{"id":35187,"date":"2024-12-01T13:00:40","date_gmt":"2024-12-01T13:00:40","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35187"},"modified":"2025-11-22T22:34:12","modified_gmt":"2025-11-22T22:34:12","slug":"%e0%a4%ab%e0%a4%bc%e0%a4%82%e0%a4%95%e0%a5%8d%e0%a4%b6%e0%a4%a8%e0%a5%8b%e0%a4%82-%e0%a4%95%e0%a5%80-%e0%a4%b8%e0%a4%82%e0%a4%b0%e0%a4%9a%e0%a4%a8%e0%a4%be-%e0%a4%95%e0%a5%87-%e0%a4%85%e0%a4%b5","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%ab%e0%a4%bc%e0%a4%82%e0%a4%95%e0%a5%8d%e0%a4%b6%e0%a4%a8%e0%a5%8b%e0%a4%82-%e0%a4%95%e0%a5%80-%e0%a4%b8%e0%a4%82%e0%a4%b0%e0%a4%9a%e0%a4%a8%e0%a4%be-%e0%a4%95%e0%a5%87-%e0%a4%85%e0%a4%b5\/","title":{"rendered":"\u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0915\u0947 \u0905\u0935\u0915\u0932\u0928 \u0915\u0947 \u0932\u093f\u090f \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e"},"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>\u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0915\u0940 \u0905\u0935\u0915\u0932\u0928 \u0915\u0947 \u0932\u093f\u090f \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e<\/h1>\n<p><em>\u0905\u092c \u0924\u0915 \u091c\u094b \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0939\u0948, \u0909\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0932\u0917\u092d\u0917 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092d\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 elementos \u0909\u092a\u0932\u092c\u094d\u0927 \u0939\u0948\u0902\u0964 \u092b\u093f\u0930 \u092d\u0940, \u0939\u092e\u0947\u0902 \u090f\u0915 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0940 posibilidad \u0914\u0930 \u0910\u0938\u0940 \u0917\u0923\u0928\u093e\u090f\u0901 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0932\u0917\u093e\u090f \u0917\u090f \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u093f\u090f, \u0914\u0930 \u092f\u0939\u0940\u0902 \u092a\u0930 \u090f\u0915 \u091a\u0930 \u0915\u0947 c\u00e1lculo \u0915\u0947 \u0932\u093f\u090f \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u091c\u0948\u0938\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0909\u092a\u092f\u094b\u0917\u0940 \u0938\u093f\u0926\u094d\u0927 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0939\u092e\u0947\u0902 \u0910\u0938\u0947 \u0905\u0935\u0915\u0932\u091c \u0915\u094b \u0924\u0947\u091c\u0940 \u0938\u0947 calcular \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0938\u0915\u094d\u0937\u092e \u092c\u0928\u093e\u090f\u0917\u093e, \u091c\u094b \u0905\u0928\u094d\u092f\u0925\u093e \u0915\u093e\u092b\u093c\u0940 \u0936\u094d\u0930\u092e\u0938\u093e\u0927\u094d\u092f \u0914\u0930 \u091c\u091f\u093f\u0932 \u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u092e\u093e\u0902\u0917 \u0915\u0930\u0924\u0947\u0964<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b><u>\u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0938\u0942\u091a\u0915\u093e\u0902\u0915<\/u><\/b><br \/>\n<b><a href=\"#1\">\u090f\u0915 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u092e\u0947\u0902 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/a><\/b><br \/>\n<a href=\"#11\">\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923<\/a><br \/>\n<a href=\"#12\">\u090f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u092e\u0947\u0902 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u0947 \u0909\u0926\u093e\u0939\u0930\u0923<\/a><br \/>\n<a href=\"#13\">\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092e\u0947\u0902 \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u0947 \u0935\u093e\u0932\u0940 \u0938\u093e\u0935\u0927\u093e\u0928\u093f\u092f\u093e\u0901<\/a><br \/>\n<b><a href=\"#2\">\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0909\u092a\u092f\u094b\u0917\u0940 resultados<\/a><\/b><br \/>\n<a href=\"#21\">\u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/a><br \/>\n<a href=\"#211\">\u0918\u093e\u0924\u093e\u0902\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u0905\u0935\u0915\u0932\u091c<\/a><br \/>\n<a href=\"#212\">\u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u093e \u0905\u0935\u0915\u0932\u091c<\/a><br \/>\n<a href=\"#22\">\u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928<\/a><br \/>\n<a href=\"#221\">\u092a\u0930\u093f\u092e\u0947\u092f \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c<\/a><br \/>\n<a href=\"#221\">\u092a\u0930\u093f\u092e\u0947\u092f \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c<\/a><br \/>\n<b><a href=\"#3\">\u0905\u092d\u094d\u092f\u093e\u0938 \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u093f\u0915\u093e<\/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>\u090f\u0915 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u092e\u0947\u0902 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=165s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0926\u094b \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948\u0902<\/span><\/a> \u091c\u094b \u0938\u0902\u092f\u094b\u091c\u0928 \u092f\u094b\u0917\u094d\u092f \u0939\u0948\u0902<\/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>\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u092e\u0947\u0902 \u0905\u0935\u0915\u0932\u0928\u0940\u092f \u0939\u0948 \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> \u092e\u0947\u0902 \u0905\u0935\u0915\u0932\u0928\u0940\u092f \u0939\u0948, \u0924\u094b \u0938\u092e\u093f\u0936\u094d\u0930 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g\\circ f<\/span><\/span> \u0938\u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in A<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0905\u0935\u0915\u0932\u0928\u0940\u092f \u0939\u094b\u0917\u093e \u0914\u0930 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0942\u0924\u094d\u0930 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0947\u0917\u093e<\/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>\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=242s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902<\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>, \u091c\u0948\u0938\u093e \u0915\u093f \u090a\u092a\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964 \u092f\u0926\u093f \u0939\u092e \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0917\u0923\u0928\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e<\/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>\u091c\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e\u0964<\/p>\n<p><a name=\"12\"><\/a><\/p>\n<h3>\u090f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u092e\u0947\u0902 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u0947 \u0909\u0926\u093e\u0939\u0930\u0923<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=423s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0915\u092e \u0938\u0947 \u0915\u092e \u092a\u0939\u0932\u0940 \u0928\u091c\u093c\u0930 \u092e\u0947\u0902 \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0932\u0917\u0924\u093e \u0939\u0948,<\/span><\/a> \u092a\u0930\u0902\u0924\u0941 \u092a\u0930\u093f\u091a\u093e\u0932\u0928\u093f\u0915 \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923 \u0938\u0947 \u092f\u0939 \u0909\u0924\u0928\u093e \u0938\u0930\u0932 \u0928\u0939\u0940\u0902 \u0939\u0948, \u0915\u093f \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0939\u092e\u0947\u0902 \u092f\u0939 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u091c\u092c \u0939\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0915\u093e \u0938\u093e\u092e\u0928\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e \u00ab\u092c\u093e\u0939\u0930 \u0938\u0947 \u092d\u0940\u0924\u0930\u00bb \u0905\u0935\u0915\u0932\u0928 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0947 \u0938\u0930\u0932\u0924\u093e \u0938\u0947 \u0938\u092e\u091d\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0909\u0926\u093e\u0939\u0930\u0923 \u0928\u093f\u0938\u094d\u0938\u0902\u0926\u0947\u0939 \u0938\u092c\u0938\u0947 \u0924\u0947\u091c\u093c \u092e\u093e\u0930\u094d\u0917 \u0939\u0948\u0902\u0964<\/p>\n<ol>\n<li>\u092f\u0926\u093f \u0939\u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = (2x^2+1)^{12}<\/span><\/span> \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u094b \u0915\u0939\u093e \u091c\u093e\u090f, \u0924\u094b \u092a\u0939\u0932\u0947 \u0939\u092e \u0918\u093e\u0924 \u0915\u094b \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930\u0924\u0947 \u0914\u0930 \u092b\u093f\u0930 \u0909\u0938 \u092c\u0921\u093c\u0947 \u092c\u0939\u0941\u092a\u0926 \u0915\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 t\u00e9rmino \u092a\u0930 \u0918\u093e\u0924 \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0932\u093e\u0917\u0942 \u0915\u0930\u0924\u0947, \u091c\u094b \u0905\u0928\u093e\u0935\u0936\u094d\u092f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0925\u0915\u093e\u090a trabajo \u0939\u0948\u0964 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u0938\u093e\u0925, \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0941\u091b \u092a\u0902\u0915\u094d\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948:<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>\u0915\u0947\u0935\u0932 \u0906\u0927\u093e\u0930\u092d\u0942\u0924 \u0905\u0935\u0915\u0932\u0928 \u0924\u0915\u0928\u0940\u0915\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\sin(\\cos(x))<\/span><\/span> \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0940 \u0915\u094b\u0936\u093f\u0936 \u0915\u0930\u0947\u0902 \u0914\u0930 \u0905\u0928\u0928\u094d\u0924 \u092a\u0940\u0921\u093c\u093e \u0915\u093e \u0938\u093e\u092e\u0928\u093e \u0915\u0930\u0947\u0902\u0964 \u0907\u0938\u0947 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0938\u0947 \u0915\u0930\u0947\u0902 \u0914\u0930 \u092a\u0930\u093f\u0923\u093e\u092e \u092c\u093f\u0928\u093e \u0906\u0901\u0938\u0941\u0913\u0902 \u0914\u0930 \u0915\u0941\u091b \u0939\u0940 \u091a\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u092a\u094d\u0930\u0915\u091f \u0939\u094b \u091c\u093e\u090f\u0917\u093e:<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>\u0906\u092a \u0915\u0908 