{"id":29716,"date":"2024-11-27T12:00:59","date_gmt":"2024-11-27T12:00:59","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29716"},"modified":"2024-11-27T17:36:09","modified_gmt":"2024-11-27T17:36:09","slug":"%e5%87%bd%e6%95%b0%e6%9e%81%e9%99%90%e7%9a%84%e5%af%bc%e6%95%b0","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e5%87%bd%e6%95%b0%e6%9e%81%e9%99%90%e7%9a%84%e5%af%bc%e6%95%b0\/","title":{"rendered":"\u51fd\u6570\u6781\u9650\u7684\u5bfc\u6570"},"content":{"rendered":"<style>\np {\n  text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">\u51fd\u6570\u6781\u9650\u7684\u5bfc\u6570<\/h1>\n<p style=\"text-align:center;\">\n  <em><strong>\u6458\u8981\uff1a<\/strong>\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u5bfc\u6570\u8fd9\u4e00\u6570\u5b66\u5de5\u5177\uff0c\u7528\u4ee5\u5206\u6790\u51fd\u6570\u7684\u53d8\u5316\u3002\u901a\u8fc7\u4ece\u5272\u7ebf\u7684\u659c\u7387\u51fa\u53d1\uff0c\u5e76\u5728\u70b9\u63a5\u8fd1\u65f6\u8ba1\u7b97\u6781\u9650\uff0c\u6211\u4eec\u5c06\u5b9a\u4e49\u5bfc\u6570\u4e3a\u5207\u7ebf\u7684\u659c\u7387\u3002\u6b64\u5916\uff0c\u6211\u4eec\u5c06\u7814\u7a76\u5176\u5173\u952e\u6027\u8d28\u548c\u89c4\u5219\uff0c\u4f8b\u5982\u6c42\u548c\u89c4\u5219\u3001\u4e58\u79ef\u89c4\u5219\u548c\u5546\u89c4\u5219\uff0c\u8fd9\u4e9b\u662f\u5c06\u5bfc\u6570\u5e94\u7528\u4e8e\u51fd\u6570\u548c\u53d8\u5316\u73b0\u8c61\u5206\u6790\u7684\u57fa\u7840\u3002<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807<\/strong><\/p>\n<p>\u901a\u8fc7\u672c\u8282\u8bfe\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u5bfc\u6570\u4f5c\u4e3a\u63cf\u8ff0\u51fd\u6570\u77ac\u65f6\u53d8\u5316\u7684\u6781\u9650\uff0c\u5e76\u7406\u89e3\u5176\u4e3a\u66f2\u7ebf\u67d0\u4e00\u70b9\u5904\u5207\u7ebf\u7684\u659c\u7387\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>\u53ef\u5bfc\u6027\u5982\u4f55\u6697\u793a\u51fd\u6570\u7684\u8fde\u7eed\u6027\u3002<\/li>\n<li><strong>\u8bc1\u660e<\/strong>\u4ece\u5f62\u5f0f\u5b9a\u4e49\u63a8\u5bfc\u7684\u57fa\u672c\u6c42\u5bfc\u89c4\u5219\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u5bfc\u6570\u7684\u4ee3\u6570\u6027\u8d28\uff08\u6c42\u548c\u3001\u4e58\u79ef\u548c\u5546\uff09\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>\uff1a<\/strong><\/p>\n<p><a href=\"#1\"><strong>\u5bfc\u6570\u7684\u6982\u5ff5<\/strong><\/a><br \/>\n<a href=\"#1\">\u5272\u7ebf\u7684\u659c\u7387<\/a><br \/>\n<a href=\"#1\">\u53d6\u6781\u9650\uff1a\u5bfc\u6570\u4e0e\u5207\u7ebf\u7684\u659c\u7387<\/a><br \/>\n<a href=\"#1\">\u66ff\u4ee3\u5b9a\u4e49<\/a><br \/>\n<a href=\"#1\"><strong>\u5bfc\u6570\u7684\u6027\u8d28<\/strong><\/a><br \/>\n<a href=\"#1\">\u53ef\u5bfc\u6027\u6697\u793a\u8fde\u7eed\u6027<\/a><br \/>\n<a href=\"#1\">\u5bfc\u6570\u7684\u4ee3\u6570<\/a><\/p>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/TFxATgmYvkY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u5bfc\u6570\u7684\u6982\u5ff5<\/h2>\n<p>\u81ea\u7136\u754c\u901a\u5e38\u662f\u53d8\u5316\u7684\uff0c\u800c\u7528\u4e8e\u8ba1\u7b97\u548c\u7406\u89e3\u53d8\u5316\u7684\u9996\u8981\u6570\u5b66\u5de5\u5177\u4fbf\u662f\u5bfc\u6570\u3002\u5b83\u7684\u51fa\u73b0\u6e90\u4e8e\u8fd9\u6837\u7684\u63d0\u95ee\uff1a\u201c\u5f53\u53d8\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u589e\u52a0\u6216\u51cf\u5c11\u5230\u4efb\u610f\u5c0f\u7684\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