{"id":292,"date":"2024-08-02T13:30:00","date_gmt":"2024-08-02T05:30:00","guid":{"rendered":"https:\/\/solost.cn\/?p=292"},"modified":"2025-02-09T19:58:25","modified_gmt":"2025-02-09T11:58:25","slug":"common-formula-zufb6","status":"publish","type":"post","link":"https:\/\/solost.cn\/?p=292","title":{"rendered":"\u5e38\u7528\u516c\u5f0f"},"content":{"rendered":"<h1>\u5e38\u7528\u516c\u5f0f<\/h1>\n<h3>\u57fa\u672c\u6c42\u5bfc\u516c\u5f0f<\/h3>\n<p>${\\left( {x}^{\\alpha }\\right) }^{\\prime } = \\alpha {x}^{\\alpha  - 1}\\left( {\\alpha \\text{ \u4e3a\u5e38\u6570 }}\\right)$<\/p>\n<p>${\\left( {a}^{x}\\right) }^{\\prime } = {a}^{x}\\ln a\\left( {a &gt; 0,a \\neq  1}\\right)$<\/p>\n<p>${\\left( {\\mathrm{e}}^{x}\\right) }^{\\prime } = {\\mathrm{e}}^{x}$<\/p>\n<p>${\\left( {\\log }_{a}x\\right) }^{\\prime } = \\frac{1}{x\\ln a}\\left( {a &gt; 0,a \\neq  1}\\right)$<\/p>\n<p>${\\left( \\ln \\left| x\\right| \\right) }^{\\prime } = \\frac{1}{x}$<\/p>\n<p>${\\left( \\sin x\\right) }^{\\prime } = \\cos x$<\/p>\n<p>${\\left( \\cos x\\right) }^{\\prime } =  - \\sin x$<\/p>\n<p>${\\left( \\arcsin x\\right) }^{\\prime } = \\frac{1}{\\sqrt{1 - {x}^{2}}}$<\/p>\n<p>${\\left( \\arccos x\\right) }^{\\prime } =  - \\frac{1}{\\sqrt{1 - {x}^{2}}}$<\/p>\n<p>${\\left( \\tan x\\right) }^{\\prime } = {\\sec }^{2}x$<\/p>\n<p>${\\left( \\cot x\\right) }^{\\prime } =  - {\\csc }^{2}x$<\/p>\n<p>${\\left( \\arctan x\\right) }^{\\prime } = \\frac{1}{1 + {x}^{2}}$<\/p>\n<p>${\\left( \\operatorname{arccot}x\\right) }^{\\prime } =  - \\frac{1}{1 + {x}^{2}}$<\/p>\n<p>${\\left( \\sec x\\right) }^{\\prime } = \\sec x\\tan x$<\/p>\n<p>${\\left( \\csc x\\right) }^{\\prime } =  - \\csc x\\cot x$<\/p>\n<p>${\\left\\lbrack  \\ln \\left( x + \\sqrt{{x}^{2} + 1}\\right) \\right\\rbrack  }^{\\prime } = \\frac{1}{\\sqrt{{x}^{2} + 1}}$<\/p>\n<p>${\\left\\lbrack  \\ln \\left( x + \\sqrt{{x}^{2} - 1}\\right) \\right\\rbrack  }^{\\prime } = \\frac{1}{\\sqrt{{x}^{2} - 1}}.$<\/p>\n<h3>\u57fa\u672c\u79ef\u5206\u516c\u5f0f<\/h3>\n<p><img decoding=\"async\" src=\"https:\/\/amarantos.oss-cn-hangzhou.aliyuncs.com\/markdown\/202408010912266.png\" alt=\"image-20240331161002315\" title=\"image-20240331161002315\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240331161037507-1711872640222-3-20240728154550-7m65j4t.png\" alt=\"image-20240331161037507\" title=\"image-20240331161037507\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240331161106424-1711872668757-5-20240728154550-vvmhsgj.png\" alt=\"image-20240331161106424\" title=\"image-20240331161106424\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240401143752112-1711953476106-1-20240728154550-wtcmn9w.png\" alt=\"image-20240401143752112\" title=\"image-20240401143752112\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240331162809839-1711873693798-10-20240728154550-ztnosk5.png\" alt=\"image-20240331