{"id":35179,"date":"2024-12-01T13:00:25","date_gmt":"2024-12-01T13:00:25","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35179"},"modified":"2025-11-22T22:33:46","modified_gmt":"2025-11-22T22:33:46","slug":"%e5%87%bd%e6%95%b0%e5%a4%8d%e5%90%88%e6%b1%82%e5%af%bc%e7%9a%84%e9%93%be%e5%bc%8f%e6%b3%95%e5%88%99","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e5%87%bd%e6%95%b0%e5%a4%8d%e5%90%88%e6%b1%82%e5%af%bc%e7%9a%84%e9%93%be%e5%bc%8f%e6%b3%95%e5%88%99\/","title":{"rendered":"\u51fd\u6570\u590d\u5408\u6c42\u5bfc\u7684\u94fe\u5f0f\u6cd5\u5219"},"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>\u51fd\u6570\u590d\u5408\u6c42\u5bfc\u7684\u94fe\u5f0f\u6cd5\u5219<\/h1>\n<p><em>\u6839\u636e\u524d\u9762\u6240\u5b66\u7684\u5185\u5bb9\uff0c\u6211\u4eec\u5df2\u7ecf\u5177\u5907\u8ba1\u7b97\u51e0\u4e4e\u4efb\u610f\u5bfc\u6570\u7684\u57fa\u672c\u5de5\u5177\u3002\u7136\u800c\uff0c\u80fd\u591f\u8ba1\u7b97\u5bfc\u6570\u4e0e\u4e3a\u6b64\u6295\u5165\u7684\u8ba1\u7b97\u5de5\u4f5c\u91cf\u662f\u4e24\u56de\u4e8b\uff0c\u800c\u8fd9\u6b63\u662f\u8bf8\u5982\u5355\u53d8\u91cf\u94fe\u5f0f\u6cd5\u5219\u7b49\u5b9a\u7406\u53d1\u6325\u4f5c\u7528\u7684\u5730\u65b9\u3002\u94fe\u5f0f\u6cd5\u5219\u80fd\u5e2e\u52a9\u6211\u4eec\u5feb\u901f\u6c42\u51fa\u4e00\u4e9b\u539f\u672c\u9700\u8981\u8017\u8d39\u5927\u91cf\u7e41\u7410\u8ba1\u7b97\u624d\u80fd\u5f97\u5230\u7684\u5bfc\u6570\u3002<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b><u>\u5185\u5bb9\u7d22\u5f15<\/u><\/b><br \/>\n<b><a href=\"#1\">\u5355\u53d8\u91cf\u94fe\u5f0f\u6cd5\u5219\u5b9a\u7406<\/a><\/b><br \/>\n<a href=\"#11\">\u94fe\u5f0f\u6cd5\u5219\u7684\u8bc1\u660e<\/a><br \/>\n<a href=\"#12\">\u5355\u53d8\u91cf\u94fe\u5f0f\u6cd5\u5219\u7684\u5e94\u7528\u793a\u4f8b<\/a><br \/>\n<a href=\"#13\">\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u65f6\u9700\u6ce8\u610f\u7684\u4e8b\u9879<\/a><br \/>\n<b><a href=\"#2\">\u7531\u94fe\u5f0f\u6cd5\u5219\u63a8\u5bfc\u51fa\u7684\u6709\u7528\u7ed3\u679c<\/a><\/b><br \/>\n<a href=\"#21\">\u53cd\u51fd\u6570\u5b9a\u7406<\/a><br \/>\n<a href=\"#211\">\u6307\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/a><br \/>\n<a href=\"#212\">\u53cd\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570<\/a><br \/>\n<a href=\"#22\">\u9690\u51fd\u6570\u6c42\u5bfc<\/a><br \/>\n<a href=\"#221\">\u6709\u7406\u5e42\u51fd\u6570\u7684\u5bfc\u6570<\/a><br \/>\n<a href=\"#221\">\u6709\u7406\u5e42\u51fd\u6570\u7684\u5bfc\u6570<\/a><br \/>\n<b><a href=\"#3\">\u7ec3\u4e60\u6307\u5357<\/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>\u5355\u53d8\u91cf\u94fe\u5f0f\u6cd5\u5219\u5b9a\u7406<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=165s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u4e3a\u4e24\u4e2a<\/span><\/a> \u53ef\u590d\u5408\u7684\u51fd\u6570<\/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>\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u4e0a\u53ef\u5bfc\uff0c\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> \u4e0a\u53ef\u5bfc\uff0c\u5219\u590d\u5408\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g\\circ f<\/span><\/span> \u5728\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in