{"id":35191,"date":"2024-12-01T13:00:20","date_gmt":"2024-12-01T13:00:20","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35191"},"modified":"2025-11-22T22:34:26","modified_gmt":"2025-11-22T22:34:26","slug":"%d0%bf%d1%80%d0%b0%d0%b2%d0%b8%d0%bb%d0%be-%d1%86%d0%b5%d0%bf%d0%be%d1%87%d0%ba%d0%b8-%d0%b4%d0%bb%d1%8f-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%be%d0%b9-%d0%ba%d0%be%d0%bc%d0%bf","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d0%bf%d1%80%d0%b0%d0%b2%d0%b8%d0%bb%d0%be-%d1%86%d0%b5%d0%bf%d0%be%d1%87%d0%ba%d0%b8-%d0%b4%d0%bb%d1%8f-%d0%bf%d1%80%d0%be%d0%b8%d0%b7%d0%b2%d0%be%d0%b4%d0%bd%d0%be%d0%b9-%d0%ba%d0%be%d0%bc%d0%bf\/","title":{"rendered":"\u041f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0439"},"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<h1>\u041f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h1>\n<p><em>\u0421 \u0443\u0447\u0451\u0442\u043e\u043c \u0432\u0441\u0435\u0433\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0440\u0430\u043d\u0435\u0435, \u043c\u044b \u0443\u0436\u0435 \u0440\u0430\u0441\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u043c \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u043c\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043c\u0438 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043b\u044e\u0431\u043e\u0439 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439. \u041e\u0434\u043d\u0430\u043a\u043e \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0440\u0430\u0437\u043b\u0438\u0447\u0430\u0442\u044c \u0441\u0430\u043c\u0443 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0438 \u0443\u0441\u0438\u043b\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043c\u044b \u0437\u0430\u0442\u0440\u0430\u0447\u0438\u0432\u0430\u0435\u043c \u043d\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u0442\u0430\u043a\u0438\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439. \u0418\u043c\u0435\u043d\u043d\u043e \u0437\u0434\u0435\u0441\u044c \u0432\u0441\u0442\u0443\u043f\u0430\u044e\u0442 \u0432 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0435 \u0442\u0435\u043e\u0440\u0435\u043c\u044b, \u043f\u043e\u0434\u043e\u0431\u043d\u044b\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u0443 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0441\u043b\u0443\u0447\u0430\u044f \u043e\u0434\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439. \u041f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u043f\u043e\u0437\u0432\u043e\u043b\u0438\u0442 \u043d\u0430\u043c \u0431\u044b\u0441\u0442\u0440\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0432 \u043f\u0440\u043e\u0442\u0438\u0432\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u043f\u043e\u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u0438 \u0431\u044b \u0434\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u0443\u0442\u043e\u043c\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0438 \u0441\u043b\u043e\u0436\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u044b.<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b><u>\u0421\u041e\u0414\u0415\u0420\u0416\u0410\u041d\u0418\u0415<\/u><\/b><br \/>\n<b><a href=\"#1\">\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0435 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439<\/a><\/b><br \/>\n<a href=\"#11\">\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/a><br \/>\n<a href=\"#12\">\u041f\u0440\u0438\u043c\u0435\u0440\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u043e\u0434\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439<\/a><br \/>\n<a href=\"#13\">\u041f\u0440\u0435\u0434\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u0442\u044c \u043f\u0440\u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0438 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/a><br \/>\n<b><a href=\"#2\">\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0438\u0437 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/a><\/b><br \/>\n<a href=\"#21\">\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/a><br \/>\n<a href=\"#211\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/a><br \/>\n<a href=\"#212\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043e\u0431\u0440\u0430\u0442\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<\/a><br \/>\n<a href=\"#22\">\u041d\u0435\u044f\u0432\u043d\u043e\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435<\/a><br \/>\n<a href=\"#221\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/a><br \/>\n<a href=\"#221\">\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/a><br \/>\n<b><a href=\"#3\">\u0421\u0431\u043e\u0440\u043d\u0438\u043a \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0439<\/a><\/b>\n<\/p>\n<p><a name=\"1\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/0y2SQpbRe3A\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0435 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=165s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\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\u0432\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/span><\/a>, \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u044e\u0449\u0438\u0435 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u044e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f: A\\subseteq \\mathbb{R} \\longmapsto B\\subseteq \\mathbb{R}<\/span>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g: B\\subseteq Dom(g) \\longmapsto D\\subseteq \\mathbb{\u0420}<\/span>\n<p>\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043d\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, \u0430 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043d\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>, \u0442\u043e \u0441\u043e\u0441\u0442\u0430\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g\\circ f<\/span><\/span> \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0434\u043b\u044f \u0432\u0441\u0435\u0445 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in A<\/span><\/span>, \u0438 \u0441\u043f\u0440\u0430\u0432\u0435\u0434\u043b\u0438\u0432\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}(g\\circ f)(x) = \\frac{d}{dx} g(f(x)) = \\frac{dg(f(x))}{df(x)} \\frac{df(x)}{dx}<\/span>\n<p><a name=\"11\"><\/a><\/p>\n<h3>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=242s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0451\u043d\u043d\u044b\u0435 \u0432\u044b\u0448\u0435. \u0415\u0441\u043b\u0438 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438, \u0442\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u043c<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\n\\dfrac{d}{dx} g(f(x))&amp; = &amp; \\displaystyle\\lim_{\\Delta x \\to 0} \\dfrac{g(f(x + \\Delta x)) - g(f(x))}{\\Delta x} \\\\ \\\\\n\n&amp;=&amp;\\displaystyle \\lim_{\\Delta x \\to 0} \\frac{g(f(x + \\Delta x)) - g(f(x))}{\\Delta x} \\cdot \\frac{f(x + \\Delta x) - f(x)}{f(x+\\Delta x) - f(x)} \\\\ \\\\\n\n&amp;=&amp; \\displaystyle \\lim_{\\Delta x \\to 0} \\frac{g(f(x + \\Delta x)) - g(f(x))}{f(x+\\Delta x) - f(x)} \\cdot \\frac{f(x + \\Delta x) - f(x)}{\\Delta x} \\\\ \\\\\n\n&amp;=&amp;\\displaystyle \\lim_{\\Delta x \\to 0} \\frac{g(f(x + \\Delta x)) - g(f(x))}{f(x+\\Delta x) - f(x)} \\cdot \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) - f(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=&amp; \\displaystyle \\lim_{f(x+\\Delta x) \\to f(x) } \\frac{g(f(x + \\Delta x)) - g(f(x))}{f(x+\\Delta x) - f(x)} \\cdot \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) - f(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=&amp; \\displaystyle \\frac{dg(f(x))}{df(x)} \\frac{df(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p>\u0427\u0442\u043e \u0438 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u043e\u0441\u044c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c.<\/p>\n<p><a name=\"12\"><\/a><\/p>\n<h3>\u041f\u0440\u0438\u043c\u0435\u0440\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u043e\u0434\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=423s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0422\u043e, \u0447\u0442\u043e \u043a\u0430\u0436\u0435\u0442\u0441\u044f \u043e\u0447\u0435\u0432\u0438\u0434\u043d\u044b\u043c \u043d\u0430 \u043f\u0435\u0440\u0432\u044b\u0439 \u0432\u0437\u0433\u043b\u044f\u0434,<\/span><\/a> \u0445\u043e\u0442\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u043e\u043d\u043d\u043e \u044d\u0442\u043e \u043d\u0435 \u0442\u0430\u043a \u0443\u0436 \u043f\u0440\u043e\u0441\u0442\u043e, \u2014 \u044d\u0442\u043e \u0442\u043e\u0442 \u0444\u0430\u043a\u0442, \u0447\u0442\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0443\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u043d\u0430\u043c: \u043a\u043e\u0433\u0434\u0430 \u043c\u044b \u0438\u043c\u0435\u0435\u043c \u0434\u0435\u043b\u043e \u0441 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u043c\u043e\u0436\u043d\u043e \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u00ab\u0441\u043d\u0430\u0440\u0443\u0436\u0438 \u0432\u043d\u0443\u0442\u0440\u044c\u00bb. \u0427\u0442\u043e\u0431\u044b \u043e\u0431\u044a\u044f\u0441\u043d\u0438\u0442\u044c \u044d\u0442\u043e \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e \u0434\u043e\u0441\u0442\u0443\u043f\u043d\u043e, \u043f\u0440\u0438\u043c\u0435\u0440\u044b \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f, \u0431\u0435\u0437\u0443\u0441\u043b\u043e\u0432\u043d\u043e, \u0441\u0430\u043c\u044b\u043c \u0431\u044b\u0441\u0442\u0440\u044b\u043c \u0441\u043f\u043e\u0441\u043e\u0431\u043e\u043c.<\/p>\n<ol>\n<li>\u0415\u0441\u043b\u0438 \u043d\u0430\u043c \u043f\u0440\u043e\u0441\u044f\u0442 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = (2x^2+1)^{12}<\/span><\/span>, \u0442\u043e \u0441\u043d\u0430\u0447\u0430\u043b\u0430 \u043f\u0440\u0438\u0448\u043b\u043e\u0441\u044c \u0431\u044b \u0440\u0430\u0437\u0432\u0435\u0440\u043d\u0443\u0442\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u044c, \u0430 \u0437\u0430\u0442\u0435\u043c \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0441\u0442\u0435\u043f\u0435\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043a \u043a\u0430\u0436\u0434\u043e\u0439 \u0447\u0430\u0441\u0442\u0438 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u043e\u0433\u043e \u0431\u043e\u043b\u044c\u0448\u043e\u0433\u043e \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u0430. \u042d\u0442\u043e \u0431\u044b\u043b\u0430 \u0431\u044b \u0438\u0437\u043b\u0438\u0448\u043d\u0435 \u0443\u0442\u043e\u043c\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u0440\u0430\u0431\u043e\u0442\u0430. \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0432 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0441\u0442\u0440\u043e\u043a:<br \/>\n<\/p>\n<p style=\"align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx} (2x^2+1)^{12} = 12(2x^2+1)^{11}(4x)= 48x(2x^2+1)^{11}<\/span>\n<\/li>\n<li>\u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0439 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\sin(\\cos(x))<\/span><\/span> \u0442\u043e\u043b\u044c\u043a\u043e \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0431\u0430\u0437\u043e\u0432\u044b\u0445 \u0442\u0435\u0445\u043d\u0438\u043a \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u2014 \u0438 \u043f\u0440\u0438\u0433\u043e\u0442\u043e\u0432\u044c\u0441\u044f \u043a \u0432\u0435\u0447\u043d\u044b\u043c \u043c\u0443\u0447\u0435\u043d\u0438\u044f\u043c. \u0421\u0434\u0435\u043b\u0430\u0439 \u044d\u0442\u043e \u0441 \u043f\u0440\u0430\u0432\u0438\u043b\u043e\u043c \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u2014 \u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043f\u043e\u044f\u0432\u0438\u0442\u0441\u044f \u0431\u0435\u0437 \u0441\u043b\u0451\u0437 \u0438 \u0437\u0430 