> \u65f6\uff0c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u503c\u4f1a\u53d1\u751f\u4ec0\u4e48\u53d8\u5316\uff1f\u201d\u901a\u8fc7\u5206\u6790\u8fd9\u4e2a\u95ee\u9898\uff0c\u5bfc\u6570\u4f5c\u4e3a\u51fd\u6570\u7684\u6781\u9650\u51fa\u73b0\u4e86\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>\u5272\u7ebf\u7684\u659c\u7387<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>\u8003\u8651\u4e00\u4e2a\u51fd\u6570<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728\u4e24\u4e2a\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span> \u5904\u7684\u503c\u3002\u8fde\u63a5\u66f2\u7ebf\u8fd9\u4e24\u4e2a\u70b9\u7684\u76f4\u7ebf\u79f0\u4e3a\u201c\u5272\u7ebf\u201d\uff0c\u5982\u56fe\u6240\u793a\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"\u5272\u7ebf\u56fe\u50cf\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"\u5272\u7ebf\u56fe\u50cf\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>\u8fd9\u6761\u5272\u7ebf\u7684\u659c\u7387\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h3>\u53d6\u6781\u9650\uff1a\u5bfc\u6570\u4e0e\u5207\u7ebf\u7684\u659c\u7387<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>\u5982\u679c\u6211\u4eec\u8003\u8651\u66f2\u7ebf<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span> \u7684\u5272\u7ebf\uff0c\u7ecf\u8fc7\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>\uff0c\u7136\u540e\u53d6 <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> \u8d8b\u4e8e\u96f6\u7684\u6781\u9650\uff0c\u6211\u4eec\u5f97\u5230\u7ecf\u8fc7\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, f(x_0))<\/span> \u7684\u5207\u7ebf\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"\u5207\u7ebf\u56fe\u50cf\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"\u5207\u7ebf\u56fe\u50cf\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>\u57fa\u4e8e\u6b64\uff0c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u7684\u5bfc\u6570\u7684\u5f62\u5f0f\u5b9a\u4e49\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{df(x_0)}{dx}:= \\lim_{\\Delta x \\to 0}\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>\u8fd9\u4e5f\u4ee3\u8868\u4e86\u901a\u8fc7\u70b9 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u7684\u5207\u7ebf\u7684\u659c\u7387\u3002<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>\u66ff\u4ee3\u5b9a\u4e49<\/h3>\n<p>\u5bfc\u6570\u4f5c\u4e3a\u6781\u9650\u7684\u53e6\u4e00\u79cd\u5b9a\u4e49\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u66ff\u4ee3\u65b9\u6cd5\u83b7\u5f97\uff1a<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nx_i &amp;= x_0\\\\\n\nx_f &amp;= x_i + \\Delta x\n\n\\end{array}\n\n<\/span>\n<p>\u56e0\u6b64\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span>\uff0c\u5bfc\u6570\u5b9a\u4e49\u53d8\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{df(x_i)}{dx} &amp;=\\displaystyle \\lim_{\\Delta x \\to 0}\\dfrac{ f(x_i + \\Delta x) - f(x_i)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{x_f - x_i \\to 0} \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\\\\ \\\\\n\n&amp;=\\displaystyle  \\lim_{x_f \\to x_i } \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\n\n\\end{array}\n\n<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"\u5bfc\u6570\u5b9a\u4e49\u4e3a\u5272\u7ebf\u659c\u7387\u6781\u9650\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"\u5bfc\u6570\u5b9a\u4e49\u4e3a\u5272\u7ebf\u659c\u7387\u6781\u9650\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>\u8fd9\u4e24\u79cd\u5b9a\u4e49\u662f\u7b49\u4ef7\u7684\uff0c\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4ea4\u66ff\u4f7f\u7528\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u5bfc\u6570\u7684\u6027\u8d28<\/h2>\n<p>\u5f53\u4ee5\u4e0b\u6781\u9650\u5b58\u5728\u65f6\uff0c\u79f0\u51fd\u6570\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u53ef\u5bfc\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>\u5982\u679c\u8be5\u6781\u9650\u5bf9\u6240\u6709 <span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I<\/span> \u90fd\u5b9a\u4e49\u826f\u597d\uff0c\u5219\u79f0\u51fd\u6570\u5728\u96c6\u5408 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u53ef\u5bfc\u3002\u53ef\u5bfc\u51fd\u6570\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u53ef\u5bfc\u6027\u6697\u793a\u8fde\u7eed\u6027<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>\u5982\u679c\u51fd\u6570\u5728<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u53ef\u5bfc\uff0c\u5219\u5b83\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u8fde\u7eed\u3002\u8fd9\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u8bba\u8bc1\u6765\u8bc1\u660e\uff1a<\/p>\n<p>\u4e3a\u4e86\u4f7f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u8fde\u7eed\uff0c\u9700\u8981\u6ee1\u8db3\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = f(x_0)<\/span>\n<p>\u5206\u6790\u6b64\u8868\u8fbe\u5f0f\u7684\u5de6\u4fa7\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to x_0} f(x) &amp;= \\displaystyle \\lim_{x\\to x_0} \\left[ f(x) + f(x_0) - f(x_0) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( f(x)  - f(x_0) \\right) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0)  \\right] \\\\ \\\\\n\n&amp;=f(x_0) +\\displaystyle \\lim_{x\\to x_0} \\left[ \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0) \\right]\n\\end{array}\n\n<\/span>\n<p>\u56e0\u6b64\uff0c\u4e3a\u4e86\u4f7f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u8fde\u7eed\uff0c\u53f3\u4fa7\u7684\u6781\u9650\u5fc5\u987b\u5b9a\u4e49\u826f\u597d\u3002\u8fd9\u4ec5\u5f53\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\dfrac{f(x) - f(x_0)}{x-x_0} = \\dfrac{df(x_0)}{dx}<\/span>\n<p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5f53\u4e14\u4ec5\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u5904\u53ef\u5bfc\u65f6 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u662f\u8fde\u7eed\u7684\u3002\u56e0\u6b64\uff0c\u53ef\u5bfc\u6027\u6697\u793a\u8fde\u7eed\u6027\u3002<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>\u5bfc\u6570\u7684\u4ee3\u6570<\/h3>\n<p>\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u4e3a\u5728\u6240\u6709 <span class=\"katex-eq\" data-katex-display=\"false\">x \\in I<\/span> \u4e0a\u53ef\u5bfc\u7684\u51fd\u6570\uff0c\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathbb{R}.