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0935\u093e\u0932\u0940 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u092d\u0940 \u0928\u093f\u0915\u093e\u0932 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\cos(\\cos(\\cos(x))),<\/span><\/span> \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df\/dx<\/span><\/span> \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u094b\u0917\u093e:<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>\u091c\u0948\u0938\u093e \u0915\u093f \u0906\u092a \u0926\u0947\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u093e \u092c\u0938 \u092c\u093e\u0939\u0930 \u0938\u0947 \u092d\u0940\u0924\u0930 \u0915\u094d\u0930\u092e\u092c\u0926\u094d\u0927 \u0905\u0935\u0915\u0932\u0928 \u0915\u0930\u0928\u093e \u0939\u0948\u0964<\/li>\n<\/ol>\n<p><a name=\"13\"><\/a><\/p>\n<h3>\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902 \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u0947 \u092f\u094b\u0917\u094d\u092f \u0938\u093e\u0935\u0927\u093e\u0928\u0940<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=607s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0938\u093e\u0939\u093f\u0924\u094d\u092f \u092e\u0947\u0902 \u0938\u092d\u0940 \u0932\u094b\u0917 \u092c\u0921\u093c\u0947 beneficios \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902<\/span><\/a> \u091c\u094b \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u0947 \u0909\u092a\u092f\u094b\u0917 \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u092a\u0930\u0902\u0924\u0941 \u092c\u0939\u0941\u0924 \u0915\u092e \u0932\u094b\u0917 \u0907\u0938 \u092c\u093e\u0924 \u092a\u0930 \u091c\u093c\u094b\u0930 \u0926\u0947\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0907\u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u0915\u093f\u0928 \u0938\u093e\u0935\u0927\u093e\u0928\u093f\u092f\u094b\u0902 \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964 \u0907\u0938 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u0940 \u0936\u0915\u094d\u0924\u093f \u0915\u0947 \u092c\u093e\u0935\u091c\u0942\u0926, \u0924\u0941\u092e\u094d\u0939\u0947\u0902 \u0939\u092e\u0947\u0936\u093e \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u0930\u0947\u0902\u091c \u092a\u0930 \u0935\u093f\u0936\u0947\u0937 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964 \u0915\u093e\u0930\u094d\u092f \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u092f\u0939 \u0938\u0941\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0915\u0930\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948 \u0915\u093f \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u0930\u0947\u0902\u091c \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0932\u093f\u090f \u0938\u0902\u0917\u0924 \u0939\u094b\u0902; \u0905\u0928\u094d\u092f\u0925\u093e \u0924\u0941\u092e \u0909\u0928 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u092a\u0930 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u093e \u091c\u094b\u0916\u093f\u092e \u0909\u0920\u093e\u0924\u0947 \u0939\u094b \u091c\u0939\u093e\u0901 \u0935\u0947 \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u092e\u0947\u0902 \u0939\u0940 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902\u0964 \u092f\u0926\u093f, \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u0924\u0941\u092e \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u0905\u0935\u0915\u0932\u0928 \u0915\u0930\u0924\u0947 \u0939\u094b<\/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>\u0914\u0930 \u092f\u0926\u093f \u0924\u0941\u092e \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u092a\u0930 \u0906\u0901\u0916 \u092c\u0902\u0926 \u0915\u0930\u0915\u0947 \u092d\u0930\u094b\u0938\u093e \u0915\u0930\u0924\u0947 \u0939\u094b, \u0924\u094b \u0924\u0941\u092e \u0915\u0941\u091b \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e\u090f\u0901 \u0915\u0930\u094b\u0917\u0947:<\/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>\u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948 \u0915\u093f \u0938\u094d\u092a\u0930\u094d\u0936\u091c\u094d\u092f\u093e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=2\\pi\/3<\/span><\/span> \u092a\u0930 \u0905\u091a\u094d\u091b\u0940 \u0924\u0930\u0939 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0909\u0938\u0915\u093e \u092e\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(2\\pi\/3) = -\\sqrt{3}<\/span><\/span> \u0939\u0948\u0964 \u0932\u0947\u0915\u093f\u0928, \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln(\\cos(x))<\/span><\/span> \u0935\u0939\u093e\u0901 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0928\u0939\u0940\u0902 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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> \u0914\u0930 \u0928\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0932\u0949\u0917\u0930\u093f\u0925\u094d\u092e \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0939\u0948! \u0910\u0938\u0947 \u092e\u093e\u092e\u0932\u094b\u0902 \u092e\u0947\u0902, \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u092f\u0939 \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0915\u0947 \u0935\u0947 \u092e\u093e\u0928 \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u092c\u0924\u093e\u090f \u091c\u093e\u090f\u0901 \u091c\u093f\u0928\u0915\u0947 \u0932\u093f\u090f \u0915\u094b\u0938\u093e\u0907\u0928 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0930\u0939\u0924\u093e \u0939\u0948 (\u0924\u093e\u0915\u093f \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0924\u0939\u0924 \u0938\u0902\u0917\u0924\u0924\u093e \u0938\u0941\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0915\u0940 \u091c\u093e \u0938\u0915\u0947), \u0914\u0930 \u0924\u092d\u0940 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u092e\u093e\u0928\u094d\u092f \u0939\u094b\u0917\u093e\u0964<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0909\u092a\u092f\u094b\u0917\u0940 resultados<\/h2>\n<p>\u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0928 \u0915\u0947\u0935\u0932 \u0910\u0938\u0947 \u0905\u0935\u0915\u0932\u0928 c\u00e1lculos \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u0948 \u091c\u094b \u0905\u0928\u094d\u092f\u0925\u093e \u0905\u0938\u0939\u0928\u0940\u092f \u0939\u094b\u0924\u0947, \u092c\u0932\u094d\u0915\u093f \u092f\u0939 \u0905\u0935\u0915\u0932\u0928 \u0924\u0915\u0928\u0940\u0915\u094b\u0902 \u0915\u094b \u0914\u0930 \u092d\u0940 \u0905\u0927\u093f\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0924\u0915 \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0915 \u0939\u0948\u0964 \u0906\u0917\u0947 \u0939\u092e \u0907\u0928 \u0924\u0915\u0928\u0940\u0915\u094b\u0902, \u0909\u0928\u0915\u0947 resultados \u0914\u0930 \u0909\u0928\u0915\u0947 \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/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>\u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU&amp;t=75s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u090f\u0915 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948<\/span><\/a> \u0914\u0930 \u0915\u093f\u0938\u0940 \u0905\u0902\u0924\u0930\u093e\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq \\mathbb{R}<\/span><\/span> \u092e\u0947\u0902 \u0905\u0935\u0915\u0932\u0928\u0940\u092f \u0939\u0948\u0964 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f, \u092a\u0939\u091a\u093e\u0928 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(f^{-1}\\circ f)(x) = f^{-1}(f(x)) = x.<\/span><\/span> \u0915\u093e \u0905\u0935\u0915\u0932\u0928 \u0915\u0930\u0928\u093e \u0938\u0902\u092d\u0935 \u0939\u0948\u0964 \u0917\u0923\u0928\u093e\u0913\u0902 \u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 resultado \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938\u0938\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df^{-1}(f(x))\/df(x)<\/span><\/span> \u0915\u094b \u0905\u0932\u0917 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0914\u0930 resultado \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938\u0947 \u0939\u0940 \u0905\u0935\u0915\u0932\u091c\u094b\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0938\u093e\u0939\u093f\u0924\u094d\u092f \u092e\u0947\u0902 \u092f\u0939 \u092a\u094d\u0930\u092e\u0947\u092f \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u0907\u0938 \u0930\u0942\u092a \u092e\u0947\u0902 \u092d\u0940 \u0932\u093f\u0916\u093e \u092e\u093f\u0932\u0924\u093e \u0939\u0948<\/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>\u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u094b \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0926\u094b\u0928\u094b\u0902 \u0930\u0942\u092a \u0938\u092e\u0924\u0941\u0932\u094d\u092f \u0939\u0948\u0902, \u0914\u0930 \u092f\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> \u0924\u0925\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=f^{-1}(y).<\/span><\/span> \u0932\u093f\u0916\u0928\u0947 \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p>\u092f\u0939\u093e\u0901 \u0924\u0915 \u0939\u092e\u0928\u0947 \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0938\u092d\u0940 \u092c\u093e\u0924\u0947\u0902 \u0926\u0947\u0916 \u0932\u0940 \u0939\u0948\u0902; \u0905\u092c \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0909\u0928 \u0905\u0935\u0915\u0932\u091c\u094b\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948 \u091c\u094b \u0905\u0928\u094d\u092f\u0925\u093e \u0905\u0924\u094d\u092f\u0902\u0924 \u0915\u0920\u093f\u0928 \u0939\u094b\u0924\u0947\u0964<\/p>\n<p><a name=\"211\"><\/a><\/p>\n<h4>\u0918\u093e\u0924\u093e\u0902\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u0905\u0935\u0915\u0932\u091c<\/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\">\u091c\u092c \u0939\u092e\u0928\u0947 \u0906\u0927\u093e\u0930\u092d\u0942\u0924 \u0905\u0935\u0915\u0932\u0928 \u0924\u0915\u0928\u0940\u0915\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e<\/a><\/span> \u0924\u092c \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0915\u093f<\/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>\u0907\u0938 resultado \u0914\u0930 \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u0947 \u0938\u093e\u0925 \u092f\u0939 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0938\u0930\u0932 \u0939\u0948 \u0915\u093f<\/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>\u092a\u094d\u0930\u092e\u093e\u0923:<\/strong><\/p>\n<p>\u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\ln(x)<\/span><\/span> \u0915\u0939\u0928\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=e^y.