u0443\u0441\u0430 \u0438 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430, \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u043e\u0441\u0442\u0430\u043b\u044c\u043d\u044b\u0445 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f \u043f\u0440\u043e\u0441\u0442\u044b\u043c.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\tan(x) &amp;= \\dfrac{d}{dx} \\left( \\dfrac{\\sin(x)}{\\cos(x)} \\right) = \\dfrac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)} = \\dfrac{1}{\\cos^2(x)} = \\color{blue}\\sec^2(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\sec(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)} \\right) = \\dfrac{\\sin(x)}{\\cos^2(x)} =\\color{blue}\\sec(x)\\tan(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\csc(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)}\\right) = -\\dfrac{cos(x)}{\\sin^2(x)} =\\color{blue} - \\csc(x)\\cot(x)\\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx} \\cot(x) &amp;= \\dfrac{d}{dx} \\left(\\dfrac{\\cos(x)}{\\sin(x)}\\right) = \\dfrac{-\\sin^2(x)-\\cos^2(x)}{\\sin^2(x)} = -\\dfrac{1}{\\sin^2(x)} =\\color{blue} -\\csc^2(x)\\color{black}\n\n\\end{array}<\/span>\n<h3>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=1331s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430<\/span><\/strong><\/a> \u0437\u0430\u0434\u0430\u0451\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(x) = \\frac{1}{x}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/span> \u0420\u0430\u0441\u0441\u0443\u0436\u0434\u0430\u044f, \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0439 \u0432\u044b\u0432\u043e\u0434:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx} \\ln(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\left [\\dfrac{\\ln(x+\\Delta x) - \\ln(x)}{\\Delta x} \\right] &amp;\\text{; \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{\\Delta x} \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right) \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\color{red}x\\color{black}} \\frac{\\color{red}x\\color{black}}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{x} \\ln \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} =\\dfrac{1}{x} \\ln \\displaystyle \\left[ \\lim_{\\Delta x \\to 0} \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n(2) &amp; n=\\dfrac{x}{\\Delta x} &amp; \\text{; \u043f\u043e\u0434\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430}\\\\ \\\\\n\n(3) &amp; (\\Delta x \\to 0^+) \\longrightarrow (n\\to +\\infty) \\\\ \\\\\n\n(4) &amp; \\dfrac{d}{dx} \\ln(x) = \\dfrac{1}{x} \\ln\\left[ \\displaystyle \\lim_{n \\to +\\infty} \\left(1 + \\dfrac{1}{n} \\right)^n \\right] = \\dfrac{1}{x} \\ln(e) = \\color{blue}\\dfrac{1}{x} \\color{black} &amp; \\text{; \u0438\u0437 (1,2,3)}\n\n\\end{array}<\/span>\n<p>\n\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u044b \u043f\u043e\u0448\u0430\u0433\u043e\u0432\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043b\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0434\u043e\u043b\u0436\u0435\u043d \u043e\u0441\u0432\u043e\u0438\u0442\u044c \u043a\u0430\u0436\u0434\u044b\u0439 \u0441\u0442\u0443\u0434\u0435\u043d\u0442: \u043e\u0442 \u043f\u0440\u043e\u0441\u0442\u0435\u0439\u0448\u0438\u0445 \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0434\u043e \u0432\u0430\u0436\u043d\u0435\u0439\u0448\u0438\u0445 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u0442\u0430\u043a\u0438\u0445 \u043a\u0430\u043a \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c. \u041e\u0441\u0432\u043e\u0438\u0432 \u044d\u0442\u0438 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430, \u0432\u044b \u0441\u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f, \u043f\u043e\u043d\u0438\u043c\u0430\u0442\u044c \u0438\u0445 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u0435 \u0438 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435. \u042d\u0442\u0438 \u0437\u043d\u0430\u043d\u0438\u044f \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0434\u043b\u044f \u0443\u0432\u0435\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0431\u043e\u043b\u0435\u0435 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447, \u0442\u0440\u0435\u0431\u0443\u044e\u0449\u0438\u0445 \u0442\u043e\u0447\u043d\u043e\u0433\u043e \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432, \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0446\u0435\u043d\u0442\u0440\u0430\u043b\u044c\u043d\u044b\u043c \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f, \u0441 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u043c\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u043c\u0438 \u0432 \u043d\u0430\u0443\u043a\u0435, \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u0438\u0438 \u0438 \u044d\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0435. \u0414\u0430\u043d\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0441\u043e\u0431\u043e\u0439 \u043f\u043e\u0448\u0430\u0433\u043e\u0432\u043e\u0435 \u0440\u0443\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u043f\u043e \u043e\u0441\u0432\u043e\u0435\u043d\u0438\u044e \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u2014 \u043e\u0442 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432 \u0434\u043e \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u041f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432 \u0438 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u044b\u0445 \u043f\u0440\u0438\u043c\u0435\u0440\u043e\u0432 \u0441\u0442\u0440\u0435\u043c\u0438\u043c\u0441\u044f \u043f\u043e\u043d\u044f\u0442\u044c \u043a\u0430\u043a \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u043f\u0440\u0430\u0432\u0438\u043b, \u0442\u0430\u043a \u0438 \u0438\u0445 \u043e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435. \u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33155,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[864,573],"tags":[],"class_list":["post-33373","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-864","category-573"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432, \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 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