<\/span> \u5219\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha \\dfrac{df(x)}{dx} \\pm \\beta\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( f(x) g(x) \\right) = \\dfrac{df(x)}{dx} g(x) + f(x)\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li>\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0<\/span>\uff0c\u5219 <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\dfrac{f(x)}{g(x)} \\right) = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx} }{\\left[g(x)\\right]^2}<\/span><\/li>\n<\/ol>\n<p>\u4ece\u4e0a\u8ff0\u6027\u8d28\u53ef\u89c1\uff0c\u5bfc\u6570\u7684\u4ee3\u6570\u53ef\u80fd\u4e0d\u50cf\u521d\u770b\u8d77\u6765\u90a3\u6837\u76f4\u89c2\u3002\u7136\u800c\uff0c\u8fd9\u4e9b\u6027\u8d28\u53ef\u4ee5\u4ece\u5bfc\u6570\u4f5c\u4e3a\u6781\u9650\u7684\u5b9a\u4e49\u4e2d\u63a8\u5bfc\u51fa\u6765\u3002<\/p>\n<p><span style=\"color: #000080;\">\u8bc1\u660e\uff1a<\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>\u5173\u4e8e\u6c42\u548c\u516c\u5f0f\u7684\u8bc1\u660e<\/strong><\/a>\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\alpha f(x) \\pm \\beta g(x) \\right) &amp; =\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\left[\\alpha f(x+\\Delta x) \\pm \\beta g(x+ \\Delta x)\\right] - \\left[\\alpha f(x) \\pm \\beta g(x) \\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\left[\\alpha f(x+\\Delta x) - \\alpha f(x)\\right] \\pm \\left[\\beta g(x+\\Delta x) - \\beta g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\alpha \\left[ f(x+\\Delta x) -  f(x)\\right] \\pm  \\beta  \\left[ g(x+\\Delta x) - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\alpha \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) -  f(x)}{\\Delta x} \\pm \\beta \\lim_{\\Delta x \\to 0} \\dfrac{ g(x+\\Delta x) -  g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\alpha \\dfrac{df(x)}{dx} \\pm \\beta \\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=1059s\" target=\"_blank\" rel=\"noopener\"><strong>\u5173\u4e8e\u4e58\u79ef\u516c\u5f0f\u7684\u8bc1\u660e<\/strong><\/a>\u7a0d\u5fae\u590d\u6742\u4e00\u4e9b\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left[f(x)g(x)\\right] &amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) -  f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) + \\color{red}f(x)g(x+\\Delta x) - f(x)g(x+\\Delta x) \\color{black} - f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\left[f(x+\\Delta x) - f(x) \\right] g(x+\\Delta x) + f(x) \\left[g(x+\\Delta x)  - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x) \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x)\\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;= g(x) \\dfrac{df(x)}{dx} + f(x)\\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p>\u8fd9\u4e00\u63a8\u5bfc\u5229\u7528\u4e86 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u662f\u53ef\u5bfc\u51fd\u6570\uff0c\u56e0\u6b64\u4e5f\u662f\u8fde\u7eed\u7684\uff0c\u5373 <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x \\to 0 } g(x+\\Delta x) = g(x)<\/span>\u3002<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>\u6700\u540e\uff0c\u5173\u4e8e\u5546\u516c\u5f0f\u7684\u8bc1\u660e\uff0c<\/strong><\/a>\u53ef\u4ee5\u501f\u52a9\u4e58\u79ef\u516c\u5f0f\u3002\u5047\u8bbe\u51fd\u6570\u5f62\u5f0f\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x),<\/span> \u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0.