162809839\" title=\"image-20240331162809839\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240331162841020-1711873722588-12-1711873740144-14-20240728154550-c0fa7wx.png\" alt=\"image-20240331162841020\" title=\"image-20240331162841020\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240331162903488-1711873744892-16-20240728154550-r9cc4bz.png\" alt=\"image-20240331162903488\" title=\"image-20240331162903488\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240401151004470-1711955405899-3-20240728154550-wlst7bz.png\" alt=\"image-20240401151004470\" title=\"image-20240401151004470\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240401151016186-1711955418889-5-20240728154550-9zvjy7x.png\" alt=\"image-20240401151016186\" title=\"image-20240401151016186\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151817994-1712215101513-1-20240728154550-klrexzo.png\" alt=\"image-20240404151817994\" title=\"image-20240404151817994\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151837933-1712215119950-3-20240728154550-6e29zl1.png\" alt=\"image-20240404151837933\" title=\"image-20240404151837933\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404170416321-1712221458185-15-20240728154550-lhusmb4.png\" alt=\"image-20240404170416321\" title=\"image-20240404170416321\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151855811-1712215136894-5-20240728154550-kg3quhj.png\" alt=\"image-20240404151855811\" title=\"image-20240404151855811\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151908168-1712215149822-7-20240728154550-yhs88sw.png\" alt=\"image-20240404151908168\" title=\"image-20240404151908168\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151922160-1712215163315-9-20240728154550-zm6d2sh.png\" alt=\"image-20240404151922160\" title=\"image-20240404151922160\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404151939722-1712215181160-11-20240728154550-cylpqjb.png\" alt=\"image-20240404151939722\" title=\"image-20240404151939722\"><\/p>\n<p><img decoding=\"async\" src=\"assets\/image-20240404152005306-1712215206437-13-20240728154550-0lmermj.png\" alt=\"image-20240404152005306\" title=\"image-20240404152005306\"><\/p>\n<h3>\u6cf0\u52d2\u516c\u5f0f\u6c42\u6781\u9650<\/h3>\n<blockquote>\n<p>\u6ce8\u610f\uff1a<\/p>\n<p>\u76f8\u51cf\u5c55\u5f00\u8fc7\u7a0b\u4e2d\uff0c\u8981\u5c55\u5230\u76f8\u540c\u6b21\u6570\uff0c\u4e0d\u80fd\u4e00\u4e2a\u5c55\u5f00\u5230 2 \u6b21\uff0c\u53e6\u4e00\u4e2a 3 \u6b21<\/p>\n<p>\u76f8\u4e58\u8fc7\u7a0b\u4e0e\u76f8\u51cf\u7c7b\u4f3c<\/p>\n<p>\u76f8\u9664\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u4ee5\u7b80\u5355\u5224\u65ad\u5206\u5b50\u6216\u8005\u5206\u6bcd\u5c55\u5f00\u7684\u9636\u6570\uff0c\u7136\u540e\u5206\u5b50\u6216\u8005\u5206\u6bcd\u76f8\u5e94\u5c55\u5230\u76f8\u540c\u9636\u6570\uff0c\u5982\u679c\u6709\u90e8\u5206\u4e0e\u5206\u6bcd\u9636\u6570\u4e0d\u540c\uff0c\u6839\u636e\u6293\u5927\u5934\u539f\u5219\u5373\u53ef\uff0c\u8d8b\u5411\u4e8e 