{"id":29727,"date":"2024-11-27T12:00:01","date_gmt":"2024-11-27T12:00:01","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29727"},"modified":"2024-11-27T17:37:10","modified_gmt":"2024-11-27T17:37:10","slug":"%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%b0%d1%8f-%d0%ba%d0%b0%d0%ba-%d0%bf%d1%80%d0%b5%d0%b4%d0%b5%d0%bb-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8","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%d0%b0%d1%8f-%d0%ba%d0%b0%d0%ba-%d0%bf%d1%80%d0%b5%d0%b4%d0%b5%d0%bb-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8\/","title":{"rendered":"\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438"},"content":{"rendered":"<style>\np {\n  text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/h1>\n<p style=\"text-align:center;\">\n  <em><strong>\u0420\u0435\u0437\u044e\u043c\u0435:<\/strong> \u041d\u0430 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435 \u043c\u044b \u0438\u0437\u0443\u0447\u0438\u043c \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u0430 \u0434\u043b\u044f \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u041c\u044b \u043d\u0430\u0447\u043d\u0435\u043c \u0441 \u043d\u0430\u043a\u043b\u043e\u043d\u0430 \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438, \u0430 \u0437\u0430\u0442\u0435\u043c, \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u044f \u043f\u0440\u0435\u0434\u0435\u043b \u043f\u0440\u0438 \u0441\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0438 \u0442\u043e\u0447\u0435\u043a, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a\u0430\u043a \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043b\u0438\u043d\u0438\u0438. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0435\u0451 \u043a\u043b\u044e\u0447\u0435\u0432\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u043f\u0440\u0430\u0432\u0438\u043b\u0430, \u0442\u0430\u043a\u0438\u0435 \u043a\u0430\u043a \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0441\u0443\u043c\u043c\u044b, \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0438 \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0434\u043b\u044f \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u043f\u0440\u0438 \u0430\u043d\u0430\u043b\u0438\u0437\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0438 \u044f\u0432\u043b\u0435\u043d\u0438\u0439 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0439.<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f<\/strong><\/p>\n<p>\u041a \u043a\u043e\u043d\u0446\u0443 \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u044b \u0441\u043c\u043e\u0433\u0443\u0442:<\/p>\n<ol>\n<li><strong>\u041f\u043e\u043d\u044f\u0442\u044c<\/strong> \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b, \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0449\u0438\u0439 \u043c\u0433\u043d\u043e\u0432\u0435\u043d\u043d\u044b\u0435 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0438 \u043a\u0430\u043a \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043a \u043a\u0440\u0438\u0432\u043e\u0439 \u0432 \u0442\u043e\u0447\u043a\u0435.<\/li>\n<li><strong>\u041e\u0431\u044a\u044f\u0441\u043d\u0438\u0442\u044c<\/strong>, \u043a\u0430\u043a \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438.<\/li>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u044c<\/strong> \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f, \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f.<\/li>\n<li><strong>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c<\/strong> \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u044b \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 (\u0441\u0443\u043c\u043c\u0430, \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438 \u0447\u0430\u0441\u0442\u043d\u043e\u0435) \u043a \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u0437\u0430\u0434\u0430\u0447\u0430\u043c.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435:<\/u><\/strong><br \/>\n<a href=\"#1\"><strong>\u041a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439<\/strong><\/a><br \/>\n<a href=\"#1\">\u041d\u0430\u043a\u043b\u043e\u043d \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438<\/a><br \/>\n<a href=\"#1\">\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0443: \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0438 \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439<\/a><br \/>\n<a href=\"#1\">\u0410\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/a><br \/>\n<a href=\"#1\"><strong>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445<\/strong><\/a><br \/>\n<a href=\"#1\">\u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c<\/a><br \/>\n<a href=\"#1\">\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445<\/a><\/p>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/TFxATgmYvkY\" title=\"YouTube \u0432\u0438\u0434\u0435\u043e \u043f\u0440\u043e\u0438\u0433\u0440\u044b\u0432\u0430\u0442\u0435\u043b\u044c\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u041a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439<\/h2>\n<p>\u041f\u0440\u0438\u0440\u043e\u0434\u0430, \u043a\u0430\u043a \u043f\u0440\u0430\u0432\u0438\u043b\u043e, \u043f\u043e\u0434\u0432\u0435\u0440\u0436\u0435\u043d\u0430 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f\u043c, \u0438 \u0433\u043b\u0430\u0432\u043d\u044b\u043c \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u043c \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0438 \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u044f \u044d\u0442\u0438\u0445 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0439 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f. \u041e\u043d\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u0438\u0437 \u0432\u043e\u043f\u0440\u043e\u0441\u0430: \u00ab\u0427\u0442\u043e \u043f\u0440\u043e\u0438\u0437\u043e\u0439\u0434\u0435\u0442 \u0441\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span>, \u0435\u0441\u043b\u0438 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u0443\u044e <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0438\u0437\u043c\u0435\u043d\u0438\u0442\u044c \u043d\u0430 \u0441\u043a\u043e\u043b\u044c \u0443\u0433\u043e\u0434\u043d\u043e \u043c\u0430\u043b\u0443\u044e \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span>?\u00bb \u041a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043f\u043e\u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0443\u044e\u0449\u0435\u0439 \u044d\u0442\u043e\u0442 \u0432\u043e\u043f\u0440\u043e\u0441.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>\u041d\u0430\u043a\u043b\u043e\u043d \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span>, \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u0443\u044e \u0432 \u0434\u0432\u0443\u0445 \u0442\u043e\u0447\u043a\u0430\u0445 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. \u041b\u044e\u0431\u0430\u044f \u043b\u0438\u043d\u0438\u044f, \u043f\u0435\u0440\u0435\u0441\u0435\u043a\u0430\u044e\u0449\u0430\u044f \u043a\u0440\u0438\u0432\u0443\u044e \u0432 \u0434\u0432\u0443\u0445 \u0442\u043e\u0447\u043a\u0430\u0445, \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u00ab\u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0435\u0439\u00bb \u0438 \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a, \u043a\u0430\u043a \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u043d\u0430 \u0440\u0438\u0441\u0443\u043d\u043a\u0435 \u043d\u0438\u0436\u0435.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"\u0413\u0440\u0430\u0444\u0438\u043a \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"\u0413\u0440\u0430\u0444\u0438\u043a \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>\u041d\u0430\u043a\u043b\u043e\u043d \u044d\u0442\u043e\u0439 \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438 \u0437\u0430\u0434\u0430\u0451\u0442\u0441\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u043e\u0439:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h3>\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0443: \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0438 \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>\u0415\u0441\u043b\u0438 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0441\u0435\u043a\u0443\u0449\u0443\u044e \u043b\u0438\u043d\u0438\u044e<\/strong><\/a> \u043a \u043a\u0440\u0438\u0432\u043e\u0439 <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span>, \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0449\u0443\u044e \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>, \u0430 \u0437\u0430\u0442\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u043c \u043f\u0440\u0435\u0434\u0435\u043b, \u043f\u0440\u0438 \u043a\u043e\u0442\u043e\u0440\u043e\u043c <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> \u0441\u0442\u0440\u0435\u043c\u0438\u0442\u0441\u044f \u043a \u043d\u0443\u043b\u044e, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u0443\u044e \u043b\u0438\u043d\u0438\u044e, \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0449\u0443\u044e \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, f(x_0)).<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"\u0413\u0440\u0430\u0444\u0438\u043a \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043b\u0438\u043d\u0438\u0438\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"\u0413\u0440\u0430\u0444\u0438\u043a \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043b\u0438\u043d\u0438\u0438\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>\u0418\u0437 \u044d\u0442\u043e\u0433\u043e \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 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0432 \u0442\u043e\u0447\u043a\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{df(x_0)}{dx}:= \\lim_{\\Delta x \\to 0}\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>\u042d\u0442\u043e \u0442\u0430\u043a\u0436\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0441\u043e\u0431\u043e\u0439 \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439, \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0449\u0435\u0439 \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>\u0410\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/h3>\n<p>\u0410\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u044b\u0439 \u0441\u043f\u043e\u0441\u043e\u0431 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0437\u0430\u043c\u0435\u043d\u044b:<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nx_i &amp;= x_0\\\\\n\nx_f &amp;= x_i + \\Delta x\n\n\\end{array}\n\n<\/span>\n<p>\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span>, \u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{df(x_i)}{dx} &amp;=\\displaystyle \\lim_{\\Delta x \\to 0}\\dfrac{ f(x_i + \\Delta x) - f(x_i)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{x_f - x_i \\to 0} \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\\\\ \\\\\n\n&amp;=\\displaystyle  \\lim_{x_f \\to x_i } \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\n\n\\end{array}\n\n<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043d\u0430\u043a\u043b\u043e\u043d\u0430 \u0441\u0435\u043a\u0443\u0449\u0438\u0445\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043d\u0430\u043a\u043b\u043e\u043d\u0430 \u0441\u0435\u043a\u0443\u0449\u0438\u0445\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>\u041e\u0431\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b \u0438 \u043c\u043e\u0433\u0443\u0442 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u0432\u0437\u0430\u0438\u043c\u043e\u0437\u0430\u043c\u0435\u043d\u044f\u0435\u043c\u043e \u0432 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442\u0438 \u043e\u0442 \u0443\u0434\u043e\u0431\u0441\u0442\u0432\u0430.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445<\/h2>\n<p>\u0413\u043e\u0432\u043e\u0440\u044f\u0442, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u0442\u043e\u0447\u043a\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, \u0435\u0441\u043b\u0438 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0439 \u043f\u0440\u0435\u0434\u0435\u043b:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>\u041c\u044b \u0442\u0430\u043a\u0436\u0435 \u0433\u043e\u0432\u043e\u0440\u0438\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043d\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, \u0435\u0441\u043b\u0438 \u043f\u0440\u0435\u0434\u0435\u043b \u0445\u043e\u0440\u043e\u0448\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0451\u043d \u0434\u043b\u044f \u0432\u0441\u0435\u0445 <span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I.<\/span> \u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0431\u043b\u0430\u0434\u0430\u044e\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u043c\u0438:<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>\u0415\u0441\u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> \u0442\u043e \u043e\u043d\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> \u042d\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p>\u0414\u043b\u044f \u0442\u043e\u0433\u043e \u0447\u0442\u043e\u0431\u044b <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0431\u044b\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e, \u0447\u0442\u043e\u0431\u044b \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u043b\u043e\u0441\u044c:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = f(x_0)<\/span>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043b\u0435\u0432\u0443\u044e \u0447\u0430\u0441\u0442\u044c \u044d\u0442\u043e\u0433\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f:<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to x_0} f(x) &amp;= \\displaystyle \\lim_{x\\to x_0} \\left[ f(x) + f(x_0) - f(x_0) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( f(x)  - f(x_0) \\right) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0)  \\right] \\\\ \\\\\n\n&amp;=f(x_0) +\\displaystyle \\lim_{x\\to x_0} \\left[ \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0) \\right]\n\\end{array}\n\n<\/span>\n<p>\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0434\u043b\u044f \u0442\u043e\u0433\u043e \u0447\u0442\u043e\u0431\u044b <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0431\u044b\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u043f\u0440\u0430\u0432\u0430 \u0434\u043e\u043b\u0436\u0435\u043d \u0431\u044b\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0451\u043d. \u042d\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0442\u043e\u0433\u0434\u0430 \u0438 \u0442\u043e\u043b\u044c\u043a\u043e \u0442\u043e\u0433\u0434\u0430, \u043a\u043e\u0433\u0434\u0430:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\dfrac{f(x) - f(x_0)}{x-x_0} = \\dfrac{df(x_0)}{dx}<\/span>\n<p>\u0414\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u0435\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445<\/h3>\n<p>\u041f\u0443\u0441\u0442\u044c <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u2014 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0434\u043b\u044f \u0432\u0441\u0435\u0445 <span class=\"katex-eq\" data-katex-display=\"false\">x \\in I,<\/span> \u0430 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathbb{R}.<\/span> \u0422\u043e\u0433\u0434\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044e\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430:<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha \\dfrac{df(x)}{dx} \\pm \\beta\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( f(x) g(x) \\right) = \\dfrac{df(x)}{dx} g(x) + f(x)\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li>\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0<\/span>, \u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\dfrac{f(x)}{g(x)} \\right) = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx} }{\\left[g(x)\\right]^2}<\/span><\/li>\n<\/ol>\n<p>\u041a\u0430\u043a \u0432\u0438\u0434\u043d\u043e, \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u043d\u0435 \u0442\u0430\u043a \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e \u043f\u043e\u043d\u044f\u0442\u043d\u0430, \u043a\u0430\u043a \u043c\u043e\u0436\u0435\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u044c\u0441\u044f \u043d\u0430 \u043f\u0435\u0440\u0432\u044b\u0439 \u0432\u0437\u0433\u043b\u044f\u0434. \u041e\u0434\u043d\u0430\u043a\u043e, \u044d\u0442\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043c\u043e\u0436\u043d\u043e \u043b\u0435\u0433\u043a\u043e \u0432\u044b\u0432\u0435\u0441\u0442\u0438 \u0438\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b\u0430.