A<\/span><\/span> \u4e0a\u53ef\u5bfc\uff0c\u5e76\u6ee1\u8db3\u516c\u5f0f<\/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>\u94fe\u5f0f\u6cd5\u5219\u7684\u8bc1\u660e<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=242s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u8003\u8651\u51fd\u6570<\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>\uff0c\u5176\u5b9a\u4e49\u5982\u524d\u3002\u82e5\u8ba1\u7b97\u590d\u5408\u51fd\u6570\u7684\u5bfc\u6570\uff0c\u5219\u6709<\/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>\u8fd9\u6b63\u662f\u6240\u8981\u8bc1\u660e\u7684\u3002<\/p>\n<p><a name=\"12\"><\/a><\/p>\n<h3>\u5355\u53d8\u91cf\u51fd\u6570\u4e2d\u94fe\u5f0f\u6cd5\u5219\u7684\u5e94\u7528\u793a\u4f8b<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=423s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u81f3\u5c11\u4ece\u8868\u9762\u4e0a\u770b\u4f3c\u4e4e\u663e\u800c\u6613\u89c1\u7684\u4e8b\u60c5\uff0c<\/span><\/a> \u4f46\u4ece\u8fd0\u7b97\u89d2\u5ea6\u6765\u770b\u5374\u5e76\u975e\u5982\u6b64\uff0c\u5373\u94fe\u5f0f\u6cd5\u5219\u544a\u8bc9\u6211\u4eec\uff1a\u5f53\u9762\u5bf9\u51fd\u6570\u7684\u590d\u5408\u65f6\uff0c\u53ef\u4ee5\u201c\u7531\u5916\u5411\u5185\u201d\u6c42\u5bfc\u3002\u4e3a\u4e86\u4ee5\u66f4\u76f4\u89c2\u7684\u65b9\u5f0f\u8bf4\u660e\u8fd9\u4e00\u70b9\uff0c\u793a\u4f8b\u65e0\u7591\u662f\u6700\u5feb\u7684\u9014\u5f84\u3002<\/p>\n<ol>\n<li>\u82e5\u8981\u6c42\u5bf9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = (2x^2+1)^{12}<\/span><\/span> \u6c42\u5bfc\uff0c\u6309\u7167\u57fa\u672c\u65b9\u6cd5\u6211\u4eec\u5fc5\u987b\u5148\u5c55\u5f00\u5e42\u6b21\uff0c\u518d\u5bf9\u5f97\u5230\u7684\u5de8\u5927\u591a\u9879\u5f0f\u9010\u9879\u6c42\u5bfc\uff0c\u8fd9\u662f\u5b8c\u5168\u6ca1\u6709\u5fc5\u8981\u4e14\u6781\u4e3a\u7e41\u7410\u7684\u5de5\u4f5c\u3002\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\uff0c\u8ba1\u7b97\u53ef\u4ee5\u5728\u51e0\u884c\u5185\u5b8c\u6210\uff1a<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>\u8bd5\u7740\u4ec5\u7528\u6700\u57fa\u672c\u7684\u6c42\u5bfc\u6280\u5de7\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\sin(\\cos(x))<\/span><\/span> \u7684\u5bfc\u6570\uff0c\u4f60\u5c06\u5760\u5165\u6c38\u6052\u7684\u75db\u82e6\u4e4b\u4e2d\u3002\u4f46\u82e5\u8fd0\u7528\u94fe\u5f0f\u6cd5\u5219\uff0c\u7ed3\u679c\u5c06\u8f7b\u677e\u4e14\u65e0\u6cea\u5730\u5728\u6570\u6b65\u5185\u51fa\u73b0\uff1a<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>\u4f60\u4e5f\u53ef\u4ee5\u6c42\u5bfc\u90a3\u4e9b\u7531\u591a\u91cd\u51fd\u6570\u590d\u5408\u800c\u6210\u7684\u51fd\u6570\u3002\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\cos(\\cos(\\cos(x))),<\/span><\/span> \u5219\u5176\u5bfc\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df\/dx<\/span><\/span> \u4e3a\uff1a<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>\u6b63\u5982\u4f60\u6240\u89c1\uff0c\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u5c31\u662f\u7531\u5916\u5411\u5185\u9010\u5c42\u6c42\u5bfc\u3002<\/li>\n<\/ol>\n<p><a