0 \u6bd4\u5206\u6bcd\u6b21\u6570\u66f4\u9ad8\u7684\u53ef\u4ee5\u76f4\u63a5\u820d\u53bb\uff0c\u8d8b\u5411\u4e8e\u6b63\u65e0\u7a77\uff0c\u6bd4\u5206\u6bcd\u9636\u6570\u4f4e\u7684\u53ef\u4ee5\u820d\u53bb<\/p>\n<\/blockquote>\n<h3>\u4e8c\u91cd\u79ef\u5206\u6c42\u6781\u503c<\/h3>\n<p><img decoding=\"async\" src=\"assets\/image-20240626094950200-20240728154550-3rz5rar.png\" alt=\"image-20240626094950200\" title=\"image-20240626094950200\"><\/p>\n<h4>\u65e0\u6761\u4ef6\u7684\u4e8c\u91cd\u79ef\u5206\u6c42\u6781\u503c<\/h4>\n<blockquote>\n<p>1.\u5148\u6c42\u51fd\u6570\u7684\u504f\u5bfc\u6570<\/p>\n<p>2.\u627e\u5230\u4ee4\u504f\u5bfc\u6570\u4e0d\u5b58\u5728\u6216\u8005\u4e3a 0 \u7684\u70b9\uff0c\u504f\u5bfc\u6570\u76f8\u4e92\u5e26\u5165\uff0c\u6c42\u51fa\u53ef\u7591\u70b9<\/p>\n<p>3.\u6c42\u4e8c\u9636\u504f\u5bfc\u6570<\/p>\n<p>4.\u628a\u53ef\u7591\u70b9\u5e26\u5165\uff0c\u5224\u65ad \u0394\uff0c\u7136\u540e\u6839\u636e\u6761\u4ef6\u5e26\u5165\u539f\u65b9\u7a0b\u5373\u53ef\u6c42\u51fa\u6781\u5927\u6216\u8005\u6781\u5c0f\u503c<\/p>\n<\/blockquote>\n<h4>\u6761\u4ef6\u6700\u503c\u4e0e\u62c9\u683c\u6717\u65e5\u6570\u4e58\u6cd5<\/h4>\n<blockquote>\n<p>\u9996\u5148\uff0c\u5982\u679c\u80fd\u5c06\u6761\u4ef6\u8f6c\u5316\u4e3a\u53d8\u91cf\u5e26\u5165\u9996\u5148\u8f6c\u5316\uff0c\u5c31\u53ef\u4ee5\u8f6c\u6210\u65e0\u6761\u4ef6\u7684\u6c42\u6cd5<\/p>\n<p>1.\u6784\u9020\u8f85\u52a9\u51fd\u6570<\/p>\n<p>2.\u6c42\u5404\u4e2a\u53d8\u91cf\u7684\u504f\u5bfc\uff0c\u4ee4\u5176\u4e3a 0<\/p>\n<p>3.\u89e3\u65b9\u7a0b<\/p>\n<p>4.\u6bd4\u8f83\u65b9\u7a0b\u5f97\u51fa\u7684\u5404\u70b9\u5927\u5c0f\uff0c\u627e\u51fa\u6700\u5927\u6700\u5c0f\u503c<\/p>\n<\/blockquote>\n<h3>\u591a\u5143\u51fd\u6570\u6c42\u6781\u503c<\/h3>\n<blockquote>\n<p>\u6ce8\u610f\uff1a<\/p>\n<p>\u6781\u503c\u53ef\u80fd\u5b58\u5728\u7684\u5730\u65b9<\/p>\n<p>1.\u51fd\u6570\u5b9a\u4e49\u57df\u5185\u90e8<\/p>\n<p>2.\u5b9a\u4e49\u57df\u8fb9\u754c<\/p>\n<\/blockquote>\n<p>\u4f8b\u5982\uff1a${(x,y)|x+2*y&lt;=1,x&gt;=0,y&gt;=0}$<\/p>\n<ol>\n<li>$x=0$ \u65f6\u5019\uff0c\u53ef\u4ee5\u6c42\u51fa\u5185\u90e8\u7684\u9a7b\u70b9(\u504f\u5bfc\u6570\u4e3a 0)\uff0c\u8fd8\u6709\u8fb9\u754c\u70b9 $(0,0),(0,1\/2)$<\/li>\n<li>$y=0$ \u65f6\u5019\uff0c\u53ef\u4ee5\u6c42\u51fa\u5185\u90e8\u7684\u9a7b\u70b9(\u504f\u5bfc\u6570\u4e3a 0)\uff0c\u8fd8\u6709\u8fb9\u754c\u70b9 $(1,0)$<\/li>\n<li>$x+2*y=1$ \u65f6\u5019\uff0c\u628a\u65b9\u7a0b\u7684 x \u6216\u8005 y \u6362\u6389\uff0c\u53c8\u4f1a\u6709\u4e00\u4e2a\u65b0\u7684\u65b9\u7a0b\uff0c\u518d\u6c42\u65b0\u65b9\u7a0b\u7684\u6781\u503c<\/li>\n<\/ol>\n<p>\u4f8b\u9898\u89c1 1000 \u9898 A \u7ec4 13 \u7ae0 18 \u9898<\/p>\n<h3>\u6781\u5750\u6807\u4e8c\u91cd\u79ef\u5206\u8ba1\u7b97<\/h3>\n<blockquote>\n<p>\u5982\u679c\u9047\u5230\u88ab\u79ef\u51fd\u6570\u662f\u692d\u5706\u7684\u60c5\u51b5 