<\/span><\/span> \u0915\u0939\u0928\u0947 \u0915\u0947 \u0938\u092e\u0924\u0941\u0932\u094d\u092f \u0939\u0948\u0964 \u092b\u093f\u0930, \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u094b \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u0947 \u092a\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0905\u0930\u094d\u0925\u093e\u0924\u094d:<\/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>\u092f\u0926\u093f \u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092e\u0947\u0902 \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0947\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u0935\u0939 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u093f\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e:<\/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>\u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u093e \u0905\u0935\u0915\u0932\u091c<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU\" target:=\"\" span=\"\" style=\"color: #ff0000;\" 0=\"\" a=\"\">\u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/a> \u0939\u092e\u0947\u0902 \u0938\u092d\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092d\u0940 \u0938\u0915\u094d\u0937\u092e \u092c\u0928\u093e\u0924\u093e \u0939\u0948\u0964 \u092f\u0947 \u0939\u0948\u0902:<\/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>\u092a\u094d\u0930\u092e\u093e\u0923<\/strong><\/p>\n<h5>\u0906\u0930\u094d\u0915\u0938\u093e\u0907\u0928<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee8ff7\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee8ff7\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u092d\u0940 \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b <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> \u0915\u0947 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 sine \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u0930\u0942\u092a \u0915\u093e \u0939\u094b<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0905\u092c, \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092a\u0939\u091a\u093e\u0928 \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u091c\u093f\u0938\u0938\u0947 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [-\\pi\/2, \\pi\/2]<\/span><\/span> \u0939\u094b, \u0924\u094b<\/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>\u0907\u0938\u0915\u094b \u0906\u0930\u094d\u0915\u0938\u093e\u0907\u0928 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092e\u0947\u0902 \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0935\u093e\u0902\u091b\u093f\u0924 resultado \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0906\u0930\u094d\u0915\u0915\u094b\u0938<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee94bf\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee94bf\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u092d\u0940 \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 coseno \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u0930\u0942\u092a \u0915\u093e \u0939\u094b<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924 \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/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>\u0905\u092c \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f identidad \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u091c\u093f\u0938\u0938\u0947 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [0, \\pi]<\/span><\/span> \u0939\u094b, \u0924\u094b:<\/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>\u0907\u0938\u0947 \u0906\u0930\u094d\u0915\u0915\u094b\u0938 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0935\u093e\u0902\u091b\u093f\u0924 resultado \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/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>\u0906\u0930\u094d\u0915\u091f\u0948\u0928\u094d\u091c\u0947\u0923\u094d\u091f<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee9764\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee9764\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b \u0907\u0938 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: <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> \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0,<\/span><\/span> \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 funci\u00f3n tangent \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u094b:<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924:<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b:<\/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>\u0905\u092c \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f identidad \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u0914\u0930 \u0907\u0938\u0938\u0947:<\/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>\u0907\u0938\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0935\u093e\u0902\u091b\u093f\u0924 resultado \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/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>\u0906\u0930\u094d\u0915\u0915\u094b\u091f\u0948\u0928\u094d\u091c\u0947\u0923\u094d\u091f<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee9993\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee9993\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">cot(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u092d\u0940 \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b \u0907\u0938 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 cotangente \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u094b:<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924:<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b:<\/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>\u0905\u092c \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f identidad \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u0914\u0930 \u0907\u0938\u0938\u0947:<\/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>\u0907\u0938\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0907\u091a\u094d\u091b\u093f\u0924 resultado \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/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>\u0906\u0930\u094d\u0915\u0938\u0947\u0915\u0947\u0928\u094d\u091f<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee9b22\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee9b22\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sec(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b \u0907\u0938 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: <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> \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 secante \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u094b:<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924 \u092f\u0939 \u0938\u0902\u092c\u0902\u0927 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b:<\/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>\u0905\u092c \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f identidad \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u0914\u0930 \u0907\u0938\u0938\u0947:<\/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>\u0907\u0938\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0907\u091a\u094d\u091b\u093f\u0924 resultado \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/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>\u0906\u0930\u094d\u0915\u0915\u094b\u0938\u0947\u0915\u0947\u0928\u094d\u091f<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f41ee9d8f\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f41ee9d8f\" class=\"collapseomatic_content \">\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\csc(x)<\/span><\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u092d\u0940 \u0907\u0938\u0915\u0947 \u0921\u094b\u092e\u0947\u0928 \u0915\u094b \u0907\u0938 \u0930\u0942\u092a \u0935\u093e\u0932\u0947 \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0924\u0915 \u0938\u0940\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: <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>, \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u093e \u0915\u0940 \u0939\u093e\u0928\u093f \u0915\u0947 \u092c\u093f\u0928\u093e \u0939\u092e \u092e\u0941\u0916\u094d\u092f caso \u092a\u0930 \u0938\u0940\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0924\u093e\u0915\u093f \u0926\u094d\u0935\u093f\u0906\u0935\u0930\u094d\u0924\u0940 cosecante \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u094b:<\/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>\u0914\u0930 \u0907\u0928 condiciones \u0915\u0947 \u0924\u0939\u0924:<\/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>\u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902, \u0924\u094b:<\/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>\u0905\u092c \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f identidad \u0915\u094b \u092f\u093e\u0926 \u0915\u0930\u0947\u0902:<\/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>\u0914\u0930 \u0907\u0938\u0938\u0947:<\/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>\u0907\u0938\u0947 \u0905\u0935\u0915\u0932\u091c \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/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>\u0914\u0930 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>\u0905\u0902\u0924\u0924\u0903, \u00aby\u00bb \u0915\u094b \u00abx\u00bb \u0938\u0947 \u092c\u0926\u0932\u0915\u0930 \u0939\u092e\u0947\u0902 \u0907\u091a\u094d\u091b\u093f\u0924 resultado \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928<\/h3>\n<p>\u0905\u092c \u0924\u0915 \u091c\u093f\u0928 \u0938\u092d\u0940 \u0905\u0935\u0915\u0932\u0928\u094b\u0902 \u0915\u0940 \u0939\u092e\u0928\u0947 \u0917\u0923\u0928\u093e \u0915\u0940 \u0939\u0948, \u0935\u0947 \u0909\u0928 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0925\u0947 \u091c\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0925\u0947: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span>\u0964 \u0939\u093e\u0932\u093e\u0902\u0915\u093f, \u0915\u0908 \u0938\u094d\u0925\u093f\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u0926\u0940 \u0917\u0908 relaci\u00f3n \u0938\u0947 \u0928 \u0924\u094b \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0940 \u0938\u094d\u092a\u0937\u094d\u091f \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948, \u0914\u0930 \u0928 \u0939\u0940 \u0915\u092d\u0940-\u0915\u092d\u0940 \u092f\u0939 \u0938\u0902\u092d\u0935 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u0910\u0938\u0947 \u092e\u093e\u092e\u0932\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928 \u0915\u0940 \u0924\u0915\u0928\u0940\u0915 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0947 fundamentos \u090f\u0915 \u092c\u093e\u0930 \u092b\u093f\u0930 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u092e\u0947\u0902 \u0928\u093f\u0939\u093f\u0924 \u0939\u0948\u0902\u0964<\/p>\n<p>\u0907\u0938 t\u00e9cnica \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0909\u0926\u093e\u0939\u0930\u0923 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u0907\u0938\u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0915\u0947 \u092c\u0940\u091a \u0928\u093f\u092e\u094d\u0928 relaci\u00f3n \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902:<\/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>\u092f\u0926\u093f \u0939\u092e \u0907\u0938 relaci\u00f3n \u0915\u093e gr\u00e1fico \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092a\u0924\u093e \u091a\u0932\u0947\u0917\u093e \u0915\u093f \u092f\u0939 \u0915\u093f\u0938\u0940 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u093e gr\u00e1fico \u0928\u0939\u0940\u0902 \u0939\u0948\u0964 \u092f\u0939 \u090f\u0915 \u0935\u0915\u094d\u0930 \u0939\u0948 \u091c\u093f\u0938\u0947 \u201c\u0939\u094b\u0939\u093e \u0926\u0947 Descartes\u201d \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/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>\u0905\u092c, \u092f\u0926\u093f \u0939\u092e \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0915\u0947 respecto \u0905\u0935\u0915\u0932\u091c \u0928\u093f\u0915\u093e\u0932\u0928\u093e \u091a\u093e\u0939\u0947\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0935\u0939 expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0916\u094b\u091c\u0928\u0947 \u092e\u0947\u0902 \u0917\u0902\u092d\u0940\u0930 \u0915\u0920\u093f\u0928\u093e\u0907\u092f\u093e\u0901 \u0939\u094b\u0902\u0917\u0940 \u091c\u094b ecuaci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0940 \u0939\u0948, \u0914\u0930 \u092b\u093f\u0930 \u0909\u0938\u0915\u093e \u0905\u0935\u0915\u0932\u0928 \u0915\u0930\u0928\u093e \u092a\u0921\u093c\u0947\u0917\u093e\u0964 \u0915\u093f\u0902\u0924\u0941 \u0939\u092e \u0909\u0938 paso \u0915\u094b \u091b\u094b\u0921\u093c \u0926\u0947\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0905\u092a\u094d\u0930\u0915\u091f \u0930\u0942\u092a \u0938\u0947 \u092e\u093e\u0928 \u0932\u0947\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0915\u093e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948, \u0905\u0930\u094d\u0925\u093e\u0924\u094d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=y(x)<\/span><\/span>\u0964 \u0910\u0938\u093e \u0915\u0930\u0928\u0947 \u092a\u0930, hoja de Descartes \u0915\u0940 relaci\u00f3n \u0907\u0938 \u0930\u0942\u092a \u092e\u0947\u0902 \u092c\u0926\u0932 \u091c\u093e\u0924\u0940 \u0939\u0948:<\/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>\u0914\u0930 \u092a\u0930\u093f\u0923\u093e\u092e\u0938\u094d\u0935\u0930\u0942\u092a, \u0939\u092e \u0938\u092e\u094d\u092a\u0942\u0930\u094d\u0923 expresi\u00f3n \u0915\u094b \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0905\u0935\u0915\u0932\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u092f\u0926\u093f \u0910\u0938\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u0928\u093f\u092e\u094d\u0928 resultado \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/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>\u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930, \u092f\u0926\u093f \u0939\u092e \u0935\u0915\u094d\u0930 \u0915\u0947 \u0915\u093f\u0938\u0940 \u092c\u093f\u0902\u0926\u0941 \u0915\u094b \u091c\u093e\u0928\u0924\u0947 \u0939\u094b\u0902, \u0924\u094b \u0909\u0938 \u092c\u093f\u0902\u0926\u0941 \u0938\u0947 \u0917\u0941\u091c\u0930\u0928\u0947 \u0935\u093e\u0932\u0940 \u0938\u094d\u092a\u0930\u094d\u0936\u0930\u0947\u0916\u093e \u0915\u0940 pendiente \u091c\u094d\u091e\u093e\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, gr\u00e1fico \u0938\u0947 \u0939\u092e \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u092c\u093f\u0902\u0926\u0941 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2,4)<\/span><\/span> \u0935\u0915\u094d\u0930 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u0948; \u0914\u0930 