{"id":33367,"date":"2024-11-30T13:00:53","date_gmt":"2024-11-30T13:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33367"},"modified":"2025-07-19T10:54:01","modified_gmt":"2025-07-19T10:54:01","slug":"%e0%a4%ac%e0%a4%b9%e0%a5%81%e0%a4%aa%e0%a4%a6%e0%a5%8b%e0%a4%82-%e0%a4%a4%e0%a5%8d%e0%a4%b0%e0%a4%bf%e0%a4%95%e0%a5%8b%e0%a4%a3%e0%a4%ae%e0%a4%bf%e0%a4%a4%e0%a5%80%e0%a4%af-%e0%a4%ab%e0%a4%b2","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%ac%e0%a4%b9%e0%a5%81%e0%a4%aa%e0%a4%a6%e0%a5%8b%e0%a4%82-%e0%a4%a4%e0%a5%8d%e0%a4%b0%e0%a4%bf%e0%a4%95%e0%a5%8b%e0%a4%a3%e0%a4%ae%e0%a4%bf%e0%a4%a4%e0%a5%80%e0%a4%af-%e0%a4%ab%e0%a4%b2\/","title":{"rendered":"\u092c\u0939\u0941\u092a\u0926\u094b\u0902, \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<style><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><br \/>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u092c\u0939\u0941\u092a\u0926, \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n\u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0905\u0935\u0915\u0932\u0928 \u0917\u0923\u093f\u0924 \u0915\u0940 \u090f\u0915 \u0915\u0947\u0902\u0926\u094d\u0930\u0940\u092f \u0935\u093f\u0927\u093f \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0935\u0948\u091c\u094d\u091e\u093e\u0928\u093f\u0915, \u0905\u092d\u093f\u092f\u093e\u0928\u094d\u0924\u094d\u0930\u093f\u0915 \u0914\u0930 \u0906\u0930\u094d\u0925\u093f\u0915 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u092e\u0947\u0902 \u092e\u0942\u0932\u092d\u0942\u0924 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0939\u0948\u0964 \u092f\u0939 \u0932\u0947\u0916 \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u092e\u0947\u0902 \u092a\u094d\u0930\u0935\u0940\u0923\u0924\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0939\u0947\u0924\u0941 \u090f\u0915 \u092a\u094d\u0930\u0917\u0924\u093f\u0936\u0940\u0932 \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u093f\u0915\u093e \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0924\u093e \u0939\u0948, \u091c\u094b \u092c\u0939\u0941\u092a\u0926\u094b\u0902 \u0938\u0947 \u0932\u0947\u0915\u0930 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0924\u0915 \u092b\u0948\u0932\u0940 \u0939\u0948\u0964 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u0928\u094b\u0902 \u0914\u0930 \u0920\u094b\u0938 \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947, \u0907\u0938\u0915\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0947 \u0938\u093e\u0925-\u0938\u093e\u0925 \u0909\u0928\u0915\u0947 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u094b \u092d\u0940 \u0938\u092e\u091d\u0928\u093e \u0939\u0948\u0964<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u0905\u0927\u093f\u0917\u092e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f<\/strong><\/p>\n<ol>\n<li><strong>\u0938\u092e\u091d\u0928\u093e<\/strong> \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0914\u0930 \u0909\u0938\u0915\u0940 \u092e\u094c\u0932\u093f\u0915 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u0915\u094b\u0964<\/li>\n<li><strong>\u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0930\u0928\u093e<\/strong> \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094b \u0906\u0927\u093e\u0930\u092d\u0942\u0924 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f\u0964<\/li>\n<li><strong>\u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e<\/strong> \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0938\u094d\u0925\u093f\u0930 \u092b\u0932\u0928\u094b\u0902 \u0914\u0930 \u092a\u0939\u091a\u093e\u0928 \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928\u0964<\/li>\n<li><strong>\u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u093e<\/strong> \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0947 \u0932\u093f\u090f \u0928\u093f\u092f\u092e, \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0938\u093e\u0907\u0928 \u0914\u0930 \u0915\u094b\u0938\u093e\u0907\u0928 \u0915\u0940 \u092e\u0942\u0932 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0938\u0947 \u0928\u093f\u0915\u093e\u0932\u093e \u0917\u092f\u093e \u0939\u0948\u0964<\/li>\n<li><strong>\u0917\u0923\u0928\u093e \u0915\u0930\u0928\u093e<\/strong> \u092f\u094c\u0917\u093f\u0915 