<\/span> \u90a3\u4e48\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{df(x)}{dx}= \\dfrac{d}{dx}(k(x)g(x)) = \\dfrac{dk(x)}{dx}g(x) + k(x)\\dfrac{dg(x)}{dx}<\/span>\n<p>\u89e3\u51fa <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}<\/span>\uff0c\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}g(x) = \\dfrac{df(x)}{dx} - k(x)\\dfrac{dg(x)}{dx} = \\dfrac{d}{dx}f(x) - \\dfrac{f(x)}{g(x)}\\dfrac{dg(x)}{dx} <\/span>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)\n\n &amp;= \\dfrac{dk(x)}{dx} =\\dfrac{1}{g(x)} \\dfrac{df(x)}{dx} - \\dfrac{f(x)}{\\left[g(x)\\right]^2}\\dfrac{dg(x)}{dx} \\\\ \\\\\n\n&amp; = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx}}{[g(x)]^2}\n\n\\end{array}\n\n<\/span>\n<p>\u8fd9\u5c31\u662f\u6211\u4eec\u60f3\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u51fd\u6570\u6781\u9650\u7684\u5bfc\u6570 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\u81ea\u7136\u754c\u901a\u5e38\u662f\u53d8\u5316\u7684\uff0c\u800c\u7528\u4e8e\u8ba1\u7b97\u548c\u7406\u89e3\u53d8\u5316\u7684\u9996\u8981\u6570\u5b66\u5de5\u5177\u4fbf\u662f\u5bfc\u6570\u3002\u5b83\u7684\u51fa\u73b0\u6e90\u4e8e\u8fd9\u6837\u7684\u63d0\u95ee\uff1a\u201c\u5f53\u53d8\u91cf \u589e\u52a0\u6216\u51cf\u5c11\u5230\u4efb\u610f\u5c0f\u7684\u91cf \u65f6\uff0c\u51fd\u6570 \u7684\u503c\u4f1a\u53d1\u751f\u4ec0\u4e48\u53d8\u5316\uff1f\u201d\u901a\u8fc7\u5206\u6790\u8fd9\u4e2a\u95ee\u9898\uff0c\u5bfc\u6570\u4f5c\u4e3a\u51fd\u6570\u7684\u6781\u9650\u51fa\u73b0\u4e86\u3002 \u5272\u7ebf\u7684\u659c\u7387 \u8003\u8651\u4e00\u4e2a\u51fd\u6570 \u5728\u4e24\u4e2a\u70b9 \u548c \u5904\u7684\u503c\u3002\u8fde\u63a5\u66f2\u7ebf\u8fd9\u4e24\u4e2a\u70b9\u7684\u76f4\u7ebf\u79f0\u4e3a\u201c\u5272\u7ebf\u201d\uff0c\u5982\u56fe\u6240\u793a\u3002 \u8fd9\u6761\u5272\u7ebf\u7684\u659c\u7387\u4e3a\uff1a \u53d6\u6781\u9650\uff1a\u5bfc\u6570\u4e0e\u5207\u7ebf\u7684\u659c\u7387 \u5982\u679c\u6211\u4eec\u8003\u8651\u66f2\u7ebf \u7684\u5272\u7ebf\uff0c\u7ecf\u8fc7\u70b9 \u548c \uff0c\u7136\u540e\u53d6 \u8d8b\u4e8e\u96f6\u7684\u6781\u9650\uff0c\u6211\u4eec\u5f97\u5230\u7ecf\u8fc7\u70b9 \u7684\u5207\u7ebf\u3002 \u57fa\u4e8e\u6b64\uff0c\u51fd\u6570 \u5728\u70b9 \u7684\u5bfc\u6570\u7684\u5f62\u5f0f\u5b9a\u4e49\u4e3a\uff1a \u8fd9\u4e5f\u4ee3\u8868\u4e86\u901a\u8fc7\u70b9 \u7684\u5207\u7ebf\u7684\u659c\u7387\u3002 \u66ff\u4ee3\u5b9a\u4e49 \u5bfc\u6570\u4f5c\u4e3a\u6781\u9650\u7684\u53e6\u4e00\u79cd\u5b9a\u4e49\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u66ff\u4ee3\u65b9\u6cd5\u83b7\u5f97\uff1a \u56e0\u6b64\uff0c\uff0c\u5bfc\u6570\u5b9a\u4e49\u53d8\u4e3a\uff1a \u8fd9\u4e24\u79cd\u5b9a\u4e49\u662f\u7b49\u4ef7\u7684\uff0c\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4ea4\u66ff\u4f7f\u7528\u3002 \u5bfc\u6570\u7684\u6027\u8d28 \u5f53\u4ee5\u4e0b\u6781\u9650\u5b58\u5728\u65f6\uff0c\u79f0\u51fd\u6570\u5728 \u5904\u53ef\u5bfc\uff1a \u5982\u679c\u8be5\u6781\u9650\u5bf9\u6240\u6709 \u90fd\u5b9a\u4e49\u826f\u597d\uff0c\u5219\u79f0\u51fd\u6570\u5728\u96c6\u5408 \u4e0a\u53ef\u5bfc\u3002\u53ef\u5bfc\u51fd\u6570\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a \u53ef\u5bfc\u6027\u6697\u793a\u8fde\u7eed\u6027 \u5982\u679c\u51fd\u6570\u5728 \u5904\u53ef\u5bfc\uff0c\u5219\u5b83\u5728 \u5904\u8fde\u7eed\u3002\u8fd9\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u8bba\u8bc1\u6765\u8bc1\u660e\uff1a \u4e3a\u4e86\u4f7f \u5728 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29706,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[858,575],"tags":[],"class_list":["post-29716","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-858","category-575"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u51fd\u6570\u6781\u9650\u7684\u5bfc\u6570 - 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