name=\"13\"><\/a><\/p>\n<h3>\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u65f6\u9700\u6ce8\u610f\u7684\u4e8b\u9879<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=0y2SQpbRe3A&amp;t=607s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u5728\u6587\u732e\u4e2d\uff0c\u5927\u5bb6\u666e\u904d\u5f3a\u8c03<\/span><\/a> \u4f7f\u7528\u94fe\u5f0f\u6cd5\u5219\u7684\u5de8\u5927\u4f18\u52bf\uff0c\u4f46\u5f88\u5c11\u6709\u4eba\u660e\u786e\u5f3a\u8c03\u5728\u4f7f\u7528\u4e4b\u524d\u5fc5\u987b\u6ce8\u610f\u7684\u4e8b\u9879\u3002\u5c3d\u7ba1\u8be5\u5b9a\u7406\u975e\u5e38\u5f3a\u5927\uff0c\u4f60\u5728\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u524d\u4ecd\u9700\u7279\u522b\u6ce8\u610f\u5404\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u548c\u503c\u57df\u3002\u5982\u679c\u4e0d\u5148\u786e\u8ba4\u8fd9\u4e9b\u96c6\u5408\u5728\u590d\u5408\u65f6\u662f\u517c\u5bb9\u7684\uff0c\u5c31\u6709\u53ef\u80fd\u8bd5\u56fe\u8ba1\u7b97\u5728\u67d0\u4e9b\u70b9\u4e0a\u5e76\u4e0d\u5b58\u5728\u7684\u5bfc\u6570\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u51fd\u6570<\/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>\u5982\u679c\u76f2\u76ee\u4fe1\u8d56\u94fe\u5f0f\u6cd5\u5219\uff0c\u4f60\u53ef\u80fd\u4f1a\u8fdb\u884c\u5982\u4e0b\u8ba1\u7b97\uff1a<\/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>\u663e\u7136\uff0c\u6b63\u5207\u51fd\u6570\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=2\\pi\/3<\/span><\/span> \u5904\u662f\u6709\u5b9a\u4e49\u7684\uff0c\u5176\u503c\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(2\\pi\/3) = -\\sqrt{3}<\/span><\/span>\u3002\u7136\u800c\uff0c\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\ln(\\cos(x))<\/span><\/span> \u5728\u8be5\u70b9\u5e76\u65e0\u5b9a\u4e49\uff0c\u56e0\u4e3a <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> \u800c\u8d1f\u6570\u662f\u6ca1\u6709\u5bf9\u6570\u7684\uff01\u5728\u6b64\u7c7b\u95ee\u9898\u4e2d\uff0c\u5fc5\u987b\u5728\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u4e4b\u524d\u6307\u51fa\uff1a\u6240\u8003\u8651\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u503c\u5fc5\u987b\u4fdd\u8bc1\u4f59\u5f26\u51fd\u6570\u4e3a\u6b63\uff08\u4ee5\u786e\u4fdd\u590d\u5408\u7684\u53ef\u884c\u6027\uff09\uff0c\u53ea\u6709\u5728\u6b64\u6761\u4ef6\u6ee1\u8db3\u65f6\u94fe\u5f0f\u6cd5\u5219\u624d\u771f\u6b63\u9002\u7528\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u7531\u94fe\u5f0f\u6cd5\u5219\u63a8\u5bfc\u51fa\u7684\u6709\u7528\u7ed3\u679c<\/h2>\n<p>\u94fe\u5f0f\u6cd5\u5219\u4e0d\u4ec5\u80fd\u5e2e\u52a9\u6211\u4eec\u5b8c\u6210\u539f\u672c\u6781\u4e3a\u7e41\u7410\u7684\u6c42\u5bfc\u4efb\u52a1\uff0c\u8fd8\u80fd\u7528\u4e8e\u6269\u5c55\u6c42\u5bfc\u6280\u5de7\uff0c\u4f7f\u5176\u9002\u7528\u4e8e\u66f4\u591a\u7c7b\u578b\u7684\u51fd\u6570\u3002\u4e0b\u9762\u6211\u4eec\u5c06\u56de\u987e\u8fd9\u4e9b\u6280\u5de7\u3001\u7ed3\u8bba\u53ca\u5176\u8bc1\u660e\u3002<\/p>\n<p><a name=\"21\"><\/a><br \/>\n<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><\/p>\n<p><a name=\"21\"><\/a><\/p>\n<h3>\u53cd\u51fd\u6570\u5b9a\u7406<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU&amp;t=75s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u662f\u4e00\u4e2a\u53cc\u5c04\u51fd\u6570\uff0c<\/span><\/a> \u4e14\u5728\u67d0\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq \\mathbb{R}<\/span><\/span> \u4e0a\u53ef\u5bfc\u3002\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\u53ef\u4ee5\u8ba1\u7b97\u6052\u7b49\u51fd\u6570\u7684\u5bfc\u6570\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(f^{-1}\\circ f)(x) = f^{-1}(f(x)) = x.