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928, \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f\u0964<\/li>\n<li><strong>\u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e<\/strong> \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0932\u0918\u0941\u0917\u0923\u0915 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u094b \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u0930\u0942\u092a \u0938\u0947\u0964<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>\u0935\u093f\u0937\u092f \u0938\u0942\u091a\u0940<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/a><br \/>\n<a href=\"#2\">\u0905\u0932\u094c\u0915\u093f\u0915 (transcendentales) \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" title=\"YouTube video player\" data-src=\"https:\/\/www.youtube.com\/embed\/LudVXLRnmLw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>\u0905\u092c \u0924\u0915 \u0939\u092e\u0928\u0947 \u0915\u0947\u0935\u0932 \u092f\u0939 \u0926\u0947\u0916\u093e \u0939\u0948 \u0915\u093f \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u094d\u092f\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u0940 \u0915\u0941\u091b \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u090f\u0901 \u0915\u094d\u092f\u093e \u0939\u0948\u0902, \u092a\u0930\u0902\u0924\u0941 \u0905\u092d\u0940 \u0924\u0915 \u092f\u0939 \u0928\u0939\u0940\u0902 \u092c\u0924\u093e\u092f\u093e \u0917\u092f\u093e \u0915\u093f \u0907\u0928\u094d\u0939\u0947\u0902 \u0915\u0948\u0938\u0947 \u0928\u093f\u0915\u093e\u0932\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0916\u0902\u0921 \u092e\u0947\u0902 \u0939\u092e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0924\u0915\u0928\u0940\u0915 \u0915\u094b \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u0938\u092e\u091d\u0947\u0902\u0917\u0947 \u0914\u0930 \u092f\u0939 \u092d\u0940 \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902 \u0909\u0928\u094d\u0939\u0947\u0902 \u0915\u0948\u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/h2>\n<h3>\u0938\u094d\u0925\u093f\u0930 \u092b\u0932\u0928 (Constante Function)<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=104s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> \u091c\u0939\u093e\u0901 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>\u0915\u094b\u0908 \u092d\u0940 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u094d\u0925\u093f\u0930\u093e\u0902\u0915 \u0939\u0948, \u0924\u094b:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{d}{dx}c = 0<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/span> \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u092f\u0939 \u0938\u093f\u0926\u094d\u0927 \u0915\u0947\u0935\u0932 \u090f\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}c &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{c - c}{\\Delta x} \\quad \\text{; $f(x)=c$ \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e} \\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{0}{\\Delta x} = 0\n\n\\end{array}\n\n<\/span>\n<h3>\u092a\u0939\u091a\u093e\u0928 \u092b\u0932\u0928 (Identity Function)<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> \u0924\u094b:<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx}{dx}=1<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/span> \u092f\u0939 \u092d\u0940 \u0932\u0917\u092d\u0917 \u092a\u093f\u091b\u0932\u0947 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948, \u090f\u0915 \u0939\u0940 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0939\u0932 \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x &amp;= \\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{(x+\\Delta x) - x}{\\Delta x} \\quad \\text{; $f(x) = x$ \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e}\\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\Delta x}{\\Delta x} = 1\n\n\\end{array}\n\n<\/span>\n<h3>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0918\u093e\u0924\u0947\u0902 (Natural Powers)<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> \u091c\u0939\u093e\u0901<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u0948, \u0924\u094b \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx^n}{dx} =nx^{n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/span> \u0907\u0938 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0939\u092e\u0947\u0902 <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">\u0928\u094d\u092f\u0942\u091f\u0928 \u0915\u0947 \u0926\u094d\u0935\u093f\u092a\u0926 \u092a\u094d\u0930\u092e\u0947\u092f<\/a> \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u093e \u0939\u094b\u0917\u093e\u0964<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}x^n = \\lim_{\\Delta x \\to 0} \\frac{(x+\\Delta x)^n -x^n}{\\Delta x} &amp;\\text{ ; $f(x)= x^n$ \u0915\u0947 \u0932\u093f\u090f \u0938\u0940\u092e\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e} \\\\ \\\\\n\n&amp; \\displaystyle \\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=0}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; \u0928\u094d\u092f\u0942\u091f\u0928 \u0915\u0947 \u0926\u094d\u0935\u093f\u092a\u0926 \u092a\u094d\u0930\u092e\u0947\u092f \u0926\u094d\u0935\u093e\u0930\u093e, \u0938\u092e\u0940\u0915\u0930\u0923 (1)} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle x^n + \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; \u092f\u094b\u0917 \u0915\u093e \u092a\u0939\u0932\u093e \u092a\u0926 \u0905\u0932\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right]}{\\Delta x} &amp; \\text{; \u0938\u092e\u093e\u0928 \u092a\u0926\u094b\u0902 \u0915\u094b \u0918\u091f\u093e\u0924\u0947 \u0939\u0941\u090f} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[ {{n}\\choose{1}} x^{n-1}(\\Delta x)^{0} + \\sum_{k=2}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\text{; \u092f\u094b\u0917 \u0915\u093e \u092a\u0939\u0932\u093e \u092a\u0926 \u0905\u0932\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f} \\\\ \\\\\n\n&amp; \\displaystyle \\color{blue} {\\displaystyle \\dfrac{d}{dx}x^n} = n x^{n-1} &amp; \\color{black}\n\n\\end{array}\n\n<\/span>\n<h3>\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0918\u093e\u0924\u0947\u0902 (Integer Powers)<\/h3>\n<p>\u0905\u092d\u0940 \u091c\u094b \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0935\u0939 \u0915\u0947\u0935\u0932 \u0909\u0928 \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0935\u0948\u0927 \u0939\u0948 \u091c\u094b \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u0948\u0902, \u0932\u0947\u0915\u093f\u0928 \u0907\u0938\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0924\u0915 \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span>, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^a}{dx} = ax^{a-1}<\/span>\n<p>\u0939\u092e\u0947\u0902 \u092a\u0939\u0932\u0947 \u0938\u0947 \u091c\u094d\u091e\u093e\u0924 \u0939\u0948 \u0915\u093f \u092f\u0939 \u0938\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0938\u0939\u0940 \u0939\u0948, \u0905\u092c \u0939\u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u092f\u0939 \u0926\u0947\u0916\u0928\u093e \u0939\u0948 \u0915\u093f \u0928\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948 \u0915\u093f \u0939\u092e \u092f\u0939 \u0926\u093f\u0916\u093e\u090f\u0901:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^{-n}}{dx} = {-n}x^{-n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/strong><\/span> \u0907\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u0947\u0935\u0932 \u0907\u0938 \u092c\u093e\u0924 \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0928\u093e \u0939\u0948 \u0915\u093f \u092f\u0939 \u090f\u0915 \u092d\u093e\u091c\u0915 \u0915\u093e \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x^{-n} &amp;= \\dfrac{d}{dx} \\left( \\dfrac{1}{x^n}\\right) \\\\ \\\\\n\n&amp; &amp;= \\dfrac{0 \\cdot nx^{n-1} - nx^{n-1} \\cdot 1}{x^{2n}}\\\\ \\\\\n\n&amp; &amp;= -nx^{n-1-2n} \\\\ \\\\\n\n&amp; &amp;= -nx^{-n-1}\n\n\\end{array}\n\n<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0905\u0932\u094c\u0915\u093f\u0915 \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 (Derivatives of Transcendental Functions)<\/h2>\n<h3>\u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928 (Trigonometric Functions)<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=667s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0907\u0928\u092e\u0947\u0902 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924<\/span><\/strong><\/a> \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0928\u093f\u092f\u092e \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0939\u0948\u0902:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sin(x) = \\cos(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sec(x) = \\sec(x)\\tan(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cos(x) = -\\sin(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\csc(x) = -\\csc(x)\\cot(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\tan(x) = \\sec^2(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cot(x) = -\\csc^2(x)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0907\u0928 \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u094b \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u093e \u0924\u0930\u0940\u0915\u093e \u092f\u0939 \u0939\u0948 \u0915\u093f \u0939\u092e \u092a\u0939\u0932\u0947 \u0938\u093e\u0907\u0928 \u0914\u0930 \u0915\u094b\u0938\u093e\u0907\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0928\u093f\u0915\u093e\u0932\u0947\u0902; \u0914\u0930 \u092b\u093f\u0930 \u0907\u0928\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947, \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0928\u093f\u092f\u092e\u094b\u0902 \u0938\u0947 \u0905\u0928\u094d\u092f \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/p>\n<h4>\u0938\u093e\u0907\u0928 \u092b\u0932\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\dfrac{d}{dx}\\sin(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x+\\Delta x) - \\sin(x)}{\\Delta x} &amp; \\text{; \u0938\u093e\u0907\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e} \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x)\\cos(\\Delta x) + \\sin(\\Delta x)\\cos(x) - \\sin(x)}{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\sin(x)\\left[\\cos(\\Delta x) -1\\right] + \\sin(\\Delta x)\\cos(x) }{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\sin(x)\\lim_{\\Delta x \\to 0} \\left[\\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\right] + \\cos(x) \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\right] &amp; \\\\ \\\\\n\n(2)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} = 1 &amp; \\text{; \u0938\u0948\u0902\u0921\u0935\u093f\u091a \u092a\u094d\u0930\u092e\u0947\u092f \u0926\u094d\u0935\u093e\u0930\u093e} \\\\ \\\\\n\n(3)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\cdot \\dfrac{\\cos(\\Delta x) + 1}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos^2(\\Delta x) - 1}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{-\\sin^2(\\Delta x)}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\cdot \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- (1)\\cdot(0) = 0 \\\\ \\\\\n\n(4) &amp;\\color{blue}\\dfrac{d}{dx}\\sin(x) = \\cos(x) \\color{black} &amp; \\text{; (1,2,3) \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924}\n\n\\end{array}<\/span>\n<h4>\u0915\u094b\u0938\u093e\u0907\u0928 \u092b\u0932\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}\\cos(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x + \\Delta x) - \\cos(x)}{\\Delta x} &amp; \\text{; \u0915\u094b\u0938\u093e\u0907\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x)\\cos(\\Delta x) - \\sin(x)\\sin(\\Delta x) - \\cos(x)}{\\Delta x} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x) [ \\cos(\\Delta x) - 1] - \\sin(x)\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ [ \\cos(\\Delta x) - 1]}{\\Delta x} - \\sin(x) \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\cdot(0) - \\sin(x)\\cdot (1)\\\\ \\\\\n\n&amp;\\color{blue}\\dfrac{d}{dx}\\cos(x) = - \\sin(x) \\color{black}\n\n\\end{array}<\/span>\n<h4>\u091f\u0948\u0928\u094d\u091c\u0947\u0902\u091f, \u0938\u0947\u0915\u0947\u0902\u091f, \u0915\u094b\u0938\u0947\u0915\u0947\u0902\u091f \u0914\u0930 \u0915\u0949\u091f\u0948\u0928\u094d\u091c\u0947\u0902\u091f \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/h4>\n<p>\u0905\u092c \u091c\u092c \u0938\u093e\u0907\u0928 \u0914\u0930 \u0915\u094b\u0938\u093e\u0907\u0928 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u091c\u094d\u091e\u093e\u0924 \u0939\u0948\u0902, \u0924\u094b \u0905\u0928\u094d\u092f \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u093e \u0938\u0930\u0932 \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\tan(x) &amp;= \\dfrac{d}{dx} \\left( \\dfrac{\\sin(x)}{\\cos(x)} \\right) = \\dfrac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)} = \\dfrac{1}{\\cos^2(x)} = \\color{blue}\\sec^2(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\sec(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)} \\right) = \\dfrac{\\sin(x)}{\\cos^2(x)} =\\color{blue}\\sec(x)\\tan(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\csc(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)}\\right) = -\\dfrac{cos(x)}{\\sin^2(x)} =\\color{blue} - \\csc(x)\\cot(x)\\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx} \\cot(x) &amp;= \\dfrac{d}{dx} \\left(\\dfrac{\\cos(x)}{\\sin(x)}\\right) = \\dfrac{-\\sin^2(x)-\\cos^2(x)}{\\sin^2(x)} = -\\dfrac{1}{\\sin^2(x)} =\\color{blue} -\\csc^2(x)\\color{black}\n\n\\end{array}<\/span>\n<p>\n\u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0939\u0940 \u0939\u092e\u0928\u0947 \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 \u0909\u0928 \u092e\u094c\u0932\u093f\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e \u0939\u0948 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091b\u093e\u0924\u094d\u0930 \u0915\u094b \u0905\u0935\u0936\u094d\u092f \u0938\u092e\u091d\u0928\u093e \u091a\u093e\u0939\u093f\u090f: \u092e\u0942\u0932 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0938\u0947 \u0932\u0947\u0915\u0930 \u092a\u094d\u0930\u092e\u0941\u0916 \u0905\u0932\u094c\u0915\u093f\u0915 \u092b\u0932\u0928\u094b\u0902 \u0924\u0915, \u091c\u0948\u0938\u0947 \u0915\u093f \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928 \u0914\u0930 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0932\u0918\u0941\u0917\u0923\u0915\u0964 \u0907\u0928 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924\u094b\u0902 \u092e\u0947\u0902 \u092a\u094d\u0930\u0935\u0940\u0923\u0924\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930 \u0932\u0947\u0928\u0947 \u0938\u0947 \u0906\u092a \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0947 \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u094b \u0928 \u0915\u0947\u0935\u0932 \u0932\u093e\u0917\u0942 \u0915\u0930 \u092a\u093e\u090f\u0901\u0917\u0947, \u092c\u0932\u094d\u0915\u093f \u0909\u0928\u0915\u0947 \u0909\u0926\u094d\u0917\u092e \u0914\u0930 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u0914\u091a\u093f\u0924\u094d\u092f \u0915\u094b \u092d\u0940 \u0938\u092e\u091d \u0938\u0915\u0947\u0902\u0917\u0947\u0964 \u092f\u0939 \u091c\u094d\u091e\u093e\u0928 \u0909\u0928 \u091c\u091f\u093f\u0932 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0928\u093f\u092a\u091f\u0928\u0947 \u0915\u0940 \u0906\u0927\u093e\u0930\u0936\u093f\u0932\u093e \u0939\u0948 \u091c\u093f\u0928\u092e\u0947\u0902 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0915\u093e \u0938\u091f\u0940\u0915 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u092c\u0939\u0941\u092a\u0926, \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0905\u0935\u0915\u0932\u0928 \u0917\u0923\u093f\u0924 \u0915\u0940 \u090f\u0915 \u0915\u0947\u0902\u0926\u094d\u0930\u0940\u092f \u0935\u093f\u0927\u093f \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0935\u0948\u091c\u094d\u091e\u093e\u0928\u093f\u0915, \u0905\u092d\u093f\u092f\u093e\u0928\u094d\u0924\u094d\u0930\u093f\u0915 \u0914\u0930 \u0906\u0930\u094d\u0925\u093f\u0915 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u092e\u0947\u0902 \u092e\u0942\u0932\u092d\u0942\u0924 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0939\u0948\u0964 \u092f\u0939 \u0932\u0947\u0916 \u092b\u0932\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u092e\u0947\u0902 \u092a\u094d\u0930\u0935\u0940\u0923\u0924\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0939\u0947\u0924\u0941 \u090f\u0915 \u092a\u094d\u0930\u0917\u0924\u093f\u0936\u0940\u0932 \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u093f\u0915\u093e \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0924\u093e \u0939\u0948, \u091c\u094b \u092c\u0939\u0941\u092a\u0926\u094b\u0902 \u0938\u0947 \u0932\u0947\u0915\u0930 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0924\u0915 \u092b\u0948\u0932\u0940 \u0939\u0948\u0964 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u0928\u094b\u0902 \u0914\u0930 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33155,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[862,577],"tags":[],"class_list":["post-33367","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-862","category-577"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u092c\u0939\u0941\u092a\u0926\u094b\u0902, \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u092b\u0932\u0928\u094b\u0902 \u0914\u0930 \u0932\u0918\u0941\u0917\u0923\u0915 \u0915\u0940 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902, \u0938\u094d\u092a\u0937\u094d\u091f 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