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0448\u0430\u0433\u043e\u0432:<br \/>\n<\/p>\n<p style=\"align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx} \\sin(\\cos(x))= -\\cos(cos(x))\\sin(x) <\/span>\n<\/li>\n<li>\u0422\u0430\u043a\u0436\u0435 \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u044f\u0432\u043b\u044f\u044e\u0449\u0438\u0445\u0441\u044f \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0435\u0439 \u0431\u043e\u043b\u044c\u0448\u043e\u0433\u043e \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u0415\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\cos(\\cos(\\cos(x))),<\/span><\/span> \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df\/dx<\/span><\/span> \u0431\u0443\u0434\u0435\u0442 \u0442\u0430\u043a\u043e\u0439:<br \/>\n<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\n\\displaystyle \\frac{d}{dx} \\cos(\\cos(\\cos(x))) &amp;=&amp; -\\sin(\\cos(\\cos(x)))\\cdot(-\\sin(\\cos(x))\\cdot(-\\sin(x)) \\\\ \\\\\n\n&amp;=&amp; -\\sin(\\cos(\\cos(x)))\\cdot\\sin(\\cos(x))\\cdot\\sin(x)\n\n\\end{array}\n\n<\/span>\n<p>\u041a\u0430\u043a \u0432\u0438\u0434\u043d\u043e, \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u2014 \u044d\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 \u043e\u0442 \u0432\u043d\u0435\u0448\u043d\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043a \u0432\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u0435\u0439.<\/li>\n<\/ol>\n<p><a name=\"13\"><\/a><\/p>\n<h3>\u041f\u0440\u0435\u0434\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u0438\u0435 \u043f\u0440\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=607s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0412 \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0435 \u0448\u0438\u0440\u043e\u043a\u043e \u043f\u043e\u0434\u0447\u0435\u0440\u043a\u0438\u0432\u0430\u044e\u0442 \u0431\u043e\u043b\u044c\u0448\u0438\u0435 \u043f\u0440\u0435\u0438\u043c\u0443\u0449\u0435\u0441\u0442\u0432\u0430<\/span><\/a> \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438, \u043e\u0434\u043d\u0430\u043a\u043e \u043b\u0438\u0448\u044c \u043d\u0435\u043c\u043d\u043e\u0433\u0438\u0435 \u0434\u0435\u043b\u0430\u044e\u0442 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0441\u0438\u043b\u044c\u043d\u044b\u0439 \u0430\u043a\u0446\u0435\u043d\u0442 \u043d\u0430 \u0442\u0435\u0445 \u043c\u0435\u0440\u0430\u0445 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u043e\u0440\u043e\u0436\u043d\u043e\u0441\u0442\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u0442\u044c \u043f\u0435\u0440\u0435\u0434 \u0435\u0433\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c. \u041d\u0435\u0441\u043c\u043e\u0442\u0440\u044f \u043d\u0430 \u043c\u043e\u0449\u044c \u044d\u0442\u043e\u0433\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u044b, \u0432\u0441\u0435\u0433\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0443\u0434\u0435\u043b\u044f\u0442\u044c \u043f\u0440\u0438\u0441\u0442\u0430\u043b\u044c\u043d\u043e\u0435 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u043e\u0431\u043b\u0430\u0441\u0442\u044f\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u043f\u0435\u0440\u0435\u0434 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435\u043c \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438. \u041f\u0440\u0435\u0436\u0434\u0435 \u0447\u0435\u043c \u043d\u0430\u0447\u0430\u0442\u044c \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f, \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0443\u0431\u0435\u0434\u0438\u0442\u044c\u0441\u044f, \u0447\u0442\u043e \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u0438\u043c\u044b \u0434\u043b\u044f \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438; \u0435\u0441\u043b\u0438 \u044d\u0442\u043e\u0433\u043e \u043d\u0435 \u0441\u0434\u0435\u043b\u0430\u0442\u044c, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0440\u0438\u0441\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432 \u0442\u043e\u0447\u043a\u0430\u0445, \u0433\u0434\u0435 \u043e\u043d\u0438 \u043d\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442. \u0415\u0441\u043b\u0438, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043f\u043e\u043f\u044b\u0442\u0430\u0442\u044c\u0441\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0432\u0438\u0434\u0430<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln(\\cos(x))<\/span>\n<p>\u0438 \u043f\u043e\u043b\u0430\u0433\u0430\u0442\u044c\u0441\u044f \u0441\u043b\u0435\u043f\u043e \u043d\u0430 \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438, \u043c\u043e\u0436\u043d\u043e \u043f\u0440\u0438\u0439\u0442\u0438 \u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u0432\u0438\u0434\u0430:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(\\cos(x)) = -\\frac{1}{\\cos(x)}\\sin(x) = -\\tan(x)<\/span>\n<p>\u041e\u0447\u0435\u0432\u0438\u0434\u043d\u043e, \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u043f\u0440\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=2\\pi\/3<\/span><\/span>, \u0442\u0430\u043a \u043a\u0430\u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(2\\pi\/3) = -\\sqrt{3}<\/span><\/span>. \u041e\u0434\u043d\u0430\u043a\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln(\\cos(x))<\/span><\/span> \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(2\\pi\/3) = \\ln(\\cos(2\\pi\/3)) = \\ln(-1\/2),<\/span><\/span> \u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u043d\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442! \u0412 \u043f\u043e\u0434\u043e\u0431\u043d\u044b\u0445 \u0441\u043b\u0443\u0447\u0430\u044f\u0445 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0437\u0430\u0440\u0430\u043d\u0435\u0435 \u0443\u043a\u0430\u0437\u0430\u0442\u044c, \u0434\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438, \u0447\u0442\u043e \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u044e\u0442\u0441\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u0442\u0430\u043a\u0438\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, \u043f\u0440\u0438 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u043e\u0441\u0442\u0430\u0451\u0442\u0441\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 (\u0442\u0435\u043c \u0441\u0430\u043c\u044b\u043c \u043e\u0431\u0435\u0441\u043f\u0435\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u0438\u043c\u043e\u0441\u0442\u044c \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438), \u0438 \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u043e\u0441\u043b\u0435 \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0431\u0443\u0434\u0435\u0442 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u0446\u0435\u043f\u043e\u0447\u043a\u0438<\/h2>\n<p>\u041f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u043f\u043e\u043b\u0435\u0437\u043d\u043e \u043d\u0435 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0438\u043d\u0430\u0447\u0435 \u0431\u044b\u043b\u0438 \u0431\u044b \u043a\u0440\u0430\u0439\u043d\u0435 \u0442\u0440\u0443\u0434\u043e\u0451\u043c\u043a\u0438\u043c\u0438; \u043e\u043d\u043e \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u0440\u0430\u0441\u0448\u0438\u0440\u0438\u0442\u044c \u043c\u0435\u0442\u043e\u0434\u044b \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u043d\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0434\u0440\u0443\u0433\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439. \u041d\u0438\u0436\u0435 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u044d\u0442\u0438 \u043c\u0435\u0442\u043e\u0434\u044b, \u0438\u0445 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0438 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430.<\/p>\n<p><a name=\"21\"><\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/5ddoUcIhgjU\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><a name=\"21\"><\/a><\/p>\n<h3>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU&amp;t=75s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u041f\u0443\u0441\u0442\u044c <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u2014 \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f<\/span><\/a> \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430\u044f \u043d\u0430 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq \\mathbb{R}<\/span><\/span>. \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438, \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(f^{-1}\\circ f)(x) = f^{-1}(f(x)) = x.<\/span><\/span> \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0442 \u043a \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c\u0443 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0443:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1 = \\displaystyle \\frac{d}{dx} x = \\frac{d}{dx} f^{-1}(f(x)) = \\frac{df^{-1}(f(x))}{df(x)}\\frac{df(x)}{dx}<\/span>\n<p>\u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0434\u0435\u043b\u0438\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df^{-1}(f(x))\/df(x)<\/span><\/span>, \u0438 \u0432 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{df^{-1}(f(x))}{df(x)}= \\frac{1}{\\frac{df(x)}{dx}}}<\/span>\n<p>\u042d\u0442\u043e \u0438 \u0435\u0441\u0442\u044c \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445. \u0412 \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0435 \u044d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043d\u0435\u0440\u0435\u0434\u043a\u043e \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442 \u0432 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c \u0432\u0438\u0434\u0435:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{dx}{dy}= \\frac{1}{\\frac{dy}{dx}}}<\/span>\n<p>\u041e\u0431\u0435 \u0444\u043e\u0440\u043c\u044b \u0437\u0430\u043f\u0438\u0441\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b \u0438 \u043f\u043e\u043b\u0443\u0447\u0430\u044e\u0442\u0441\u044f, \u0435\u0441\u043b\u0438 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=f^{-1}(y).<\/span><\/span><\/p>\n<p>\u0414\u043e \u044d\u0442\u043e\u0433\u043e \u043c\u043e\u043c\u0435\u043d\u0442\u0430 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043b\u0438 \u0432\u0441\u0451, \u0447\u0442\u043e \u043a\u0430\u0441\u0430\u0435\u0442\u0441\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u043e\u0432\u043a\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438; \u0442\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c, \u043a\u0430\u043a \u0435\u0451 \u043c\u043e\u0436\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0438\u043d\u0430\u0447\u0435 \u0431\u044b\u043b\u043e \u0431\u044b \u0434\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u0441\u043b\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c.<\/p>\n<p><a name=\"211\"><\/a><\/p>\n<h4>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/h4>\n<p><span style=\"color: #ff0000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU&amp;t=215s\" target=\"_blank\" style=\"color: #ff0000;\" rel=\"noopener\">\u041a\u043e\u0433\u0434\u0430 \u043c\u044b \u0438\u0437\u0443\u0447\u0430\u043b\u0438 \u0431\u0430\u0437\u043e\u0432\u044b\u0435 \u0442\u0435\u0445\u043d\u0438\u043a\u0438<\/a><\/span> \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f, \u043c\u044b \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u043b\u0438, \u0447\u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(x) = \\frac{1}{x}<\/span>\n<p>\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u044d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043b\u0435\u0433\u043a\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}e^x = e^x<\/span>\n<p style=\"text-align: justify;color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e:<\/strong><\/p>\n<p>\u041e\u0447\u0435\u0432\u0438\u0434\u043d\u043e, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\ln(x)<\/span><\/span> \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=e^y.