<\/span><\/span> \u5176\u8ba1\u7b97\u5f97\u5230\uff1a<\/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>\u7531\u6b64\u53ef\u4ee5\u89e3\u51fa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">df^{-1}(f(x))\/df(x)<\/span><\/span>\uff0c\u5f97\u5230\uff1a<\/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>\u8fd9\u5c31\u662f\u7528\u4e8e\u6c42\u5bfc\u7684\u53cd\u51fd\u6570\u5b9a\u7406\u3002\u5728\u6587\u732e\u4e2d\uff0c\u7ecf\u5e38\u53ef\u4ee5\u770b\u5230\u5b83\u5199\u6210\uff1a<\/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>\u4e24\u79cd\u8868\u8fbe\u65b9\u5f0f\u662f\u7b49\u4ef7\u7684\uff0c\u53ea\u9700\u5199\u51fa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=f^{-1}(y).<\/span><\/span><\/p>\n<p>\u5230\u6b64\u4e3a\u6b62\uff0c\u6211\u4eec\u5df2\u89e3\u91ca\u5b8c\u53cd\u51fd\u6570\u5b9a\u7406\u7684\u5185\u5bb9\u3002\u4e0b\u9762\u5c06\u5c55\u793a\u5982\u4f55\u5229\u7528\u8be5\u5b9a\u7406\u6765\u8ba1\u7b97\u4e00\u4e9b\u5426\u5219\u5341\u5206\u56f0\u96be\u7684\u5bfc\u6570\u3002<\/p>\n<p><a name=\"211\"><\/a><\/p>\n<h4>\u6307\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/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\">\u5728\u4e4b\u524d\u5b66\u4e60\u57fa\u672c\u6c42\u5bfc\u6280\u5de7\u65f6\uff0c<\/a><\/span> \u6211\u4eec\u5df2\u7ecf\u770b\u5230\uff1a<\/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>\u5229\u7528\u8fd9\u4e00\u7ed3\u679c\u53ca\u53cd\u51fd\u6570\u5b9a\u7406\uff0c\u53ef\u4ee5\u8f7b\u677e\u8bc1\u660e\uff1a<\/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>\u8bc1\u660e\uff1a<\/strong><\/p>\n<p>\u663e\u7136 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\ln(x)<\/span><\/span> \u7b49\u4ef7\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x=e^y.<\/span><\/span> \u4e8e\u662f\uff0c\u6839\u636e\u53cd\u51fd\u6570\u5b9a\u7406\uff1a<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}e^y = \\frac{dx}{dy} = \\frac{1}{\\frac{dy}{dx}} = \\frac{1}{\\frac{d}{dx}\\ln(x)} = x = e^y<\/span>\n<p>Es decir:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dy}e^y = e^y<\/span>\n<p>Si en esta ultima expresi\u00f3n remplazamos las \u00aby\u00bb por \u00abx\u00bb, obtenemos lo que se quer\u00eda demostrar:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}e^x = e^x.<\/span>\n<p><a name=\"212\"><\/a><\/p>\n<h4>\u53cd\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=5ddoUcIhgjU\" target:=\"\" span=\"\" style=\"color: #ff0000;\" 0=\"\" a=\"\">\u53cd\u51fd\u6570\u5b9a\u7406<\/a> \u4e5f\u53ef\u4ee5\u7528\u6765\u6c42\u51fa\u6240\u6709\u53cd\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570\u3002\u5b83\u4eec\u4e3a\uff1a<\/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>\u8bc1\u660e<\/strong><\/p>\n<h5>\u53cd\u6b63\u5f26\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dc974\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dc974\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <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> \u7684\u533a\u95f4\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