{"id":35183,"date":"2024-12-01T13:00:00","date_gmt":"2024-12-01T13:00:00","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35183"},"modified":"2025-11-22T22:33:57","modified_gmt":"2025-11-22T22:33:57","slug":"%d9%82%d8%a7%d8%b9%d8%af%d8%a9-%d8%a7%d9%84%d8%b3%d9%84%d8%b3%d9%84%d8%a9-%d9%84%d9%85%d8%b4%d8%aa%d9%82%d8%a9-%d8%aa%d8%b1%d9%83%d9%8a%d8%a8-%d8%a7%d9%84%d8%af%d9%88%d8%a7%d9%84","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d9%82%d8%a7%d8%b9%d8%af%d8%a9-%d8%a7%d9%84%d8%b3%d9%84%d8%b3%d9%84%d8%a9-%d9%84%d9%85%d8%b4%d8%aa%d9%82%d8%a9-%d8%aa%d8%b1%d9%83%d9%8a%d8%a8-%d8%a7%d9%84%d8%af%d9%88%d8%a7%d9%84\/","title":{"rendered":"\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0644\u0645\u0634\u062a\u0642\u0629 \u062a\u0631\u0643\u064a\u0628 \u0627\u0644\u062f\u0648\u0627\u0644"},"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>\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0644\u0645\u0634\u062a\u0642\u0629 \u062a\u0631\u0643\u064a\u0628 \u0627\u0644\u062f\u0648\u0627\u0644<\/h1>\n<p><em>\u0627\u0633\u062a\u0646\u0627\u062f\u064b\u0627 \u0625\u0644\u0649 \u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u0647 \u062d\u062a\u0649 \u0627\u0644\u0622\u0646\u060c \u0628\u0627\u062a \u0644\u062f\u064a\u0646\u0627 \u0643\u0644 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0627\u062a \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062d\u0633\u0627\u0628 \u0645\u0639\u0638\u0645 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u062a\u0642\u0631\u064a\u0628\u064b\u0627. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u064a\u0646\u0628\u063a\u064a \u0627\u0644\u062a\u0645\u064a\u064a\u0632 \u0628\u064a\u0646 \u0627\u0644\u0642\u062f\u0631\u0629 \u0639\u0644\u0649 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0629 \u0648\u0628\u064a\u0646 \u0627\u0644\u062c\u0647\u062f \u0627\u0644\u0630\u064a \u0646\u0628\u0630\u0644\u0647 \u0641\u064a \u0625\u062c\u0631\u0627\u0621 \u062a\u0644\u0643 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a\u060c \u0648\u0647\u0646\u0627 \u062a\u0628\u0631\u0632 \u0623\u0647\u0645\u064a\u0629 \u0627\u0644\u0646\u0638\u0631\u064a\u0627\u062a \u0645\u062b\u0644 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0648\u0627\u062d\u062f. \u0625\u0630 \u062a\u062a\u064a\u062d \u0644\u0646\u0627 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u0628\u0633\u0631\u0639\u0629\u060c \u0628\u064a\u0646\u0645\u0627 \u0642\u062f \u064a\u062a\u0637\u0644\u0651\u0628 \u0630\u0644\u0643 \u0639\u0645\u0644\u064b\u0627 \u0637\u0648\u064a\u0644\u064b\u0627 \u0648\u0645\u064f\u0631\u0647\u0642\u064b\u0627 \u0625\u0630\u0627 \u0627\u062a\u0651\u064f\u0628\u0639 \u0628\u0637\u0631\u064a\u0642\u0629 \u0645\u0628\u0627\u0634\u0631\u0629.<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b><u>\u0641\u0647\u0631\u0633 \u0627\u0644\u0645\u062d\u062a\u0648\u064a\u0627\u062a<\/u><\/b><br \/>\n<b><a href=\"#1\">\u0645\u0628\u0631\u0647\u0646\u0629 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u0645\u062a\u063a\u064a\u0631 \u062d\u0642\u064a\u0642\u064a \u0648\u0627\u062d\u062f<\/a><\/b><br \/>\n<a href=\"#11\">\u0628\u0631\u0647\u0627\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/a><br \/>\n<a href=\"#12\">\u0623\u0645\u062b\u0644\u0629 \u0639\u0644\u0649 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0645\u062a\u063a\u064a\u0631 \u0648\u0627\u062d\u062f<\/a><br \/>\n<a href=\"#13\">\u062a\u0646\u0628\u064a\u0647 \u064a\u062c\u0628 \u0623\u062e\u0630\u0647 \u0641\u064a \u0627\u0644\u0627\u0639\u062a\u0628\u0627\u0631 \u0639\u0646\u062f \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/a><br \/>\n<b><a href=\"#2\">\u0646\u062a\u0627\u0626\u062c \u0645\u0641\u064a\u062f\u0629 \u0645\u0633\u062a\u062e\u0644\u0635\u0629 \u0645\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/a><\/b><br \/>\n<a href=\"#21\">\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629<\/a><br \/>\n<a href=\"#211\">\u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u064a\u0629<\/a><br \/>\n<a href=\"#212\">\u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629<\/a><br \/>\n<a href=\"#22\">\u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0636\u0645\u0646\u064a<\/a><br \/>\n<a href=\"#221\">\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u0642\u0648\u0649 \u0627\u0644\u0643\u0633\u0631\u064a\u0629<\/a><br \/>\n<a href=\"#221\">\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u0642\u0648\u0649 \u0627\u0644\u0643\u0633\u0631\u064a\u0629<\/a><br \/>\n<b><a href=\"#3\">\u062f\u0644\u064a\u0644 \u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646<\/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>\u0645\u0628\u0631\u0647\u0646\u0629 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u0645\u062a\u063a\u064a\u0631 \u062d\u0642\u064a\u0642\u064a \u0648\u0627\u062d\u062f<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=165s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0648 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u062f\u0627\u0644\u062a\u064a\u0646<\/span><\/a> \u0642\u0627\u0628\u0644\u062a\u064a\u0646 \u0644\u0644\u062a\u0631\u0643\u064a\u0628<\/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{R}<\/span>\n<p>\u0625\u0630\u0627 \u0643\u0627\u0646\u062a <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0648\u0643\u0627\u0646\u062a <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>\u060c \u0641\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0645\u0631\u0643\u0628\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g\\circ f<\/span><\/span> \u062a\u0643\u0648\u0646 \u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in A<\/span><\/span>\u060c \u0648\u062a\u062a\u062d\u0642\u0642 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629<\/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>\u0628\u0631\u0647\u0627\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=242s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0644\u0646\u0646\u0638\u0631 \u0641\u064a \u0627\u0644\u062f\u0627\u0644\u062a\u064a\u0646<\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0648 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0643\u0645\u0627 \u0633\u064f\u0628\u0642\u062a \u0627\u0644\u0625\u0634\u0627\u0631\u0629. \u0625\u0630\u0627 \u0642\u0645\u0646\u0627 \u0628\u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0645\u0631\u0643\u0628\u0629\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649<\/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>\u0648\u0647\u0648 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647.<\/p>\n<p><a name=\"12\"><\/a><\/p>\n<h3>\u0623\u0645\u062b\u0644\u0629 \u0639\u0644\u0649 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0645\u062a\u063a\u064a\u0631 \u0648\u0627\u062d\u062f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=423s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0645\u0646 \u0627\u0644\u0623\u0645\u0648\u0631 \u0627\u0644\u062a\u064a \u062a\u0628\u062f\u0648 \u0648\u0627\u0636\u062d\u0629\u060c \u0639\u0644\u0649 \u0627\u0644\u0623\u0642\u0644 \u0644\u0644\u0648\u0647\u0644\u0629 \u0627\u0644\u0623\u0648\u0644\u0649\u060c<\/span><\/a> \u0644\u0643\u0646\u0647\u0627 \u0644\u064a\u0633\u062a \u0643\u0630\u0644\u0643 \u0645\u0646 \u0645\u0646\u0638\u0648\u0631 \u0639\u0645\u0644\u064a\u060c \u0647\u0648 \u0623\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u062a\u062e\u0628\u0631\u0646\u0627 \u0628\u0623\u0646\u0647 \u0639\u0646\u062f\u0645\u0627 \u0646\u0635\u0627\u062f\u0641 \u062a\u0631\u0643\u064a\u0628 \u062f\u0648\u0627\u0644\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u00ab\u0645\u0646 \u0627\u0644\u062e\u0627\u0631\u062c \u0646\u062d\u0648 \u0627\u0644\u062f\u0627\u062e\u0644\u00bb. \u0648\u0644\u062a\u0648\u0636\u064a\u062d \u0630\u0644\u0643 \u0628\u0637\u0631\u064a\u0642\u0629 \u064a\u0633\u0647\u0644 \u0641\u0647\u0645\u0647\u0627\u060c \u062a\u0638\u0644 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0627\u0644\u0637\u0631\u064a\u0642 \u0627\u0644\u0623\u0633\u0631\u0639 \u0628\u0644\u0627 \u0645\u0646\u0627\u0632\u0639.<\/p>\n<ol>\n<li>\u0625\u0630\u0627 \u0637\u064f\u0644\u0628 \u0645\u0646\u0627 \u0627\u0634\u062a\u0642\u0627\u0642 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = (2x^2+1)^{12}<\/span><\/span> \u0641\u0633\u0646\u0642\u0648\u0645 \u0623\u0648\u0644\u064b\u0627 \u0628\u062a\u0637\u0648\u064a\u0631 \u0627\u0644\u0642\u0648\u0649 \u062b\u0645 \u0646\u0637\u0628\u0642 \u0642\u0627\u0639\u062f\u0629 \u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0642\u0648\u0629 \u0639\u0644\u0649 \u0643\u0644 \u062c\u0632\u0621 \u0645\u0646 \u0630\u0644\u0643 \u0643\u062b\u064a\u0631 \u0627\u0644\u062d\u062f\u0648\u062f \u0627\u0644\u0643\u0628\u064a\u0631 \u0627\u0644\u0630\u064a \u0633\u0646\u062d\u0635\u0644 \u0639\u0644\u064a\u0647. \u0648\u0647\u0648 \u0639\u0645\u0644 \u0645\u064f\u0631\u0647\u0642 \u062f\u0648\u0646 \u062f\u0627\u0639\u064d. \u0623\u0645\u0627 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a\u0645\u0643\u0646 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0629 \u0641\u064a \u0628\u0636\u0639\u0629 \u0623\u0633\u0637\u0631 \u0641\u0642\u0637:<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>\u062d\u0627\u0648\u0644 \u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\sin(\\cos(x))<\/span><\/span> \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062a\u0642\u0646\u064a\u0627\u062a \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u0642\u0637\u060c \u0648\u0633\u062a\u062c\u062f \u0646\u0641\u0633\u0643 \u0641\u064a \u0645\u0639\u0627\u0646\u0627\u0629 \u0623\u0628\u062f\u064a\u0629. \u0642\u0645 \u0628\u0630\u0644\u0643 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0648\u0633\u064a\u0638\u0647\u0631 \u0627\u0644\u0646\u0627\u062a\u062c \u0628\u0644\u0627 \u062f\u0645\u0648\u0639 \u0648\u0641\u064a \u062e\u0637\u0648\u0627\u062a \u0642\u0644\u064a\u0644\u0629:<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>\u064a\u0645\u0643\u0646\u0643 \u0623\u064a\u0636\u064b\u0627 \u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u062a\u064a \u062a\u0643\u0648\u0646 \u062a\u0631\u0643\u064a\u0628\u064b\u0627 \u0644\u0639\u062f\u0629 \u062f\u0648\u0627\u0644. \u0625\u0630\u0627 \u0643\u0627\u0646\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\cos(\\cos(\\cos(x))),<\/span><\/span> \u0641\u0625\u0646 \u0645\u0634\u062a\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df\/dx<\/span><\/span> \u062a\u0635\u0628\u062d \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<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>\u0648\u0643\u0645\u0627 \u062a\u0631\u0649\u060c \u0641\u0625\u0646 \u062a\u0637\u0628\u064a\u0642 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0647\u0648 \u0628\u0628\u0633\u0627\u0637\u0629 \u0627\u0634\u062a\u0642\u0627\u0642 \u0645\u062a\u0633\u0644\u0633\u0644 \u0645\u0646 \u0627\u0644\u062e\u0627\u0631\u062c \u0646\u062d\u0648 \u0627\u0644\u062f\u0627\u062e\u0644.<\/li>\n<\/ol>\n<p><a name=\"13\"><\/a><\/p>\n<h3>\u0627\u062d\u062a\u064a\u0627\u0637 \u064a\u062c\u0628 \u0623\u062e\u0630\u0647 \u0641\u064a \u0627\u0644\u0627\u0639\u062a\u0628\u0627\u0631 \u0639\u0646\u062f \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=607s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u062a\u0628\u064a\u0651\u0646 \u0627\u0644\u0623\u062f\u0628\u064a\u0627\u062a \u062c\u0645\u064a\u0639\u0647\u0627 \u0627\u0644\u0641\u0648\u0627\u0626\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629<\/span><\/a> \u0644\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629\u060c \u0644\u0643\u0646 \u0642\u0644\u0651\u0629 \u0642\u0644\u064a\u0644\u0629 \u062a\u062a\u0646\u0627\u0648\u0644 \u0628\u0648\u0636\u0648\u062d \u0627\u0644\u0627\u062d\u062a\u064a\u0627\u0637\u0627\u062a \u0627\u0644\u0648\u0627\u062c\u0628 \u0627\u062a\u062e\u0627\u0630\u0647\u0627 \u0642\u0628\u0644 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u0647\u0627. \u0641\u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0642\u0648\u0629 \u0647\u0630\u0647 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0629\u060c \u064a\u062c\u0628 \u0639\u0644\u064a\u0643 \u062f\u0627\u0626\u0645\u064b\u0627 \u0627\u0644\u0627\u0646\u062a\u0628\u0627\u0647 \u0628\u062f\u0642\u0629 \u0625\u0644\u0649 \u0645\u062c\u0627\u0644\u0627\u062a \u0627\u0644\u062f\u0648\u0627\u0644 \u0648\u0645\u062f\u0627\u0647\u0627 \u0642\u0628\u0644 \u062a\u0637\u0628\u064a\u0642 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629. \u0625\u0630 \u064a\u0646\u0628\u063a\u064a \u0627\u0644\u062a\u0623\u0643\u062f \u0645\u0633\u0628\u0642\u064b\u0627 \u0645\u0646 \u0623\u0646\u0651 \u0645\u062c\u0627\u0644\u0627\u062a \u0627\u0644\u062f\u0648\u0627\u0644 \u0648\u0645\u062f\u0649 \u0643\u0644 \u0645\u0646\u0647\u0627 \u0645\u062a\u0648\u0627\u0641\u0642\u0627\u0646 \u0645\u0646 \u0623\u062c\u0644 \u0627\u0644\u062a\u0631\u0643\u064a\u0628\u061b \u0644\u0623\u0646 \u0625\u0647\u0645\u0627\u0644 \u0630\u0644\u0643 \u0642\u062f \u064a\u0624\u062f\u064a \u0625\u0644\u0649 \u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0627\u062a \u0641\u064a \u0646\u0642\u0627\u0637 \u0644\u0627 \u062a\u0643\u0648\u0646 \u0641\u064a\u0647\u0627 \u0627\u0644\u0645\u0634\u062a\u0642\u0629 \u0645\u064f\u0639\u0631\u0651\u0641\u0629 \u0623\u0635\u0644\u064b\u0627. \u0641\u0625\u0630\u0627 \u0627\u0634\u062a\u0642\u0642\u062a \u0645\u062b\u0644\u064b\u0627 \u062f\u0627\u0644\u0629 \u0645\u0646 \u0627\u0644\u0646\u0648\u0639<\/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>\u0648\u0627\u0639\u062a\u0645\u062f\u062a \u0639\u0644\u0649 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u062f\u0648\u0646 \u062a\u062d\u0642\u0642\u060c \u0641\u0633\u062a\u0642\u0648\u0645 \u0628\u062d\u0633\u0627\u0628\u0627\u062a \u0645\u062b\u0644 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0645\u0646 \u0627\u0644\u0648\u0627\u0636\u062d \u0623\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0638\u0644\u064a\u0629 (\u0627\u0644\u0638\u0644) \u0645\u064f\u0639\u0631\u0651\u0641\u0629 \u0639\u0646\u062f \u0627\u0644\u0642\u064a\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=2\\pi\/3<\/span><\/span> \u0644\u0623\u0646\u0651 \u0642\u064a\u0645\u062a\u0647\u0627 \u0647\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(2\\pi\/3) = -\\sqrt{3}<\/span><\/span>. \u0644\u0643\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln(\\cos(x))<\/span><\/span> \u063a\u064a\u0631 \u0645\u064f\u0639\u0631\u0651\u0641\u0629 \u0647\u0646\u0627\u0643 \u0644\u0623\u0646 <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> \u0648\u0644\u0627 \u064a\u0648\u062c\u062f \u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0644\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0633\u0627\u0644\u0628\u0629! \u0641\u064a \u0645\u062b\u0644 \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0627\u062a \u064a\u062c\u0628 \u062a\u062d\u062f\u064a\u062f \u0642\u064a\u0645 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0627\u0644\u0645\u0633\u0645\u0648\u062d \u0628\u0647\u0627 \u0642\u0628\u0644 \u062a\u0637\u0628\u064a\u0642 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0628\u062d\u064a\u062b \u062a\u062c\u0639\u0644 \u062f\u0627\u0644\u0629 \u0627\u0644\u0643\u0648\u0633\u064a\u0646 \u0645\u0648\u062c\u0628\u0629 (\u0628\u0645\u0627 \u064a\u0636\u0645\u0646 \u0627\u0644\u062a\u0648\u0627\u0641\u0642 \u0641\u064a \u0627\u0644\u062a\u0631\u0643\u064a\u0628)\u060c \u0648\u0639\u0646\u062f\u0647\u0627 \u0641\u0642\u0637 \u062a\u0643\u0648\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0635\u062d\u064a\u062d\u0629.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0646\u062a\u0627\u0626\u062c \u0645\u0641\u064a\u062f\u0629 \u0645\u0633\u062a\u062e\u0644\u0635\u0629 \u0645\u0646 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629<\/h2>\n<p>\u0644\u0627 \u062a\u0642\u062a\u0635\u0631 \u0641\u0627\u0626\u062f\u0629 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0639\u0644\u0649 \u062a\u0645\u0643\u064a\u0646\u0646\u0627 \u0645\u0646 \u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0627\u062a \u0643\u0627\u0646\u062a \u0633\u062a\u0628\u062f\u0648 \u0634\u0627\u0642\u0629 \u0628\u0637\u0631\u064a\u0642\u0629 \u0623\u062e\u0631\u0649\u061b \u0628\u0644 \u062a\u0633\u0627\u0647\u0645 \u0623\u064a\u0636\u064b\u0627 \u0641\u064a \u062a\u0648\u0633\u064a\u0639 \u062a\u0642\u0646\u064a\u0627\u062a \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0644\u062a\u0634\u0645\u0644 \u0639\u062f\u062f\u064b\u0627 \u0623\u0643\u0628\u0631 \u0645\u0646 \u0627\u0644\u062f\u0648\u0627\u0644. \u0633\u0646\u0633\u062a\u0639\u0631\u0636 \u0641\u064a\u0645\u0627 \u064a\u0644\u064a \u0647\u0630\u0647 \u0627\u0644\u062a\u0642\u0646\u064a\u0627\u062a \u0648\u0646\u062a\u0627\u0626\u062c\u0647\u0627 \u0648\u0628\u0631\u0627\u0647\u064a\u0646\u0647\u0627.<\/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>\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU&amp;t=75s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u062f\u0627\u0644\u0629 \u062a\u0628\u0627\u064a\u0646\u064a\u0629<\/span><\/a> \u0648\u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0641\u064a \u0641\u062a\u0631\u0629 \u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq \\mathbb{R}<\/span><\/span>. \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u064a\u0645\u0643\u0646 \u062d\u0633\u0627\u0628 \u0645\u0634\u062a\u0642\u0629 \u062f\u0627\u0644\u0629 \u0627\u0644\u0647\u0648\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(f^{-1}\\circ f)(x) = f^{-1}(f(x)) = x.