{"id":33363,"date":"2024-11-30T13:00:57","date_gmt":"2024-11-30T13:00:57","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33363"},"modified":"2025-07-19T10:44:08","modified_gmt":"2025-07-19T10:44:08","slug":"%e5%a4%9a%e9%a1%b9%e5%bc%8f%e5%87%bd%e6%95%b0%e3%80%81%e4%b8%89%e8%a7%92%e5%87%bd%e6%95%b0%e4%b8%8e%e5%af%b9%e6%95%b0%e5%87%bd%e6%95%b0%e7%9a%84%e5%af%bc%e6%95%b0","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e5%a4%9a%e9%a1%b9%e5%bc%8f%e5%87%bd%e6%95%b0%e3%80%81%e4%b8%89%e8%a7%92%e5%87%bd%e6%95%b0%e4%b8%8e%e5%af%b9%e6%95%b0%e5%87%bd%e6%95%b0%e7%9a%84%e5%af%bc%e6%95%b0\/","title":{"rendered":"\u591a\u9879\u5f0f\u51fd\u6570\u3001\u4e09\u89d2\u51fd\u6570\u4e0e\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570"},"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>\u591a\u9879\u5f0f\u51fd\u6570\u3001\u4e09\u89d2\u51fd\u6570\u4e0e\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n\u5bfc\u6570\u662f\u5fae\u79ef\u5206\u4e2d\u7684\u6838\u5fc3\u5de5\u5177\uff0c\u5728\u79d1\u5b66\u3001\u5de5\u7a0b\u548c\u7ecf\u6d4e\u5b66\u4e2d\u5177\u6709\u57fa\u7840\u6027\u7684\u5e94\u7528\u3002\u672c\u6587\u63d0\u4f9b\u4e86\u4e00\u4efd\u5faa\u5e8f\u6e10\u8fdb\u7684\u6307\u5357\uff0c\u65e8\u5728\u638c\u63e1\u51fd\u6570\u6c42\u5bfc\u7684\u6280\u5de7\uff0c\u4ece\u591a\u9879\u5f0f\u51fd\u6570\u4e00\u76f4\u5ef6\u4f38\u81f3\u4e09\u89d2\u51fd\u6570\u4e0e\u5bf9\u6570\u51fd\u6570\u3002\u901a\u8fc7\u63a8\u5bfc\u4e0e\u5177\u4f53\u793a\u4f8b\uff0c\u529b\u6c42\u6df1\u5165\u7406\u89e3\u5bfc\u6570\u6cd5\u5219\u7684\u5e94\u7528\u53ca\u5176\u7406\u8bba\u4f9d\u636e\u3002<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807<\/strong><\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u5bfc\u6570\u7684\u4e00\u822c\u6982\u5ff5\u53ca\u5176\u57fa\u672c\u6027\u8d28\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u5bfc\u6570\u7684\u6b63\u5f0f\u5b9a\u4e49\u4ee5\u8ba1\u7b97\u57fa\u672c\u5bfc\u6570\u3002<\/li>\n<li><strong>\u901a\u8fc7\u6781\u9650\u8bc1\u660e<\/strong>\u5e38\u6570\u51fd\u6570\u548c\u6052\u7b49\u51fd\u6570\u7684\u5bfc\u6570\u3002<\/li>\n<li><strong>\u63a8\u5bfc<\/strong>\u4ece\u6b63\u5f26\u548c\u4f59\u5f26\u51fd\u6570\u7684\u57fa\u672c\u5bfc\u6570\u51fa\u53d1\uff0c\u5f97\u5230\u4e09\u89d2\u51fd\u6570\u7684\u6c42\u5bfc\u6cd5\u5219\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong>\u590d\u5408\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570\uff0c\u4f7f\u7528\u4ee3\u6570\u89c4\u5219\u3002<\/li>\n<li><strong>\u5f62\u5f0f\u5316\u8bc1\u660e<\/strong>\u81ea\u7136\u5bf9\u6570\u7684\u5bfc\u6570\uff0c\u57fa\u4e8e\u6781\u9650\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>\u5185\u5bb9\u7d22\u5f15<\/u>\uff1a<\/strong><br \/>\n<a href=\"#1\">\u4ee3\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/a><br \/>\n<a href=\"#2\">\u8d85\u8d8a\u51fd\u6570\u7684\u5bfc\u6570<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" data-src=\"https:\/\/www.youtube.com\/embed\/LudVXLRnmLw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u4ec5\u4ec5\u8ba8\u8bba\u4e86\u5bfc\u6570\u7684\u5b9a\u4e49\u53ca\u5176\u4e00\u4e9b\u4ee3\u6570\u6027\u8d28\uff0c\u4f46\u5c1a\u672a\u8bb2\u89e3\u5982\u4f55\u5b9e\u9645\u8ba1\u7b97\u5bfc\u6570\u3002\u672c\u6587\u5c06\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\uff0c\u9010\u4e00\u5c55\u793a\u5404\u79cd\u6c42\u5bfc\u6280\u5de7\u53ca\u5176\u5728\u5404\u7c7b\u51fd\u6570\u4e2d\u7684\u5177\u4f53\u5e94\u7528\u3002<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u4ee3\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/h2>\n<h3>\u5e38\u6570\u51fd\u6570<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=104s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> \u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>\u4e3a\u4efb\u610f\u5b9e\u6570\u5e38\u6570\uff0c\u5219\u6709\uff1a<\/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>\u8bc1\u660e\uff1a<\/strong><\/span> \u5b9e\u9645\u4e0a\uff0c\u8be5\u8bc1\u660e\u4ec5\u9700\u4e00\u6b65\u5b8c\u6210\uff1a<\/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{\uff1b$f(x)=c$ \u7684\u5bfc\u6570\u5b9a\u4e49} \\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{0}{\\Delta x} = 0\n\n\\end{array}\n\n<\/span>\n<h3>\u6052\u7b49\u51fd\u6570<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> \u5219\u6709\uff1a<\/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>\u8bc1\u660e\uff1a<\/strong><\/span> \u4e0e\u4e0a\u4e00\u4e2a\u51e0\u4e4e\u76f8\u540c\uff0c\u4e5f\u53ea\u9700\u4e00\u6b65\u5373\u53ef\u5b8c\u6210\uff1a<\/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{\uff1b$f(x) = x$ \u7684\u5bfc\u6570\u5b9a\u4e49}\\\\ \\\\\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>\u81ea\u7136\u6570\u5e42\u51fd\u6570<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> \u5176\u4e2d<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u4e3a\u4efb\u610f\u81ea\u7136\u6570\uff0c\u5219\u6709\uff1a<\/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>\u8bc1\u660e\uff1a<\/strong><\/span> \u4e3a\u4e86\u8bc1\u660e\u8be5\u5b9a\u7406\uff0c\u6211\u4eec\u9700\u8981\u4f7f\u7528 <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">\u725b\u987f\u4e8c\u9879\u5f0f\u5b9a\u7406<\/a>\uff1a<\/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{\uff1b\u5bf9 $f(x)= x^n$ \u4f7f\u7528\u5bfc\u6570\u5b9a\u4e49} \\\\ \\\\\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{\uff1b\u6839\u636e\u725b\u987f\u4e8c\u9879\u5f0f\u5b9a\u7406\u5c55\u5f00 $(1)$} \\\\ \\\\\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{\uff1b\u5c06\u7b2c\u4e00\u4e2a\u9879\u4ece\u548c\u5f0f\u4e2d\u5206\u79bb} \\\\ \\\\\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{\uff1b\u6d88\u53bb\u76f8\u540c\u9879} \\\\ \\\\\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{\uff1b\u63d0\u53d6\u548c\u5f0f\u7684\u7b2c\u4e00\u4e2a\u9879} \\\\ \\\\\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>\u6574\u6570\u6b21\u5e42\u51fd\u6570<\/h3>\n<p>\u524d\u9762\u7684\u8bc1\u660e\u4ec5\u9002\u7528\u4e8e\u5e42\u4e3a\u81ea\u7136\u6570\u7684\u60c5\u5f62\uff0c\u4f46\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u63a8\u5e7f\u81f3\u4efb\u610f\u6574\u6570\u3002\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span> \u65f6\uff0c\u6709\uff1a<\/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>\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u8fd9\u4e2a\u7ed3\u8bba\u5bf9\u4e8e\u6b63\u6574\u6570\u6210\u7acb\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u9a8c\u8bc1\u5f53\u5e42\u4e3a\u8d1f\u6574\u6570\u7684\u60c5\u5f62\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u9700\u8981\u8bc1\u660e\uff1a<\/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>\u8bc1\u660e\uff1a<\/strong><\/span> \u53ea\u9700\u8003\u8651\u4e00\u4e2a\u5546\u7684\u5bfc\u6570\u5373\u53ef\uff1a<\/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>\u8d85\u8d8a\u51fd\u6570\u7684\u5bfc\u6570<\/h2>\n<h3>\u4e09\u89d2\u51fd\u6570<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=667s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u8fd9\u4e9b\u51fd\u6570\u9075\u5faa\u4ee5\u4e0b<\/span><\/strong><\/a>\u6c42\u5bfc\u6cd5\u5219\uff1a<\/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>\u4e3a\u4e86\u63a8\u5bfc\u51fa\u6bcf\u4e00\u6761\u89c4\u5219\uff0c\u6700\u4f73\u7684\u505a\u6cd5\u662f\u9996\u5148\u4ece\u6b63\u5f26\u51fd\u6570\u4e0e\u4f59\u5f26\u51fd\u6570\u7684\u5bfc\u6570\u5165\u624b\uff0c\u7136\u540e\u5229\u7528\u5bfc\u6570\u7684\u4ee3\u6570\u89c4\u5219\uff0c\u63a8\u5bfc\u51fa\u5176\u4f59\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570\u3002<\/p>\n<h4>\u6b63\u5f26\u51fd\u6570\u5bfc\u6570\u7684\u8bc1\u660e<\/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{\uff1b\u6b63\u5f26\u51fd\u6570\u5bfc\u6570\u7684\u5b9a\u4e49} \\\\ \\\\\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{\uff1b\u7531\u5939\u903c\u5b9a\u7406\u5f97\u51fa} \\\\ \\\\\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{\uff1b\u7531 (1,2,3) \u5f97\u51fa}\n\n\\end{array}<\/span>\n<h4>\u4f59\u5f26\u51fd\u6570\u5bfc\u6570\u7684\u8bc1\u660e<\/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{\uff1b\u4f59\u5f26\u51fd\u6570\u5bfc\u6570\u7684\u5b9a\u4e49} \\\\ \\\\\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>\u6b63\u5207\u3001\u6b63\u5272\u3001\u4f59\u5272\u4e0e\u4f59\u5207\u51fd\u6570\u7684\u5bfc\u6570<\/h4>\n<p>\u6709\u4e86\u6b63\u5f26\u4e0e\u4f59\u5f26\u51fd\u6570\u7684\u5bfc\u6570\u7ed3\u679c\u540e\uff0c\u63a8\u5bfc\u5176\u4f59\u4e09\u89d2\u51fd\u6570\u7684\u5bfc\u6570\u5c31\u53d8\u5f97\u975e\u5e38\u76f4\u63a5\uff1a<\/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}{\\sin(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<h3>\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=1331s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u81ea\u7136\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570<\/span><\/strong><\/a>\u4e3a\uff1a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(x) = \\frac{1}{x}<\/span>\n<p><span style=\"color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/span> \u4ece\u5bfc\u6570\u7684\u5b9a\u4e49\u51fa\u53d1\uff0c\u53ef\u4ee5\u5f97\u5230\u4ee5\u4e0b\u63a8\u7406\u8fc7\u7a0b\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx} \\ln(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\left [\\dfrac{\\ln(x+\\Delta x) - \\ln(x)}{\\Delta x} \\right] &amp;\\text{\uff1b\u81ea\u7136\u5bf9\u6570\u7684\u5bfc\u6570\u5b9a\u4e49} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{\\Delta x} \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right) \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\color{red}x\\color{black}} \\cdot \\frac{\\color{red}x\\color{black}}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{x} \\ln \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} =\\dfrac{1}{x} \\ln \\displaystyle \\left[ \\lim_{\\Delta x \\to 0} \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n(2) &amp; n=\\dfrac{x}{\\Delta x} &amp; \\text{\uff1b\u53d8\u91cf\u66ff\u6362} \\\\ \\\\\n\n(3) &amp; (\\Delta x \\to 0^+) \\longrightarrow (n\\to +\\infty) \\\\ \\\\\n\n(4) &amp; \\dfrac{d}{dx} \\ln(x) = \\dfrac{1}{x} \\ln\\left[ \\displaystyle \\lim_{n \\to +\\infty} \\left(1 + \\dfrac{1}{n} \\right)^n \\right] = \\dfrac{1}{x} \\ln(e) = \\color{blue}\\dfrac{1}{x} \\color{black} &amp; \\text{\uff1b\u7531 (1,2,3) \u5f97\u51fa}\n\n\\end{array}<\/span>\n<p>\n\u81f3\u6b64\uff0c\u6211\u4eec\u5df2\u7ecf\u4e00\u6b65\u6b65\u56de\u987e\u4e86\u6bcf\u4f4d\u5b66\u751f\u90fd\u5e94\u638c\u63e1\u7684\u57fa\u672c\u5bfc\u6570\u5185\u5bb9\uff1a\u4ece\u57fa\u7840\u7684\u4ee3\u6570\u51fd\u6570\u5230\u4e3b\u8981\u7684\u8d85\u8d8a\u51fd\u6570\uff0c\u5982\u4e09\u89d2\u51fd\u6570\u548c\u81ea\u7136\u5bf9\u6570\u3002\u638c\u63e1\u8fd9\u4e9b\u63a8\u5bfc\u8fc7\u7a0b\uff0c\u5c06\u4f7f\u4f60\u80fd\u591f\u81ea\u5982\u5730\u8fd0\u7528\u5bfc\u6570\u6cd5\u5219\uff0c\u7406\u89e3\u5176\u6765\u6e90\u4e0e\u5f62\u5f0f\u5316\u7684\u8bba\u8bc1\u4f9d\u636e\u3002\u8fd9\u4e00\u77e5\u8bc6\u6784\u6210\u4e86\u575a\u5b9e\u7684\u57fa\u7840\uff0c\u4f7f\u4f60\u6709\u4fe1\u5fc3\u5e94\u5bf9\u66f4\u4e3a\u590d\u6742\u7684\u95ee\u9898\uff0c\u5e76\u80fd\u5bf9\u53d8\u5316\u8fdb\u884c\u7cbe\u786e\u5206\u6790\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u591a\u9879\u5f0f\u51fd\u6570\u3001\u4e09\u89d2\u51fd\u6570\u4e0e\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570 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