u6b63\u5f26\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u7531\u6b64\u53ef\u63a8\u77e5\uff0c\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [-\\pi\/2, \\pi\/2]<\/span><\/span> \u65f6\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u6b63\u5f26\u5bfc\u6570\u4e2d\u5f97\u5230\uff1a<\/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>\u800c\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sin(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u6b64\u5f0f\u4e2d\u7684 \u201cy\u201d \u6362\u4e3a \u201cx\u201d \u5373\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u53cd\u4f59\u5f26\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dcaed\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dcaed\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u7684\u533a\u95f4\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u4f59\u5f26\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u7531\u6b64\u53ef\u63a8\u77e5\uff0c\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in [0, \\pi]<\/span><\/span> \u65f6\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u4f59\u5f26\u5bfc\u6570\u53ef\u5f97\uff1a<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\cos(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u201cy\u201d\u6362\u4e3a\u201cx\u201d\u5373\u53ef\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u53cd\u6b63\u5207\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dcc13\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dcc13\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tan(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <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> \u7684\u533a\u95f4\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u6b63\u5207\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u4ece\u800c\u6709\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u6b63\u5207\u5bfc\u6570\u53ef\u5f97\uff1a<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\tan(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u6b64\u5f0f\u4e2d\u7684\u201cy\u201d\u6362\u4e3a\u201cx\u201d\u5373\u53ef\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u53cd\u4f59\u5207\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dcdb9\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dcdb9\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">cot(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left[0+k\\pi , \\pi+ k\\pi \\right],<\/span><\/span> \u7684\u533a\u95f4\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u4f59\u5207\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u7531\u6b64\u53ef\u63a8\u77e5\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u4f59\u5207\u5bfc\u6570\u53ef\u5f97\uff1a<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ctg(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u201cy\u201d\u6362\u4e3a\u201cx\u201d\u5373\u53ef\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u53cd\u6b63\u5272\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dceaa\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dceaa\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sec(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <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> \u7684\u96c6\u5408\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u6b63\u5272\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u7531\u6b64\u53ef\u63a8\u77e5\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u6b63\u5272\u5bfc\u6570\u53ef\u5f97\uff1a<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sec(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u6b64\u5f0f\u4e2d\u7684\u201cy\u201d\u6362\u4e3a\u201cx\u201d\u5373\u53ef\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u53cd\u4f59\u5272\u51fd\u6570<\/h5>\n<span class=\"collapseomatic \" id=\"id69e3f916dd036\"  tabindex=\"0\" title=\"Mostrar Demostraci\u00f3n\"    >Mostrar Demostraci\u00f3n<\/span><div id=\"target-id69e3f916dd036\" class=\"collapseomatic_content \">\n<p>\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\csc(x)<\/span><\/span> \u5728\u5c06\u5176\u5b9a\u4e49\u57df\u9650\u5236\u4e3a\u5f62\u5982 <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> \u7684\u533a\u95f4\uff08\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e3a\u4efb\u610f\u6574\u6570\uff09\u65f6\u4e3a\u53cc\u5c04\u3002\u65e0\u9700\u5931\u4e00\u822c\u6027\uff0c\u6211\u4eec\u53ef\u53ea\u8003\u8651\u4e3b\u503c\u533a\u95f4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u6b64\u65f6\u53cc\u5c04\u7684\u4f59\u5272\u51fd\u6570\u4e3a\uff1a<\/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>\u5728\u6b64\u6761\u4ef6\u4e0b\uff0c\u6709\uff1a<\/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>\u5e94\u7528\u53cd\u51fd\u6570\u5b9a\u7406\u53ef\u5f97\uff1a<\/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>\u56de\u5fc6\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/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>\u7531\u6b64\u53ef\u63a8\u77e5\uff1a<\/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>\u5c06\u5176\u4ee3\u5165\u53cd\u4f59\u5272\u5bfc\u6570\u53ef\u5f97\uff1a<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=\\csc(x)<\/span><\/span>\uff0c\u5219\uff1a<\/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>\u6700\u540e\uff0c\u5c06\u6b64\u5f0f\u4e2d\u7684\u201cy\u201d\u6362\u4e3a\u201cx\u201d\u5373\u53ef\u5f97\u5230\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/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>\u9690\u51fd\u6570\u6c42\u5bfc<\/h3>\n<p>\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u6240\u8ba1\u7b97\u7684\u5bfc\u6570\u90fd\u5bf9\u5e94\u4e8e\u4ee5\u663e\u5f0f\u65b9\u5f0f\u5b9a\u4e49\u7684\u51fd\u6570\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span>\u3002\u7136\u800c\uff0c\u5728\u67d0\u4e9b\u60c5\u5f62\u4e0b\uff0c\u7531\u4e8e\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u60f3\u8981\u5f97\u5230\u51fd\u6570\u7684\u663e\u5f0f\u8868\u8fbe\u5f0f\u8981\u4e48\u975e\u5e38\u56f0\u96be\uff0c\u8981\u4e48\u6839\u672c\u65e0\u6cd5\u505a\u5230\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9690\u5f0f\u6c42\u5bfc\u6280\u672f\u4fbf\u6d3e\u4e0a\u7528\u573a\uff0c\u800c\u5176\u57fa\u7840\u540c\u6837\u6e90\u4e8e\u94fe\u5f0f\u6cd5\u5219\u3002<\/p>\n<p>\u4e3a\u4e86\u7406\u89e3\u8fd9\u79cd\u6280\u5de7\uff0c\u793a\u4f8b\u5f80\u5f80\u6bd4\u8bc1\u660e\u66f4\u6709\u6548