<\/span><\/span> \u0648\u062a\u0639\u0637\u064a \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u0648\u0645\u0646 \u0647\u0646\u0627 \u064a\u0645\u0643\u0646 \u062d\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df^{-1}(f(x))\/df(x)<\/span><\/span> \u0641\u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0647\u0630\u0627 \u0645\u0627 \u064a\u064f\u0639\u0631\u0641 \u0628\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0641\u064a \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a. \u0648\u0645\u0646 \u0627\u0644\u0634\u0627\u0626\u0639 \u0641\u064a \u0627\u0644\u0623\u062f\u0628\u064a\u0627\u062a \u0643\u062a\u0627\u0628\u0629 \u0647\u0630\u0647 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0629 \u0639\u0644\u0649 \u0627\u0644\u0635\u0648\u0631\u0629<\/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>\u0648\u0643\u0644\u062a\u0627 \u0627\u0644\u0635\u064a\u063a\u062a\u064a\u0646 \u0645\u062a\u0643\u0627\u0641\u0626\u062a\u0627\u0646\u060c \u0648\u064a\u062a\u0645 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u064a\u0647\u0645\u0627 \u0628\u0643\u062a\u0627\u0628\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=f^{-1}(y).<\/span><\/span><\/p>\n<p>\u0625\u0644\u0649 \u0647\u0646\u0627 \u0631\u0623\u064a\u0646\u0627 \u0645\u0636\u0645\u0648\u0646 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629\u060c \u0648\u0627\u0644\u0622\u0646 \u0633\u0646\u0631\u0649 \u0643\u064a\u0641 \u064a\u0645\u0643\u0646 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u0647\u0627 \u0644\u062d\u0633\u0627\u0628 \u0628\u0639\u0636 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u062a\u064a \u0642\u062f \u062a\u0643\u0648\u0646 \u0635\u0639\u0628\u0629 \u0628\u063a\u064a\u0631 \u0630\u0644\u0643.<\/p>\n<p><a name=\"211\"><\/a><\/p>\n<h4>\u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u064a\u0629<\/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\">\u0639\u0646\u062f\u0645\u0627 \u062f\u0631\u0633\u0646\u0627 \u062a\u0642\u0646\u064a\u0627\u062a<\/a><\/span> \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0631\u0623\u064a\u0646\u0627 \u0623\u0646<\/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>\u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0648\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u064a\u0645\u0643\u0646 \u0628\u0633\u0647\u0648\u0644\u0629 \u0625\u062b\u0628\u0627\u062a \u0623\u0646<\/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>\u0627\u0644\u0628\u0631\u0647\u0627\u0646:<\/strong><\/p>\n<p>\u0645\u0646 \u0627\u0644\u0648\u0627\u0636\u062d \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\ln(x)<\/span><\/span> \u064a\u0643\u0627\u0641\u0626 \u0627\u0644\u0642\u0648\u0644 \u0628\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=e^y.<\/span><\/span> \u062b\u0645 \u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU\" target:=\"\" span=\"\" style=\"color: #ff0000;\" 0=\"\" a=\"\">\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629<\/a> \u062a\u062a\u064a\u062d \u0644\u0646\u0627 \u0623\u064a\u0636\u064b\u0627 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0645\u0634\u062a\u0642\u0627\u062a \u062c\u0645\u064a\u0639 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629. \u0648\u0647\u0630\u0647 \u0647\u064a:<\/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>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/strong><\/p>\n<h5>\u0642\u0648\u0633 \u0627\u0644\u062c\u064a\u0628 (Arcsin)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52da2da\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52da2da\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064a\u062f\u0646\u0627 \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <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> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0643\u062a\u0641\u0627\u0621 \u0628\u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0639\u0646\u062f\u0645\u0627 \u064a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\u060c \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u062c\u064a\u0628 \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u062a\u062d\u062a \u0647\u0630\u0647 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [-\\pi\/2, \\pi\/2]<\/span><\/span> \u0641\u0625\u0646\u0647 \u064a\u062a\u062d\u0642\u0642:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u062c\u064a\u0628 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sin(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0642\u0648\u0633 \u0627\u0644\u062c\u064a\u0628 \u062a\u0645\u0627\u0645 (Arccos)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52da539\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52da539\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064f\u064a\u0651\u062f \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0642\u062a\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\u060c \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u0628\u062a\u062d\u062a \u0647\u0630\u0647 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [0, \\pi]<\/span><\/span> \u0641\u0625\u0646\u0647 \u064a\u062a\u062d\u0642\u0642:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u062c\u064a\u0628 \u062a\u0645\u0627\u0645 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\cos(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0642\u0648\u0633 \u0627\u0644\u0638\u0644 (Arctan)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52da6b1\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52da6b1\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064f\u064a\u0651\u062f \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <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> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0642\u062a\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0,<\/span><\/span> \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u0638\u0644 \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u062a\u062d\u0642\u064a\u0642 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u0639\u0637\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u0638\u0644 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\tan(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0642\u0648\u0633 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a (Arccot)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52dae0c\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52dae0c\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">cot(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064f\u064a\u0651\u062f \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0642\u062a\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\u060c \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u0628\u062a\u062d\u062a \u0647\u0630\u0647 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ctg(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0642\u0648\u0633 \u0627\u0644\u0642\u0627\u0637\u0639 (Arcsec)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52daf75\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52daf75\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sec(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064f\u064a\u0651\u062f \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <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> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0642\u062a\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\u060c \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u0642\u0627\u0637\u0639 \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u0628\u062a\u062d\u062a \u0647\u0630\u0647 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u0642\u0627\u0637\u0639 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sec(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0642\u0648\u0633 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a (Arccsc)<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3fb52db0ba\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3fb52db0ba\" class=\"collapseomatic_content \">\n<p>\u0625\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\csc(x)<\/span><\/span> \u062a\u0643\u0648\u0646 \u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0625\u0630\u0627 \u0642\u064f\u064a\u0651\u062f \u0645\u062c\u0627\u0644\u0647\u0627 \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 <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> \u062d\u064a\u062b <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0639\u062f\u062f \u0635\u062d\u064a\u062d. \u0648\u062f\u0648\u0646 \u0641\u0642\u062f\u0627\u0646 \u0644\u0644\u0639\u0645\u0648\u0645\u064a\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u0627\u0642\u062a\u0635\u0627\u0631 \u0639\u0644\u0649 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\u060c \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a \u0627\u0644\u062a\u0628\u0627\u064a\u0646\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644:<\/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>\u0648\u0628\u062a\u062d\u062a \u0647\u0630\u0647 \u0627\u0644\u0634\u0631\u0648\u0637 \u064a\u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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>\u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646 \u0646\u062a\u0630\u0643\u0631 \u0627\u0644\u0647\u0648\u064a\u0629 \u0627\u0644\u0645\u062b\u0644\u062b\u064a\u0629:<\/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>\u0648\u0645\u0646\u0647\u0627 \u0646\u0633\u062a\u0646\u062a\u062c:<\/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>\u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a \u0645\u0634\u062a\u0642\u0629 \u0642\u0648\u0633 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0643\u0633\u064a \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\csc(x)<\/span><\/span><\/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>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0628\u0627\u0633\u062a\u0628\u062f\u0627\u0644 \u00aby\u00bb \u0628\u0640 \u00abx\u00bb \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0635\u0644 \u0625\u0644\u0649 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647:<\/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>\u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0636\u0645\u0646\u064a<\/h3>\n<p>\u062c\u0645\u064a\u0639 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u062a\u064a \u062d\u0633\u0628\u0646\u0627\u0647\u0627 \u062d\u062a\u0649 \u0627\u0644\u0622\u0646 \u0643\u0627\u0646\u062a \u0644\u062f\u0648\u0627\u0644 \u0639\u064f\u0631\u0651\u0641\u062a \u0628\u0634\u0643\u0644 \u0635\u0631\u064a\u062d: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span>. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0647\u0646\u0627\u0643 \u062d\u0627\u0644\u0627\u062a \u0644\u0627 \u064a\u0643\u0648\u0646 \u0645\u0646 \u0627\u0644\u0633\u0647\u0644 \u0641\u064a\u0647\u0627 \u2014 \u0623\u0648 \u0642\u062f \u064a\u0643\u0648\u0646 \u0645\u0646 \u0627\u0644\u0645\u0633\u062a\u062d\u064a\u0644 \u2014 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0635\u064a\u063a\u0629 \u0635\u0631\u064a\u062d\u0629 \u0644\u0644\u062f\u0627\u0644\u0629 \u0627\u0646\u0637\u0644\u0627\u0642\u064b\u0627 \u0645\u0646 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a. \u0648\u0644\u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0627\u062a \u0628\u0627\u0644\u0630\u0627\u062a \u062a\u064f\u0633\u062a\u062e\u062f\u0645 \u062a\u0642\u0646\u064a\u0629 \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0636\u0645\u0646\u064a\u060c \u0627\u0644\u062a\u064a \u062a\u0639\u062a\u0645\u062f \u0641\u064a \u0623\u0633\u0627\u0633\u0647\u0627\u060c \u0645\u0631\u0629 \u0623\u062e\u0631\u0649\u060c \u0639\u0644\u0649 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629.<\/p>\n<p>\u0648\u0644\u062a\u0641\u0647\u0651\u0645 \u0647\u0630\u0647 \u0627\u0644\u062a\u0642\u0646\u064a\u0629\u060c \u062a\u064f\u0639\u062f \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0623\u0643\u062b\u0631 \u0641\u0627\u0626\u062f\u0629 \u0645\u0646 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646. \u0644\u0630\u0644\u0643 \u0646\u0639\u062a\u0628\u0631 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u064a\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0627\u0644\u0645\u0639\u0637\u0627\u0629 \u0628\u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629:<\/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>\u0625\u0630\u0627 \u0642\u0645\u0646\u0627 \u0628\u062a\u0645\u062b\u064a\u0644 \u0647\u0630\u0647 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0627\u0646\u064a\u064b\u0627 \u0633\u0646\u0644\u0627\u062d\u0638 \u0623\u0646\u0647\u0627 \u0644\u0627 \u062a\u0645\u062b\u0644 \u062f\u0627\u0644\u0629\u060c \u0628\u0644 \u062a\u0645\u062b\u0644 \u0645\u0646\u062d\u0646\u0649 \u064a\u064f\u0639\u0631\u0641 \u0628\u0627\u0633\u0645 \u00ab\u0648\u0631\u0642\u0629 \u062f\u064a\u0643\u0627\u0631\u062a\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>A partir de esto podemos calcular, si conocemos un punto de la curva, la pendiente de la recta tangente que pasa por ese punto. Por ejemplo, a partir del gr\u00e1fico podemos intuir que el punto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2,4)<\/span><\/span> est\u00e1 sobre la curva; y de hecho, esto se corrobora porque <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> Sabiendo esto podemos decir r\u00e1pidamente que la pendiente de la recta tangente que pasa por ese punto ser\u00e1:<\/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>\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u0642\u0648\u0649 \u0627\u0644\u0643\u0633\u0631\u064a\u0629<\/h4>\n<p>\u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0627\u0644\u0636\u0645\u0646\u064a \u064a\u0645\u0643\u0646 \u062a\u0648\u0633\u064a\u0639 \u0646\u0637\u0627\u0642 \u0625\u062d\u062f\u0649 \u0627\u0644\u062a\u0642\u0646\u064a\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a \u0627\u0644\u0627\u0634\u062a\u0642\u0627\u0642\u060c \u0648\u0647\u064a \u0645\u0634\u062a\u0642\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0645\u0646 \u0627\u0644\u0646\u0648\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^n<\/span><\/span> \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}<\/span><\/span>. \u0648\u064a\u0645\u0643\u0646 \u0627\u0644\u0622\u0646 \u0627\u0644\u0627\u0646\u062a\u0642\u0627\u0644 \u0645\u0646 \u0627\u0644\u0623\u0633\u0633 \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0625\u0644\u0649 \u0627\u0644\u0623\u0633\u0633 \u0627\u0644\u0643\u0633\u0631\u064a\u0629\u060c \u0648\u0625\u062b\u0628\u0627\u062a \u0628\u0633\u0647\u0648\u0644\u0629 \u0623\u0646:<\/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>\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p,q\\in\\mathbb{Z}<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span><\/span>.<\/p>\n<p>\u0648\u0644\u0625\u062b\u0628\u0627\u062a \u0630\u0644\u0643 \u0646\u0642\u0648\u0644: \u0644\u064a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=x^{p\/q}<\/span><\/span> \u0648\u0646\u0637\u0628\u0651\u0642 \u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064b\u0627 \u0637\u0628\u064a\u0639\u064a\u064b\u0627 \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649:<\/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>\u0648\u0627\u0644\u0622\u0646\u060c \u0628\u0627\u0634\u062a\u0642\u0627\u0642 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0636\u0645\u0646\u064a\u064b\u0627 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u062f\u0644\u064a\u0644 \u0627\u0644\u062a\u0645\u0627\u0631\u064a\u0646:<\/h2>\n<h4>\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u0645\u062a\u063a\u064a\u0631 \u0648\u0627\u062d\u062f<\/h4>\n<ol>\n<li>\u0627\u062d\u0633\u0628 \u0645\u0634\u062a\u0642\u0627\u062a \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<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>\u0627\u062d\u0633\u0628 \u0645\u0634\u062a\u0642\u0629 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<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>\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0644\u0645\u0634\u062a\u0642\u0629 \u062a\u0631\u0643\u064a\u0628 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0633\u062a\u0646\u0627\u062f\u064b\u0627 \u0625\u0644\u0649 \u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u0647 \u062d\u062a\u0649 \u0627\u0644\u0622\u0646\u060c \u0628\u0627\u062a \u0644\u062f\u064a\u0646\u0627 \u0643\u0644 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0627\u062a \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062d\u0633\u0627\u0628 \u0645\u0639\u0638\u0645 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u062a\u0642\u0631\u064a\u0628\u064b\u0627. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u064a\u0646\u0628\u063a\u064a \u0627\u0644\u062a\u0645\u064a\u064a\u0632 \u0628\u064a\u0646 \u0627\u0644\u0642\u062f\u0631\u0629 \u0639\u0644\u0649 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0629 \u0648\u0628\u064a\u0646 \u0627\u0644\u062c\u0647\u062f \u0627\u0644\u0630\u064a \u0646\u0628\u0630\u0644\u0647 \u0641\u064a \u0625\u062c\u0631\u0627\u0621 \u062a\u0644\u0643 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a\u060c \u0648\u0647\u0646\u0627 \u062a\u0628\u0631\u0632 \u0623\u0647\u0645\u064a\u0629 \u0627\u0644\u0646\u0638\u0631\u064a\u0627\u062a \u0645\u062b\u0644 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0648\u0627\u062d\u062f. \u0625\u0630 \u062a\u062a\u064a\u062d \u0644\u0646\u0627 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35164,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[860,565],"tags":[],"class_list":["post-35183","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-860","category-565"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0644\u0645\u0634\u062a\u0642\u0629 \u062a\u0631\u0643\u064a\u0628 \u0627\u0644\u062f\u0648\u0627\u0644 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0641\u064a 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