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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> \u092f\u0939 \u091c\u093e\u0928\u0924\u0947 \u0939\u0941\u090f \u0939\u092e \u0924\u0941\u0930\u0902\u0924 \u0915\u0939 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0909\u0938 \u092c\u093f\u0902\u0926\u0941 \u092a\u0930 \u0938\u094d\u092a\u0930\u094d\u0936\u0930\u0947\u0916\u093e \u0915\u0940 pendiente \u0939\u094b\u0917\u0940:<\/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>\u092a\u0930\u093f\u092e\u0947\u092f \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c<\/h4>\n<p>\u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0939\u092e \u0905\u0935\u0915\u0932\u0928 \u0915\u0940 \u090f\u0915 \u0906\u0927\u093e\u0930\u092d\u0942\u0924 t\u00e9cnica \u0915\u0947 \u0926\u093e\u092f\u0930\u0947 \u0915\u093e \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u092f\u0939 t\u00e9cnica \u0909\u0928 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u0948 \u091c\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^n<\/span><\/span> \u0939\u094b\u0902, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}<\/span><\/span>\u0964 \u0905\u092c \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0938\u0947 \u0906\u0917\u0947 \u092c\u0922\u093c\u0915\u0930 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0924\u0915 \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u092c\u093f\u0928\u093e \u0915\u0920\u093f\u0928\u093e\u0908 \u0915\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f<\/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>\u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p,q\\in\\mathbb{Z}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span><\/span>\u0964<\/p>\n<p>\u0907\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092e\u093e\u0928 \u0932\u0947\u0902: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=x^{p\/q}<\/span><\/span> \u0914\u0930 \u0926\u094b\u0928\u094b\u0902 \u092a\u0915\u094d\u0937\u094b\u0902 \u092a\u0930 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0932\u0918\u0941\u0917\u0923\u0915 \u0932\u0917\u093e\u090f\u0901, \u091c\u093f\u0938\u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e:<\/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>\u0905\u092c, \u0907\u0938 expresi\u00f3n \u0915\u093e \u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928 \u0915\u0930\u0928\u0947 \u092a\u0930:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\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}}\n\n<\/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>\u0935\u094d\u092f\u093e\u092f\u093e\u092e \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u093f\u0915\u093e:<\/h2>\n<h4>\u090f\u0915 \u091a\u0930 \u092e\u0947\u0902 \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e<\/h4>\n<ol>\n<li>\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0947 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902:<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>\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902:<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>\u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0915\u0940 \u0905\u0935\u0915\u0932\u0928 \u0915\u0947 \u0932\u093f\u090f \u0936\u094d\u0930\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0905\u092c \u0924\u0915 \u091c\u094b \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0939\u0948, \u0909\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0932\u0917\u092d\u0917 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092d\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 elementos \u0909\u092a\u0932\u092c\u094d\u0927 \u0939\u0948\u0902\u0964 \u092b\u093f\u0930 \u092d\u0940, \u0939\u092e\u0947\u0902 \u090f\u0915 \u0905\u0935\u0915\u0932\u091c \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0940 posibilidad \u0914\u0930 \u0910\u0938\u0940 \u0917\u0923\u0928\u093e\u090f\u0901 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0932\u0917\u093e\u090f \u0917\u090f \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 [&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":0,"footnotes":""},"categories":[862,577],"tags":[],"class_list":["post-35187","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-862","category-577"],"yoast_head":"<!-- This site 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\u0909\u0926\u093e\u0939\u0930\u0923 y \u0935\u094d\u092f\u093e\u092f\u093e\u092e\u0964 \u0907\u0938\u092e\u0947\u0902 \u0905\u0935\u094d\u092f\u0915\u094d\u0924 \u0905\u0935\u0915\u0932\u0928 \u0914\u0930 \u092a\u094d\u0930\u0924\u093f\u0932\u094b\u092e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/hi\/\u092b\u093c\u0902\u0915\u094d\u0936\u0928\u094b\u0902-\u0915\u0940-\u0938\u0902\u0930\u091a\u0928\u093e-\u0915\u0947-\u0905\u0935\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-12-01T13:00:40+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-11-22T22:34:12+00:00\" \/>\n<meta property=\"og:image\" 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