\uff0c\u56e0\u6b64\u8003\u8651\u7531\u65b9\u7a0b\u7ed9\u51fa\u7684\u53d8\u91cf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/span> \u4e4b\u95f4\u7684\u5173\u7cfb\uff1a<\/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>\u82e5\u5c06\u5176\u4f5c\u56fe\uff0c\u53ef\u4ee5\u53d1\u73b0\u5b83\u5e76\u4e0d\u662f\u67d0\u4e2a\u51fd\u6570\u7684\u56fe\u50cf\uff0c\u800c\u662f\u4e00\u6761\u79f0\u4e3a\u201c\u7b1b\u5361\u5c14\u53f6\u7ebf\u201d\u7684\u66f2\u7ebf\u3002<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-l30tAMcTkk0\/YLCIuWcDueI\/AAAAAAAAFIY\/K7uSR44DepgIjBlSVV7mCQO-Z0iy_RnRQCLcBGAsYHQ\/s0\/hojaDeDescartes.PNG\" alt=\"hoja de descartes\" class=\"alignnone size-full lazyload\" width=\"690\" height=\"515\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-l30tAMcTkk0\/YLCIuWcDueI\/AAAAAAAAFIY\/K7uSR44DepgIjBlSVV7mCQO-Z0iy_RnRQCLcBGAsYHQ\/s0\/hojaDeDescartes.PNG\" alt=\"hoja de descartes\" class=\"alignnone size-full lazyload\" width=\"690\" height=\"515\" \/><\/noscript><\/center><\/p>\n<p>Ahora, si quisi\u00e9ramos calcular, por ejemplo: la derivada de <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> con respecto a <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, entonces tendr\u00edamos serias dificultades con encontrar de forma explicita expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> que satisface la ecuaci\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> para luego derivar. Lo que hacemos, sin embargo, es saltarnos ese paso y asumimos impl\u00edcitamente que <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> es funci\u00f3n de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, es decir: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=y(x)<\/span><\/span>. Haciendo esto, la relaci\u00f3n de la hoja de Descartes se transforma en:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x^3 +y^3(x)- 9xy(x)=0<\/span>\n<p>Y podemos, en consecuencia, derivar todo utilizando la regla de la cadena. Si lo hacemos, llegaremos al siguiente resultado:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\n\\displaystyle 3x^{2} + 3\\,y(x)^{2}\\,\\frac{dy}{dx} - \\left(9\\,y(x) + 9x\\,\\frac{dy}{dx}\\right) &amp;=&amp; 0 \\\\ \\\\\n\n\\displaystyle 3x^{2} + 3\\,y(x)^{2}\\,\\frac{dy}{dx} - 9\\,y(x) - 9x\\,\\frac{dy}{dx} &amp;=&amp; 0 \\\\ \\\\\n\n\\displaystyle \\frac{dy}{dx}\\,\\big(3\\,y(x)^{2} - 9x\\big) &amp;=&amp; 9\\,y(x) - 3x^{2} \\\\ \\\\\n\n\\displaystyle \\frac{dy}{dx} &amp;=&amp; \\dfrac{9\\,y(x) - 3x^{2}}{3\\,y(x)^{2} - 9x} \\\\ \\\\\n\n\\displaystyle \\color{blue}{\\frac{dy}{dx}} &amp;\\color{blue}{=}&amp; \\color{blue}{\\dfrac{3\\,y(x) - x^{2}}{y(x)^{2} - 3x}}\n\n\\end{array}\n\n<\/span>\n<p>A partir de esto podemos calcular, si conocemos un punto de la curva, la pendiente de la recta tangente que pasa por ese punto. Por ejemplo, a partir del gr\u00e1fico podemos intuir que el punto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2,4)<\/span><\/span> est\u00e1 sobre la curva; y de hecho, esto se corrobora porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^3 + 4^3 - 9\\cdot 2\\cdot 4 = 8+64 - 72 = 0. <\/span><\/span> Sabiendo esto podemos decir r\u00e1pidamente que la pendiente de la recta tangente que pasa por ese punto ser\u00e1:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\left.