<\/span><\/span> \u0422\u043e\u0433\u0434\u0430, \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}e^y = \\frac{dx}{dy} = \\frac{1}{\\frac{dy}{dx}} = \\frac{1}{\\frac{d}{dx}\\ln(x)} = x = e^y<\/span>\n<p>Es decir:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}e^y = e^y<\/span>\n<p>Si en esta ultima expresi\u00f3n remplazamos las \u00aby\u00bb por \u00abx\u00bb, obtenemos lo que se quer\u00eda demostrar:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}e^x = e^x.<\/span>\n<p><a name=\"212\"><\/a><\/p>\n<h4>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043e\u0431\u0440\u0430\u0442\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<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU\" target:=\"\" span=\"\" style=\"color: #ff0000;\" 0=\"\" a=\"\">\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/a> \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432\u0441\u0435\u0445 \u043e\u0431\u0440\u0430\u0442\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. \u042d\u0442\u0438 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 son:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ccccccc}\n\n\\dfrac{d}{dx}\\text{Arcsin}(x) &amp;=&amp; \\dfrac{1}{\\sqrt{1-x^2}} &amp;\\phantom{asd}&amp;\\dfrac{d}{dx}\\text{Arccos}(x) &amp;=&amp; \\dfrac{-1}{\\sqrt{1-x^2}} \\\\ \\\\\n\n\\dfrac{d}{dx}\\text{Arctan}(x) &amp;=&amp; \\dfrac{1}{1+x^2} &amp;\\phantom{asd}&amp;\\dfrac{d}{dx}\\text{Arccot}(x) &amp;=&amp; \\dfrac{-1}{1-x^2} \\\\ \\\\\n\n\\dfrac{d}{dx}\\text{Arcsec}(x) &amp;=&amp; \\dfrac{1}{x\\sqrt{x^2-1}} &amp;\\phantom{asd}&amp;\\dfrac{d}{dx}\\text{Arccsc}(x) &amp;=&amp; \\dfrac{-1}{x\\sqrt{x^2-1}}\n\n\\end{array}<\/span>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u042c\u0421\u0422\u0412\u041e<\/strong><\/p>\n<h5>\u0410\u0440\u043a\u0441\u0438\u043d\u0443\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb24013c1\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb24013c1\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[\\frac{-\\pi}{2}+k\\pi , \\frac{\\pi}{2}+ k\\pi \\right],<\/span><\/span> \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0438\u043d\u0443\u0441\u0430 \u043f\u0440\u0438\u043d\u0438\u043c\u0430\u0435\u0442 \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\sin : \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] \\longrightarrow [-1,1]<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sin(x) \\longleftrightarrow x=arcsin(y).<\/span>\n<p>\u0415\u0441\u043b\u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u0442\u044c \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0438\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcsin(y) = \\frac{1}{\\frac{d}{dx}\\sin(x)} = \\frac{1}{\\cos(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442, \u0447\u0442\u043e \u0434\u043b\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [-\\pi\/2, \\pi\/2]<\/span><\/span> \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\cos(x) = \\sqrt{1 - \\sin^2(x)}<\/span>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u0441\u0438\u043d\u0443\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcsin(y) = \\frac{1}{\\cos(x)} = \\frac{1}{ \\sqrt{1 - \\sin^2(x)}}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sin(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcsin(y) = \\frac{1}{ \\sqrt{1 - y^2}}<\/span>\n<p>\u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}arcsin(x) = \\frac{1}{ \\sqrt{1 - x^2}}}<\/span>\n<\/div>\n<h5>\u0410\u0440\u043a\u043a\u043e\u0441\u0438\u043d\u0443\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb24014bf\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb24014bf\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span>, \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u0431\u0443\u0434\u0435\u0442 \u0438\u043c\u0435\u0442\u044c \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos : \\left[0, \\pi\\right] \\longrightarrow [-1,1]<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\cos(x) \\longleftrightarrow x=arccos(y).<\/span>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arccos(y) = \\frac{1}{\\frac{d}{dx}\\cos(x)} = \\frac{-1}{\\sin(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [0, \\pi]<\/span><\/span>, \u0442\u043e\u0433\u0434\u0430<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sin(x) = \\sqrt{1 - \\cos^2(x)}<\/span>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arccos(y) = \\frac{-1}{\\sin(x)} = \\frac{-1}{ \\sqrt{1 - \\cos^2(x)}}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\cos(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arccos(y) = \\frac{-1}{ \\sqrt{1 - y^2}}<\/span>\n<p>\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}arccos(x) = \\frac{-1}{ \\sqrt{1 - x^2}}}<\/span>\n<\/div>\n<h5>\u0410\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb240158f\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb240158f\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[-\\frac{\\pi}{2}+k\\pi , \\frac{\\pi}{2}+ k\\pi \\right],<\/span><\/span>, \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0,<\/span><\/span> \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u0431\u0443\u0434\u0435\u0442 \u0438\u043c\u0435\u0442\u044c \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\tan : \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] \\longrightarrow \\mathbb{R}<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\tan(x) \\longleftrightarrow x=arctan(y).