<\/p>\n<p><span style=\"color: #000080;\">\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0441\u0443\u043c\u043c\u044b<\/strong><\/a> \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\alpha f(x) \\pm \\beta g(x) \\right) &amp; =\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\left[\\alpha f(x+\\Delta x) \\pm \\beta g(x+ \\Delta x)\\right] - \\left[\\alpha f(x) \\pm \\beta g(x) \\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\left[\\alpha f(x+\\Delta x) - \\alpha f(x)\\right] \\pm \\left[\\beta g(x+\\Delta x) - \\beta g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\alpha \\left[ f(x+\\Delta x) -  f(x)\\right] \\pm  \\beta  \\left[ g(x+\\Delta x) - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\alpha \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) -  f(x)}{\\Delta x} \\pm \\beta \\lim_{\\Delta x \\to 0} \\dfrac{ g(x+\\Delta x) -  g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\alpha \\dfrac{df(x)}{dx} \\pm \\beta \\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=1059s\" target=\"_blank\" rel=\"noopener\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f<\/strong><\/a> \u043d\u0435\u043c\u043d\u043e\u0433\u043e \u0441\u043b\u043e\u0436\u043d\u0435\u0435:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left[f(x)g(x)\\right] &amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) -  f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) + \\color{red}f(x)g(x+\\Delta x) - f(x)g(x+\\Delta x) \\color{black} - f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\left[f(x+\\Delta x) - f(x) \\right] g(x+\\Delta x) + f(x) \\left[g(x+\\Delta x)  - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x) \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x)\\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;= g(x) \\dfrac{df(x)}{dx} + f(x)\\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p>\u0417\u0434\u0435\u0441\u044c \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>, \u0431\u0443\u0434\u0443\u0447\u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439, \u0442\u0430\u043a\u0436\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439, \u0442\u043e \u0435\u0441\u0442\u044c <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x\\to 0 } g(x+\\Delta x) = g(x).<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e<\/strong><\/a> \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c, \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0432\u0448\u0438\u0441\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u043e\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f. \u041f\u0443\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438\u043c\u0435\u0435\u0442 \u0432\u0438\u0434 <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x),<\/span> \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0.<\/span> \u0422\u043e\u0433\u0434\u0430:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{df(x)}{dx}= \\dfrac{d}{dx}(k(x)g(x)) = \\dfrac{dk(x)}{dx}g(x) + k(x)\\dfrac{dg(x)}{dx}<\/span>\n<p>\u0420\u0435\u0448\u0430\u044f \u0434\u043b\u044f <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx},<\/span> \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}g(x) = \\dfrac{df(x)}{dx} - k(x)\\dfrac{dg(x)}{dx} = \\dfrac{d}{dx}f(x) - \\dfrac{f(x)}{g(x)}\\dfrac{dg(x)}{dx} <\/span>\n<p>\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)\n\n &amp;= \\dfrac{dk(x)}{dx} =\\dfrac{1}{g(x)} \\dfrac{df(x)}{dx} - \\dfrac{f(x)}{\\left[g(x)\\right]^2}\\dfrac{dg(x)}{dx} \\\\ \\\\\n\n&amp; = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx}}{[g(x)]^2}\n\n\\end{array}\n\n<\/span>\n<p>\u042d\u0442\u043e \u0442\u043e, \u0447\u0442\u043e \u0438 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u043e\u0441\u044c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0420\u0435\u0437\u044e\u043c\u0435: \u041d\u0430 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435 \u043c\u044b \u0438\u0437\u0443\u0447\u0438\u043c \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u0430\u043a \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u0430 \u0434\u043b\u044f \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u041c\u044b \u043d\u0430\u0447\u043d\u0435\u043c \u0441 \u043d\u0430\u043a\u043b\u043e\u043d\u0430 \u0441\u0435\u043a\u0443\u0449\u0435\u0439 \u043b\u0438\u043d\u0438\u0438, \u0430 \u0437\u0430\u0442\u0435\u043c, \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u044f \u043f\u0440\u0435\u0434\u0435\u043b \u043f\u0440\u0438 \u0441\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0438 \u0442\u043e\u0447\u0435\u043a, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a\u0430\u043a \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043b\u0438\u043d\u0438\u0438. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0435\u0451 \u043a\u043b\u044e\u0447\u0435\u0432\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u043f\u0440\u0430\u0432\u0438\u043b\u0430, \u0442\u0430\u043a\u0438\u0435 \u043a\u0430\u043a \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0441\u0443\u043c\u043c\u044b, \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0438 \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 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