\\frac{dy}{dx}\\right|_{(2,4)}= \\frac{3\\cdot 4 - 2^2}{4^2 - 3\\cdot 2}= \\frac{8}{10}= \\frac{4}{5}}<\/span>\n<p><a name=\"221\"><\/a><\/p>\n<h4>\u6709\u7406\u6570\u5e42\u7684\u5bfc\u6570<\/h4>\n<p>\u901a\u8fc7\u9690\u5f0f\u6c42\u5bfc\uff0c\u6211\u4eec\u53ef\u4ee5\u6269\u5c55\u57fa\u672c\u6c42\u5bfc\u6280\u5de7\u4e4b\u4e00\u2014\u2014\u5373\u5bf9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^n<\/span><\/span>\uff08\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}<\/span><\/span>\uff09\u6c42\u5bfc\u7684\u65b9\u6cd5\u3002\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u5c06\u6574\u6570\u6307\u6570\u63a8\u5e7f\u5230\u6709\u7406\u6307\u6570\uff0c\u5e76\u5bb9\u6613\u8bc1\u660e\uff1a<\/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>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p,q\\in\\mathbb{Z}<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span><\/span>\u3002<\/p>\n<p>\u4e3a\u8bc1\u660e\u6b64\u5f0f\uff0c\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=x^{p\/q}<\/span><\/span>\uff0c\u5e76\u53d6\u81ea\u7136\u5bf9\u6570\uff0c\u5f97\u5230\uff1a<\/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>\u5bf9\u8be5\u5f0f\u8fdb\u884c\u9690\u5f0f\u6c42\u5bfc\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{y}\\frac{dy}{dx} = \\frac{p}{q}\\frac{1}{x}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\frac{dy}{dx} = \\frac{p}{q}\\frac{1}{x}y(x)= \\frac{p}{q}\\frac{1}{x}x^{p\/q} = \\frac{p}{q}x^{(p\/q) - 1}}<\/span>\n<p><a name=\"3\"><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KwJ5Bb5Ch_o\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/a><\/p>\n<h2>\u7ec3\u4e60\u6307\u5357\uff1a<\/h2>\n<h4>\u5355\u53d8\u91cf\u94fe\u5f0f\u6cd5\u5219<\/h4>\n<ol>\n<li>\u8ba1\u7b97\u4e0b\u5217\u51fd\u6570\u7684\u5bfc\u6570\uff1a<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>\u8ba1\u7b97\u4e0b\u5217\u51fd\u6570\u7684\u5bfc\u6570\uff1a<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>\u51fd\u6570\u590d\u5408\u6c42\u5bfc\u7684\u94fe\u5f0f\u6cd5\u5219 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\u6c42\u5bfc\uff0c\u6309\u7167\u57fa\u672c\u65b9\u6cd5\u6211\u4eec\u5fc5\u987b\u5148\u5c55\u5f00\u5e42\u6b21\uff0c\u518d\u5bf9\u5f97\u5230\u7684\u5de8\u5927\u591a\u9879\u5f0f\u9010\u9879\u6c42\u5bfc\uff0c\u8fd9\u662f\u5b8c\u5168\u6ca1\u6709\u5fc5\u8981\u4e14\u6781\u4e3a\u7e41\u7410\u7684\u5de5\u4f5c\u3002\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\uff0c\u8ba1\u7b97\u53ef\u4ee5\u5728\u51e0\u884c\u5185\u5b8c\u6210\uff1a \u8bd5\u7740\u4ec5\u7528\u6700\u57fa\u672c\u7684\u6c42\u5bfc\u6280\u5de7\u8ba1\u7b97 \u7684\u5bfc\u6570\uff0c\u4f60\u5c06\u5760\u5165\u6c38\u6052\u7684\u75db\u82e6\u4e4b\u4e2d\u3002\u4f46\u82e5\u8fd0\u7528\u94fe\u5f0f\u6cd5\u5219\uff0c\u7ed3\u679c\u5c06\u8f7b\u677e\u4e14\u65e0\u6cea\u5730\u5728\u6570\u6b65\u5185\u51fa\u73b0\uff1a \u4f60\u4e5f\u53ef\u4ee5\u6c42\u5bfc\u90a3\u4e9b\u7531\u591a\u91cd\u51fd\u6570\u590d\u5408\u800c\u6210\u7684\u51fd\u6570\u3002\u82e5 \u5219\u5176\u5bfc\u6570 \u4e3a\uff1a 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