<\/span>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arctan(y) = \\frac{1}{\\frac{d}{dx}\\tan(x)} = \\frac{1}{\\sec^2(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0441\u043b\u0435\u0434\u0443\u0435\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sec^2(x) =1+\\tan^2(x)<\/span>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u044d\u0442\u043e \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arctan(y) = \\frac{1}{\\sec^2(x)} = \\frac{1}{ 1+\\tan^2(x)}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\tan(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arctan(y) = \\frac{1}{1 + y^2}<\/span>\n<p>\u0418 \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 resultado:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}arctan(x) = \\frac{1}{1+ x^2}}<\/span>\n<\/div>\n<h5>\u0410\u0440\u043a\u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb2401649\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb2401649\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">cot(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span>, \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430 \u0431\u0443\u0434\u0435\u0442 \u0438\u043c\u0435\u0442\u044c \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ctg : \\left[0, \\pi\\right] \\longrightarrow \\mathbb{R}<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ctg(x) \\longleftrightarrow x=arcctg(y).<\/span>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcctg(y) = \\frac{1}{\\frac{d}{dx}ctg(x)} = \\frac{-1}{\\csc^2(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\csc^2(x) =1+ctg^2(x)<\/span>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u044d\u0442\u043e \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u043a\u043e\u0442\u0430\u043d\u0433\u0435\u043d\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcctg(y) = \\frac{-1}{\\csc^2(x)} = \\frac{-1}{ 1+ctg^2(x)}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ctg(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}arcctg(y) = \\frac{-1}{1 + y^2}<\/span>\n<p>\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 resultado:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}arcctg(x) = \\frac{-1}{1+ x^2}}<\/span>\n<\/div>\n<h5>\u0410\u0440\u043a\u0441\u0435\u043a\u0430\u043d\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb2401701\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb2401701\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sec(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[0+k\\pi , \\pi+ k\\pi \\right]\\setminus\\left\\{\\frac{\\pi}{2} + k\\pi\\right\\},<\/span><\/span> \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0435\u043a\u0430\u043d\u0441\u0430 \u0431\u0443\u0434\u0435\u0442 \u0438\u043c\u0435\u0442\u044c \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sec : \\left[0, \\pi\\right]\\setminus\\{\\pi\/2\\} \\longrightarrow \\mathbb{R}\\setminus]-1,1[<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sec(x) \\longleftrightarrow x={arcsec}(y).<\/span>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arcsec}(y) = \\frac{1}{\\frac{d}{dx}\\sec(x)} = \\frac{1}{\\sec(x)\\tan(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\tan^2(x) =\\sec^2(x)-1<\/span>\n<p>\u0422\u043e\u0433\u0434\u0430, \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0432 \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u0441\u0435\u043a\u0430\u043d\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arcsec}(y) = \\frac{1}{\\sec(x)\\tan(x)} = \\frac{1}{sec(x)\\sqrt{\\sec^2(x)-1}}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sec(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arcsec}(y) = \\frac{1}{y\\sqrt{y^2-1}}<\/span>\n<p>\u0418 \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 resultado:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}{arcsec}(x) = \\frac{1}{x\\sqrt{x^2-1}}}<\/span>\n<\/div>\n<h5>\u0410\u0440\u043a\u043a\u043e\u0441\u0435\u043a\u0430\u043d\u0441<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb24017b3\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb24017b3\" class=\"collapseomatic_content \">\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\csc(x)<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0439, \u0435\u0441\u043b\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0442\u044c \u0435\u0451 \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e\u043c \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[-\\frac{\\pi}{2}+k\\pi , \\frac{\\pi}{2} + k\\pi \\right]\\setminus\\left\\{0+k\\pi\\right\\}<\/span><\/span>, \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u2014 \u043b\u044e\u0431\u043e\u0435 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u0411\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u043e\u0431\u0449\u043d\u043e\u0441\u0442\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0438\u0435\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043e\u0441\u0435\u043a\u0430\u043d\u0441\u0430 \u0431\u0443\u0434\u0435\u0442 \u0438\u043c\u0435\u0442\u044c \u0432\u0438\u0434<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\csc : \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]\\setminus\\{0\\} \\longrightarrow \\mathbb{R}\\setminus]-1,1[<\/span>\n<p>\u0438 \u043f\u0440\u0438 \u044d\u0442\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\csc(x) \\longleftrightarrow x={arccsc}(y).<\/span>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arccsc}(y) = \\frac{1}{\\frac{d}{dx}\\csc(x)} = \\frac{-1}{\\csc(x)ctg(x)}<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin^2(x) + \\cos^2(x) = 1<\/span>\n<p>\u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ctg^2(x) =\\csc^2(x)-1<\/span>\n<p>\u0422\u043e\u0433\u0434\u0430, \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0432 \u044d\u0442\u043e \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0430\u0440\u043a\u043a\u043e\u0441\u0435\u043a\u0430\u043d\u0441\u0430, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arcsec}(y) = \\frac{-1}{\\csc(x)ctg(x)} = \\frac{-1}{csc(x)\\sqrt{\\csc^2(x)-1}}<\/span>\n<p>\u0418 \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\csc(x)<\/span><\/span>, \u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}{arccsc}(y) = \\frac{-1}{y\\sqrt{y^2-1}}<\/span>\n<p>\u0418 \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0437\u0430\u043c\u0435\u043d\u0438\u0432 \u00aby\u00bb \u043d\u0430 \u00abx\u00bb \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 resultado:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{d}{dx}{arccsc}(x) = \\frac{-1}{x\\sqrt{x^2-1}}}<\/span>\n<\/div>\n<p><a name=\"22\"><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hOAydWcd6zw\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/a><\/p>\n<h3>\u041d\u0435\u044f\u0432\u043d\u043e\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435<\/h3>\n<p>\u0412\u0441\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u043b\u0438 \u0434\u043e \u044d\u0442\u043e\u0433\u043e \u043c\u043e\u043c\u0435\u043d\u0442\u0430, \u0431\u044b\u043b\u0438 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u044b \u0434\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u0437\u0430\u0434\u0430\u043d\u043d\u044b\u0445 \u0432 \u044f\u0432\u043d\u043e\u043c \u0432\u0438\u0434\u0435: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span>. \u041e\u0434\u043d\u0430\u043a\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u0438, \u043a\u043e\u0433\u0434\u0430, \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u043c\u0438, \u043b\u0438\u0431\u043e \u0442\u0440\u0443\u0434\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u044f\u0432\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043b\u0438\u0431\u043e \u044d\u0442\u043e \u0432\u043e\u043e\u0431\u0449\u0435 \u043d\u0435\u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e. \u0414\u043b\u044f \u0442\u0430\u043a\u0438\u0445 \u0441\u043b\u0443\u0447\u0430\u0435\u0432 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \u0442\u0435\u0445\u043d\u0438\u043a\u0430 \u043d\u0435\u044f\u0432\u043d\u043e\u0433\u043e \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f, \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u043d\u0430\u044f, \u043a\u0430\u043a \u0438 \u043f\u0440\u0435\u0436\u0434\u0435, \u043d\u0430 \u043f\u0440\u0430\u0432\u0438\u043b\u0435 \u0446\u0435\u043f\u043e\u0447\u043a\u0438.<\/p>\n<p>\u0427\u0442\u043e\u0431\u044b \u043f\u043e\u043d\u044f\u0442\u044c \u044d\u0442\u043e\u0442 m\u00e9todo, \u043f\u043e\u043b\u0435\u0437\u043d\u0435\u0435 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043f\u0440\u0438\u043c\u0435\u0440\u044b, \u0447\u0435\u043c \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u044b\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430. \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u043c\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435\u043c<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x^3 +y^3- 9xy=0<\/span>\n<p>\u0415\u0441\u043b\u0438 \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0433\u0440\u0430\u0444\u0438\u043a \u044d\u0442\u043e\u0433\u043e \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f, \u0442\u043e \u0441\u0442\u0430\u043d\u0435\u0442 \u044f\u0441\u043d\u043e, \u0447\u0442\u043e \u044d\u0442\u043e \u043d\u0435 \u0433\u0440\u0430\u0444\u0438\u043a \u043a\u0430\u043a\u043e\u0439-\u043b\u0438\u0431\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u042d\u0442\u043e \u0433\u0440\u0430\u0444\u0438\u043a \u043a\u0440\u0438\u0432\u043e\u0439, \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u043c\u043e\u0439 \u00ab\u043b\u0438\u0441\u0442 \u0414\u0435\u043a\u0430\u0440\u0442\u0430\u00bb.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-l30tAMcTkk0\/YLCIuWcDueI\/AAAAAAAAFIY\/K7uSR44DepgIjBlSVV7mCQO-Z0iy_RnRQCLcBGAsYHQ\/s0\/hojaDeDescartes.PNG\" alt=\"hoja de descartes\" class=\"alignnone size-full lazyload\" width=\"690\" height=\"515\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-l30tAMcTkk0\/YLCIuWcDueI\/AAAAAAAAFIY\/K7uSR44DepgIjBlSVV7mCQO-Z0iy_RnRQCLcBGAsYHQ\/s0\/hojaDeDescartes.PNG\" alt=\"hoja de descartes\" class=\"alignnone size-full lazyload\" width=\"690\" height=\"515\" \/><\/noscript><\/center><\/p>\n<p>Ahora, si quisi\u00e9ramos calcular, por ejemplo: la derivada de <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> con respecto a <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, entonces tendr\u00edamos serias dificultades con encontrar de forma explicita expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> que satisface la ecuaci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> para luego derivar. Lo que hacemos, sin embargo, es saltarnos ese paso y asumimos impl\u00edcitamente que <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> es funci\u00f3n de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, es decir: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=y(x)<\/span><\/span>. Haciendo esto, la relaci\u00f3n de la hoja de Descartes se transforma en:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x^3 +y^3(x)- 9xy(x)=0<\/span>\n<p>Y podemos, en consecuencia, derivar todo utilizando la regla de la cadena. Si lo hacemos, llegaremos al siguiente resultado:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\n\\displaystyle 3x^{2} + 3\\,y(x)^{2}\\,\\frac{dy}{dx} - \\left(9\\,y(x) + 9x\\,\\frac{dy}{dx}\\right) &amp;=&amp; 0 \\\\ \\\\\n\n\\displaystyle 3x^{2} + 3\\,y(x)^{2}\\,\\frac{dy}{dx} - 9\\,y(x) - 9x\\,\\frac{dy}{dx} &amp;=&amp; 0 \\\\ \\\\\n\n\\displaystyle \\frac{dy}{dx}\\,\\big(3\\,y(x)^{2} - 9x\\big) &amp;=&amp; 9\\,y(x) - 3x^{2} \\\\ \\\\\n\n\\displaystyle \\frac{dy}{dx} &amp;=&amp; \\dfrac{9\\,y(x) - 3x^{2}}{3\\,y(x)^{2} - 9x} \\\\ \\\\\n\n\\displaystyle \\color{blue}{\\frac{dy}{dx}} &amp;\\color{blue}{=}&amp; \\color{blue}{\\dfrac{3\\,y(x) - x^{2}}{y(x)^{2} - 3x}}\n\n\\end{array}\n\n<\/span>\n<p>\u0410 \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u044d\u0442\u043e\u0433\u043e, \u0435\u0441\u043b\u0438 \u043c\u044b \u0437\u043d\u0430\u0435\u043c \u0442\u043e\u0447\u043a\u0443 \u043a\u0440\u0438\u0432\u043e\u0439, \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0440\u044f\u043c\u043e\u0439, \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0449\u0435\u0439 \u0447\u0435\u0440\u0435\u0437 \u044d\u0442\u0443 \u0442\u043e\u0447\u043a\u0443. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0433\u043b\u044f\u0434\u044f \u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u043a, \u043c\u043e\u0436\u043d\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u0442\u044c, \u0447\u0442\u043e \u0442\u043e\u0447\u043a\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2,4)<\/span><\/span> \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u0438\u0442 \u043a\u0440\u0438\u0432\u043e\u0439; \u0438 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e, \u044d\u0442\u043e \u043f\u043e\u0434\u0442\u0432\u0435\u0440\u0436\u0434\u0430\u0435\u0442\u0441\u044f, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^3 + 4^3 - 9\\cdot 2\\cdot 4 = 8+64 - 72 = 0. <\/span><\/span> \u0417\u043d\u0430\u044f \u044d\u0442\u043e, \u043c\u043e\u0436\u043d\u043e \u0431\u044b\u0441\u0442\u0440\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043d\u0430\u043a\u043b\u043e\u043d \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0440\u044f\u043c\u043e\u0439 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\left.\\frac{dy}{dx}\\right|_{(2,4)}= \\frac{3\\cdot 4 - 2^2}{4^2 - 3\\cdot 2}= \\frac{8}{10}= \\frac{4}{5}}<\/span>\n<p><a name=\"221\"><\/a><\/p>\n<h4>\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/h4>\n<p>\u041d\u0435\u044f\u0432\u043d\u043e\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u0440\u0430\u0441\u0448\u0438\u0440\u0438\u0442\u044c \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u043e\u0434\u043d\u043e\u0439 \u0438\u0437 \u0431\u0430\u0437\u043e\u0432\u044b\u0445 \u0442\u0435\u0445\u043d\u0438\u043a \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u2014 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432\u0438\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^n<\/span><\/span>, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}<\/span><\/span>. \u0422\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043e\u0442 \u0446\u0435\u043b\u044b\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u043a \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u043c \u0438 \u0431\u0435\u0437 \u0442\u0440\u0443\u0434\u0430 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}x^{p\/q}= \\frac{p}{q}x^{(p\/q) -1}<\/span>\n<p>\u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p,q\\in\\mathbb{Z}<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span><\/span>.<\/p>\n<p>\u0427\u0442\u043e\u0431\u044b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u044d\u0442\u043e, \u043f\u043e\u043b\u043e\u0436\u0438\u043c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=x^{p\/q}<\/span><\/span> \u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c, \u043f\u043e\u043b\u0443\u0447\u0430\u044f:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(y) = \\displaystyle \\frac{p}{q}\\ln(x)<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u044f \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043d\u0435\u044f\u0432\u043d\u043e, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{y}\\frac{dy}{dx} = \\frac{p}{q}\\frac{1}{x}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{dy}{dx} = \\frac{p}{q}\\frac{1}{x}y(x)= \\frac{p}{q}\\frac{1}{x}x^{p\/q} = \\frac{p}{q}x^{(p\/q) - 1}}<\/span>\n<p><a name=\"3\"><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KwJ5Bb5Ch_o\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/a><\/p>\n<h2>Gu\u00eda de Ejercicios:<\/h2>\n<h4>Regla de la Cadena Una Variable<\/h4>\n<ol>\n<li>Calcule las derivadas del siguiente grupo de funciones:<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"20px\">a.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=(x^2-3)^{12}<\/span><\/span><\/td>\n<td width=\"20px\">b.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\left(\\frac{4x^3 - x\\cos(2x) - 1}{\\sin(2x) + 2} \\right)^5<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td width=\"20px\">c.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\cos(1-x^2)<\/span><\/span><\/td>\n<td width=\"20px\">d.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\tan(x\\cos(3-x^2))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td width=\"20px\">e.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\frac{1}{(\\sec(2x)-1)^{3\/2}}<\/span><\/span><\/td>\n<td width=\"20px\">f.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\frac{\\tan(2x)}{1-\\cot(2x)}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td width=\"20px\">g.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\ln\\left(\\frac{\\tan(x)}{x^2+1}\\right)<\/span><\/span><\/td>\n<td width=\"20px\">h.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=3^{\\csc(4x)}<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calcule la derivada del siguiente grupo de funciones:<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"20px\">a.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\frac{1}{\\sqrt{x}arctan\\left(x^3\\right)}<\/span><\/span><\/td>\n<td width=\"20px\">b.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\displaystyle \\frac{{arcsec}(x^2-x+2)}{\\sqrt{x^2+1}}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td width=\"20px\">c.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^x<\/span><\/span><\/td>\n<td width=\"20px\">d.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)={arccsc}\\left(x^{\\ln(x)}\\right)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td width=\"20px\">e.<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln\\left(arctan(e^x)\\right)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u041f\u0440\u0430\u0432\u0438\u043b\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0421 \u0443\u0447\u0451\u0442\u043e\u043c \u0432\u0441\u0435\u0433\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0440\u0430\u043d\u0435\u0435, \u043c\u044b \u0443\u0436\u0435 \u0440\u0430\u0441\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u043c 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