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}$$=$$r^{2}$<\/p>\n<p>\u4ee4 $x=arcost$, $y=brsint$<\/p>\n<p>\u6839\u636e\u9898\u76ee\u7ed9\u51fa\u7684\u6761\u4ef6\uff0c\u8ba1\u7b97\u51fa t \u7684\u8303\u56f4\u548c r \u7684\u8303\u56f4<\/p>\n<p>\u7136\u540e <strong>$dxdy=abrdr$<\/strong> \u5199\u65b9\u7a0b\u8fc7\u7a0b\u4e00\u5b9a\u8bb0\u5f97\u6362\u4e86\uff0c\u4e0d\u662f $rdr$<\/p>\n<\/blockquote>\n<h3>\u6c42\u6e10\u8fd1\u7ebf<\/h3>\n<blockquote>\n<p>\u6c42\u6e10\u8fd1\u7ebf\u8fc7\u7a0b<\/p>\n<p>1.\u5782\u76f4\u6e10\u8fd1\u7ebf\uff08$x\\to x_0$\uff09<\/p>\n<p>2.\u6c34\u5e73\u6e10\u8fd1\u7ebf\uff08$x\\to \\infty$\uff09<\/p>\n<p>3.\u6c42\u659c\u6e10\u8fd1\u7ebf\u65f6\uff0c\u5148\u6c42\u51fa$\\frac{f(x)}{x},$\u4e5f\u5c31\u662f\u659c\u7387\uff0c\u5224\u65ad$x\\to +\\infty\u548cx \\to -\\infty$\u7684\u6781\u9650\u503c\u662f\u5426\u76f8\u540c\uff0c\u5982\u679c\u5b58\u5728\u4e14\u6781\u9650\u503c\u4e0d\u540c\uff0c\u5219\u659c\u6e10\u8fd1\u7ebf\u6761\u6570\u53d6\u51b3\u4e8e\u4e0d\u540c\u6781\u9650\u503c\u4e2a\u6570<\/p>\n<p>\u6ce8\u610f\uff1a\u5982\u679c\u9047\u5230\u8d8b\u5411\u4e8e\u8d1f\u65e0\u7a77\uff0c\u7a33\u59a5\u8d77\u89c1\u7528-t \u6362\u5143 x\uff0ct \u8d8b\u5411\u4e8e\u6b63\u65e0\u7a77,\u8d8b\u5411\u4e8e\u8d1f\u65e0\u7a77\u65f6\u4ece\u7edd\u5bf9\u503c\u63d0\u51fa $x^2$ \u65f6\u5019\u8981\u52a0\u8d1f\u53f7<\/p>\n<\/blockquote>\n<h3>\u9047\u5230\u4ee5\u4e0b\u60c5\u51b5\u53ef\u4ee5\u7528\u5bfc\u6570\u5b9a\u4e49\u6765\u6c42\u5bfc\u6570<\/h3>\n<ol>\n<li>\u88ab\u6c42\u5bfc\u7684\u51fd\u6570\u8fc7\u4e8e\u590d\u6742<\/li>\n<li>\u88ab\u6c42\u5bfc\u7684\u51fd\u6570\u4e2d\u542b\u6709\u62bd\u8c61\u51fd\u6570<\/li>\n<li>\u88ab\u6c42\u5bfc\u7684\u51fd\u6570\u4e0d\u7b26\u5408\u6c42\u5bfc\u6cd5\u5219<\/li>\n<li>\u5206\u6bb5\u51fd\u6570\u5206\u6bb5\u70b9\u6c42\u5bfc\u7528\u5b9a\u4e49<\/li>\n<li>\u8ba9\u51fd\u6570\u503c\u4e3a 0 \u7684\u70b9\uff08\u5982\u679c\u51fa\u73b0\u8868\u8fbe\u5f0f\u4e00\u90e8\u5206\u8d8b\u4e8e $x_0$ \u65f6 $f(x)$ \u4e3a 0\uff0c\u53ef\u4ee5\u62c6\u6210 $(x-x0)*g(x)$ \u4f7f\u7528\u6781\u9650\u8fd0\u7b97\u6cd5\u5219\u62c6\u5f00\u7b97\uff09<\/li>\n<\/ol>\n<h3>\u8fde\u7eed\u76f8\u5173\u6ce8\u610f\u70b9<\/h3>\n<ul>\n<li>\n<p>\u95ee $f&#039;(x)$ \u662f\u5426\u8fde\u7eed\uff1f<\/p>\n<p>\u5206\u6bb5\u51fd\u6570\u5206\u6bb5\u70b9\u5904\uff0c\u6c42\u975e\u5206\u6bb5\u70b9\u8d8b\u5411\u4e8e\u5206\u6bb5\u70b9\u7684\u6781\u9650\u503c\u662f\u5426\u4e0e\u5206\u6bb5\u70b9\u5904\u51fd\u6570\u503c\u76f8\u7b49\uff0c\u76f8\u7b49\u5373\u8fde\u7eed\uff08\u90e8\u5206\u60c5\u51b5\u9700\u8981\u5de6\u5bfc\u6570\u53f3\u5bfc\u6570\u548c\u51fd\u6570\u503c\u90fd\u8fdb\u884c\u5224\u65ad\uff09<\/p>\n<blockquote>\n<p>\u4e00\u822c\u60c5\u51b5\u5148\u6c42 $f(x)$ \u5728\u5206\u6bb5\u70b9\u548c\u4e0d\u5728\u5206\u6bb5\u70b9\u7684\u5bfc\u6570\u7684\u8868\u8fbe\u5f0f<\/p>\n<p>\u5728\u8ba9\u975e\u5206\u6bb5\u70b9\u7684\u5bfc\u6570\u8d8b\u4e8e\u5206\u6bb5\u70b9\uff0c\u6bd4\u8f83\u5206\u6bb5\u70b9\u548c\u975e\u5206\u6bb5\u70b9\u7684\u51fd\u6570\u503c\u662f\u5426\u76f8\u7b49<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<p>\u6c42\u5206\u6bb5\u51fd\u6570\u7684\u5bfc\u6570<\/p>\n<ul>\n<li>\u4e0d\u5728\u5206\u6bb5\u70b9\uff1a\u4f7f\u7528\u5bfc\u6570\u6c42\u5bfc\u6cd5\u5219\u76f4\u63a5\u6c42<\/li>\n<li>\u5728\u5206\u6bb5\u70b9\uff1a\u4f7f\u7528\u5bfc\u6570\u7684\u5b9a\u4e49\u8fdb\u884c\u6c42\u89e3<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>\u51fd\u6570\u6709\u754c\u6027\u5224\u65ad\u95ee\u9898<\/h3>\n<ol>\n<li>\n<p>\u8fde\u7eed\u51fd\u6570\u6709\u754c\u95ee\u9898<\/p>\n<ul>\n<li>\u95ed\u533a\u95f4\u51fd\u6570 $f(x)$ \u8fde\u7eed\uff0c\u5219 $f(x)$ \u6709\u754c<\/li>\n<li>\u5f00\u533a\u95f4\u51fd\u6570\u8fde\u7eed\uff0c\u4e14\u5728\u7aef\u70b9\u5904\u6781\u9650\u5b58\u5728\uff0c\u51fd\u6570\u6709\u754c<\/li>\n<li><strong>\u5bfc\u51fd\u6570<\/strong> \u5728\u6709\u9650\u533a\u95f4\u6709\u754c\uff0c\u5219 $f(x)$ \u6709\u754c<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c \u51fd\u6570\u7684\u5355\u8c03\u6ca1\u529e\u6cd5\u76f4\u63a5\u63a8\u51fa $f\u2019(x)&gt;0$ \u7684<\/p>\n<p>\u4f8b\u5982\uff0c\u51fd\u6570\u5728\u67d0\u90bb\u57df\u5355\u8c03\u589e\uff0c\u53ea\u80fd\u63a8\u51fa\u5bfc\u51fd\u6570 &gt;=0\uff0c\u4e0d\u80fd\u63a8\u51fa &gt;0\uff0c\u6bd4\u5982 $x^{3}$<\/p>\n<p>\u884d\u751f\uff1a$f(x)$ \u5728 $x_0$ \u7684\u67d0\u90bb\u57df\u5185\u662f\u51f9\u51fd\u6570\uff0c\u53ea\u80fd\u63a8\u51fa $f&#039;&#039;(x)&gt;=0$,\u4f8b\u5982 $x^{4}$ \u4e0d\u80fd\u63a8\u51fa $f&#039;&#039;(x)&gt;0$<\/p>\n<p>\u5bfc\u51fd\u6570\u5728\u67d0\u70b9\u5927\u4e8e $0$\uff0c\u4e0d\u80fd\u63a8\u51fa\u51fd\u6570\u5728\u90bb\u57df\u5185\u5355\u8c03\u589e<\/p>\n<p>\u5728\u4e8c\u9636\u53ef\u5bfc\u7684\u60c5\u51b5\u4e0b\uff0c$f\u2019(x_0)&gt;0$\uff0c\u53ef\u4ee5\u63a8\u51fa\u5728 $x_0$ \u7684\u90bb\u57df\u5185\u51fd\u6570\u5355\u589e(\u4e8c\u9636\u5bfc\u6570\u5b58\u5728\uff0c\u4e00\u9636\u5bfc\u51fd\u6570\u8fde\u7eed)\uff0c\u4e8c\u9636\u5bfc\u6570\u5b58\u5728\uff0c\u4e00\u9636\u5bfc\u6570\u8fde\u7eed\uff08\u8fde\u7eed\u5c31\u662f\u4e0d\u4f1a\u5bfc\u6570\u7a81\u53d8\u65b9\u5411\uff09\uff0c\u5c31\u53ef\u4ee5\u63a8\u51fa\u4e00\u70b9\u5bfc\u6570\u503c\u5927\u4e8e 0\uff0c\u4e00\u5b9a\u662f\u90bb\u57df\u5185\u5176\u4ed6\u70b9\u5bfc\u6570\u503c\u4e5f\u662f\u5927\u4e8e 0 \u7684<\/p>\n<\/li>\n<\/ol>\n<h3>\u6781\u9650\u503c\u548c\u51fd\u6570\u503c\u65e0\u5173<\/h3>\n<ol>\n<li>\u8bbe\u51fd\u6570 $f(x)$ \u5728\u533a\u95f4 $(-1,1)$ \u5185\u6709\u5b9a\u4e49\uff0c\u4e14 0 \u70b9\u7684\u6781\u9650\u503c\u7b49\u4e8e 0<\/li>\n<\/ol>\n<p>$(A) \u5f53  \\mathop{\\lim }\\limits_{{x \\rightarrow  0}}\\frac{f\\left( x\\right) }{\\sqrt{\\left| x\\right| }} = 0,f\\left( x\\right)   \u5728  x = 0  \u5904\u53ef\u5bfc;$<\/p>\n<p>$(B) \u5f53  \\mathop{\\lim }\\limits_{{x \\rightarrow  0}}\\frac{f\\left( x\\right) }{{x}^{2}} = 0,f\\left( x\\right)   \u5728  x = 0  \u5904\u53ef\u5bfc;$<\/p>\n<p>A,B \u9009\u9879\u4e2d\uff0c\u53ef\u4ee5\u5f97\u51fa\uff0c\u51fd\u6570\u5728 0 \u70b9\u7684\u6781\u9650\u503c\u7b49\u4e8e 0\uff0c\u4f46\u662f\u5bfc\u6570\u7684\u5b9a\u4e49\u5f0f\u662f\uff0c$lim_{x-&gt;0}\\frac{f(x)-f(0)}{x}$\uff0c\u9700\u8981\u77e5\u9053\u7684\u662f 0 \u70b9\u7684\u51fd\u6570\u503c\uff0c\u4f46\u662f\u9009\u9879\u4e2d\u53ea\u80fd\u5f97\u51fa\u6781\u9650\u503c\uff0c\u7531\u4e8e\u6781\u9650\u503c\u548c\u51fd\u6570\u503c\u65e0\u5173\uff0c\u6240\u4ee5\u9009\u9879\u5168\u9519<\/p>\n<ul>\n<li>\u65e0\u7a77\u5c0f\u4e0e\u6709\u754c\u91cf\u7684\u4e58\u79ef\u662f\u4ecd\u7136\u662f\u65e0\u7a77\u5c0f\uff0c\u4f8b\u5982 $xsin(\\frac{1}{x})$<\/li>\n<li>\u4f8b\u5982 $xe^{\\frac{1}{x}}$\uff0c\u5f53 x \u8d8b\u5411\u4e8e 0 \u65f6\uff0c\u6781\u9650\u503c\u4e0d\u5b58\u5728\uff0c\u56e0\u4e3a\u662f\u65e0\u7a77\u5c0f\u4e58\u65e0\u754c\u91cf<\/li>\n<li>\u6781\u9650\u5b58\u5728\u7684\u5145\u8981\u6761\u4ef6\u662f\u5de6\u53f3\u6781\u9650\u5b58\u5728\u4e14\u76f8\u7b49<\/li>\n<li>\u51fd\u6570\u5bfc\u6570\u7684\u5145\u8981\u6761\u4ef6\u662f\u5de6\u53f3\u5bfc\u6570\u5b58\u5728\u4e14\u76f8\u7b49<\/li>\n<li>\u5bfc\u6570\u5b58\u5728\u7684\u5fc5\u8981\u6761\u4ef6\u662f\u51fd\u6570\u8fde\u7eed<\/li>\n<\/ul>\n<h3>\u51fd\u6570\u7684\u6027\u6001<\/h3>\n<h4>\u51fd\u6570\u5355\u8c03\u6027<\/h4>\n<p>$f&#039;(x)&gt;0 =&gt; f(x)\u5355\u8c03\u589e$<\/p>\n<p>$f&#039;(x)&gt;=0 &lt;=&gt; f(x)\u5355\u8c03\u4e0d\u51cf$<\/p>\n<p>$f(x)\u5355\u8c03\u589e\u52a0 =&gt; f&#039;(x)&gt;=0$<\/p>\n<p>\u6781\u9650\u503c\u5927\u4e8e 0 \u53ef\u4ee5\u63a8\u51fa\u51fd\u6570\u503c\u5927\u4e8e 0<\/p>\n<p>\u6781\u9650\u503c &gt;=0\uff0c\u4e0d\u80fd\u63a8\u51fa\u51fd\u6570\u503c\u5927\u4e8e\u7b49\u4e8e 0($x^3$)<\/p>\n<p>\u51fd\u6570\u503c &gt;=0 \u6216\u8005 &gt;0\uff0c\u53ef\u4ee5\u63a8\u51fa\u6781\u9650\u503c\u5927\u4e8e 0<\/p>\n<h3>\u6570\u5217\u6536\u655b\u95ee\u9898<\/h3>\n<p>\u8bc1\u660e\u6570\u5217\u6536\u655b\u6709\u4e24\u79cd\u65b9\u6cd5<\/p>\n<ol>\n<li>\n<p>\u5355\u8c03\u6709\u754c\u6027\u51c6\u5219<\/p>\n<p>\u8bc1\u660e\u6570\u5217\u5355\u8c03\u589e\u6709\u4e0a\u754c\uff0c\u6216\u8005\u5355\u8c03\u51cf\u6709\u4e0b\u754c\uff0c\u5355\u8c03\u6027\u8bc1\u660e\u7528\u505a\u5dee\u6216\u8005\u505a\u6bd4<\/p>\n<\/li>\n<li>\n<p>\u5939\u903c\u5b9a\u7406<\/p>\n<p>\u901a\u8fc7\u653e\u7f29\u6765\u8ba1\u7b97\u5de6\u53f3\u8fb9\u754c\u7684\u6781\u9650\u8d8b\u4e8e\u540c\u4e00\u4e2a\u503c<\/p>\n<\/li>\n<\/ol>\n<h3>$1^{\u221e}$<\/h3>\n<p>\u5f53\u4f7f\u7528 $1^{\u221e}$ \u65f6\u5019\uff0c\u6ce8\u610f $limu^v$ \u4f7f\u7528\u8fc7\u7a0b\u4e2d\u9700\u8981\u4fdd\u8bc1\u6c42\u51fa\u7684\u7ed3\u679c\u4e0d\u662f\u65e0\u7a77\uff0c\u4e00\u822c\u4e0d\u8981\u7528\u5728\u5206\u6bcd\u5355\u72ec\u4f7f\u7528<\/p>\n<h3>\u6570\u5217\u6781\u9650<\/h3>\n<ul>\n<li>\u6570\u5217\u6c42\u6781\u9650\u8fc7\u7a0b\u4e2d\u8981\u628a $n-&gt;x$\uff0c\u628a\u6570\u5217\u6781\u9650\u6539\u5199\u6210\u51fd\u6570\u6781\u9650\uff0c\u624d\u80fd\u7528\u6d1b\u5fc5\u8fbe<\/li>\n<li>\u4e0d\u8981\u968f\u4fbf\u5728 $e$ \u4e0a\u505a\u7b49\u4ef7\u4ee3\u6362\uff0c\u56e0\u4e3a\u5728\u5206\u5f0f\u4e2d\uff0c\u5206\u5b50\u548c\u5206\u6bcd\u90fd\u6709 $e$\uff0c\u5982\u679c\u4e0d\u7edf\u4e00\u7684\u8bdd\uff0c\u76f4\u63a5\u6362\u5206\u5b50\u7684 $e$ \u7684\u6307\u6570\uff0c\u5c31\u76f8\u5f53\u4e8e\u7b49\u4ef7\u540e\u66f4\u6362 $e$ \u6307\u6570\u4e2d\u52a0\u51cf\u8fdb\u884c\u76f4\u63a5\u7b49\u4ef7\u4ee3\u6362\u4e86<\/li>\n<li>\u53ea\u6709\u5728\u6781\u9650\u7684\u975e\u96f6\u56e0\u5b50(\u4e0e\u6574\u4e2a\u4e16\u5b50\u662f\u76f8\u4e58\u7684)\u7684\u6781\u9650\u53ef\u4ee5\u5148\u6c42\u51fa\u6765<\/li>\n<\/ul>\n<ol>\n<li>\n<p>\u5224\u65ad\u6c42\u6570\u5217\u548c\u7684\u6781\u9650\u7528\u5939\u903c\u51c6\u5219\u8fd8\u662f\u5b9a\u79ef\u5206\u5b9a\u4e49<\/p>\n<p>\u5206\u6bcd\u7684\u53d8\u5316\u90e8\u5206\u548c\u4e3b\u4f53\u90e8\u5206\u4f5c\u6bd4\uff0c\u5982\u679c\u8d8b\u5411\u4e8e 0 \u662f\u6b21\u91cf\u7ea7\u90a3\u5c31\u4f7f\u7528\u5939\u903c\u5b9a\u7406\uff0c\u5982\u679c\u8d8b\u5411\u4e8e 1 \u90a3\u5c31\u662f\u540c\u91cf\u7ea7\u4f7f\u7528\u5b9a\u79ef\u5206\u7684\u5b9a\u4e49<\/p>\n<p>\u6b21\u91cf\u7ea7\u548c\u4e3b\u4f53\u90e8\u5206\u76f8\u6bd4\uff0c\u53ef\u4ee5\u5ffd\u7565\uff0c\u4e0d\u5f71\u54cd\u6781\u9650\u503c\uff0c\u4f46\u662f\u5982\u679c\u662f\u540c\u91cf\u7ea7\uff0c\u5c31\u4f1a\u5f71\u54cd\u6781\u9650\u503c\uff0c\u5939\u903c\u51c6\u5219\u65e0\u6cd5\u7cbe\u51c6\u6c42\u51fa<\/p>\n<\/li>\n<li>\n<p>$\\mathop{\\lim }\\limits_{{n \\rightarrow  \\infty }}\\sqrt[n]{{a}_{1}^{n} + {a}_{2}^{n} + \\cdots  + {a}_{m}^{n}} = \\mathop{\\max }\\limits_{{1 \\leq  i \\leq  m}}{a}_{i}$<\/p>\n<p>$\\mathop{\\lim }\\limits_{{n \\rightarrow  \\infty }}\\sqrt[n]{1 + {x}^{n} + {\\left( \\frac{{x}^{2}}{2}\\right) }^{n}},\\left( {x \\geq  0}\\right)$ \u4e2d\uff0c\u628a 1 \u53ef\u4ee5\u6362\u6210 $1,2,3.....,10000$ \u5c31\u7b49\u4ef7\u4e8e $1^n+1^n+....1^n$ \u7ed3\u679c\u8fd8\u662f\u7b49\u4ef7\u4e8e\u5728 $1,x,\\frac{x^2}{2}$ \u5728\u4e0d\u540c\u533a\u57df\u7684\u5927\u5c0f<\/p>\n<\/li>\n<\/ol>\n<h4>\u6570\u5217\u6781\u9650\u8bc1\u660e\u65b9\u6cd5\u603b\u7ed3<\/h4>\n<h5>\u6570\u5217\u6781\u9650\u8bc1\u660e\u5e38\u7528<\/h5>\n<ol>\n<li>\n<p>\u4e0d\u5b9a\u5f0f<\/p>\n<p>\u5bf9\u6570\u5217\u6c42\u6781\u9650\u8fc7\u7a0b\u548c\u51fd\u6570\u4e0d\u5b9a\u5f0f\u6781\u9650\u76f8\u540c\uff0c\u4f46\u4e00\u5b9a <strong>\u8981\u628a\u6570\u5217\u6781\u9650\u8f6c\u6362\u4e3a\u51fd\u6570\u6781\u9650\uff08\u5f52\u7ed3\u539f\u7406\uff09<\/strong> \uff0c\u5426\u5219\u4e0d\u80fd\u76f4\u63a5\u7528\u6d1b\u5fc5\u8fbe<\/p>\n<\/li>\n<li>\n<p>n \u9879\u548c\u7684\u6570\u5217\u6781\u9650<\/p>\n<ol>\n<li>\u5939\u903c\u539f\u7406<\/li>\n<li>\u5b9a\u79ef\u5206\u5b9a\u4e49<\/li>\n<\/ol>\n<\/li>\n<li>\n<p>n \u9879\u8fde\u4e58\u7684\u6570\u5217\u6781\u9650<\/p>\n<ol>\n<li>\u5939\u903c\u539f\u7406<\/li>\n<li>\u53d6\u5bf9\u6570\u5316\u4e3a n \u9879\u548c<\/li>\n<\/ol>\n<\/li>\n<li>\n<p>\u9012\u63a8\u5173\u7cfb\u5b9a\u4e49\u7684\u6570\u5217<\/p>\n<ol>\n<li>\u5148\u8bc1\u6570\u5217\u6536\u655b\uff0c\u5728\u6c42\u6781\u9650<\/li>\n<li>\n<p>\u5148\u6c42\u6781\u9650\uff0c\u518d\u8bc1\u6781\u9650\u4e3a\u6240\u6c42\u6781\u9650\uff08\u4e00\u822c\u4f7f\u7528\u538b\u7f29\u6620\u5c04\u539f\u7406\uff09<\/p>\n<p>\u4f7f\u7528\u538b\u7f29\u6620\u5c04\u539f\u7406\u65f6\uff0c\u82e5\u6c42\u51fa\u7684\u6781\u9650\u503c\u6bd4\u8f83\u590d\u6742\uff0c\u5219\u4e0d\u628a\u6781\u9650\u5e26\u5165\uff0c\u800c\u662f\u4f7f\u7528 a \u5e26\u5165<\/p>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h5>\u5355\u8c03\u6027\u5224\u5b9a\u5e38\u7528\u65b9\u6cd5<\/h5>\n<p>\u5355\u8c03\u6027\u5224\u5b9a\u5e38\u7528\u6709\u4e09\u79cd\u65b9\u6cd5:<\/p>\n<p>(1) \u82e5 ${x}_{n + 1} - {x}_{n} \\geq  0\\left( { \\leq  0}\\right)$ ,\u5219 $\\left\\{  {x}_{n}\\right\\}$ \u5355\u8c03\u589e (\u5355\u8c03\u51cf);<\/p>\n<p>(2) \u8bbe $\\left\\{  {x}_{n}\\right\\}$ \u4e0d\u53d8\u53f7,<\/p>\n<p>\u82e5 ${x}_{n} &gt; 0$ ,\u5219\u5f53 $\\frac{{x}_{n + 1}}{{x}_{n}} \\geq  1\\left( { \\leq  1}\\right)$ \u65f6, $\\left\\{  {x}_{n}\\right\\}$ \u5355\u8c03\u589e (\u5355\u8c03\u51cf);<\/p>\n<p>\u82e5 ${x}_{n} &lt; 0$ ,\u5219\u5f53 $\\frac{{x}_{n + 1}}{{x}_{n}} \\geq  1\\left( { \\leq  1}\\right)$ \u65f6, $\\left\\{  {x}_{n}\\right\\}$ \u5355\u8c03\u51cf (\u5355\u8c03\u589e);<\/p>\n<p>(3) \u8bbe\u6570\u5217 $\\left\\{  {x}_{n}\\right\\}$ \u7531 ${x}_{1} = a,{x}_{n + 1} = f\\left( {x}_{n}\\right) \\left( {n = 1,2,\\cdots }\\right) ,{x}_{n} \\in  I$ \u6240\u786e\u5b9a,<\/p>\n<p>\u82e5 $f\\left( x\\right)$ \u5728 $I$ \u4e0a\u5355\u8c03\u589e,\u5219<\/p>\n<p>\u5f53 ${x}_{1} \\leq  {x}_{2}$ \u65f6, $\\left\\{  {x}_{n}\\right\\}$ \u5355\u8c03\u589e; \u5f53 ${x}_{1} \\geq  {x}_{2}$ \u65f6, $\\left\\{  {x}_{n}\\right\\}$ \u5355\u8c03\u51cf;<\/p>\n<p>\u82e5 $f\\left( x\\right)$ \u5728 $I$ \u4e0a\u5355\u8c03\u51cf,\u5219 $\\left\\{  {x}_{n}\\right\\}$ \u4e0d\u5355\u8c03.<\/p>\n<h3>\u65e0\u7a77\u5c0f\u9636\u7684\u6bd4\u8f83<\/h3>\n<ol>\n<li>\u5b9a\u4e49\u6cd5\u76f4\u63a5\u6bd4\u8f83\uff08\u51fd\u6570\u4e24\u4e24\u8fdb\u884c\u6bd4\u9636\uff09<\/li>\n<li>\u7b49\u4ef7\u65e0\u7a77\u5c0f\u66ff\u6362\u540e\u8fdb\u884c\u7b80\u5355\u6bd4\u9636<\/li>\n<li>\u6d1b\u5fc5\u8fbe<\/li>\n<li>\u6cf0\u52d2\u5c55\u5f00<\/li>\n<li>\u5229\u7528\u5b9a\u4e49\u6c42\u51fa\u51fd\u6570\u7684\u9636\u6570(\u7528 $x^k$ \u76f8\u6bd4\u4e3a 1\uff0c\u90a3\u4e48\u539f\u51fd\u6570\u4e3a k \u9636\u6570)<\/li>\n<li>\u6c42\u5bfc\u5b9a\u9636\u6570\uff08\u6bcf\u4e2a\u90fd\u8fdb\u884c\u6c42\u5bfc\uff0c\u76f8\u5f53\u4e8e\u964d\u4e86\u4e00\u9636\uff0c\u518d\u7528\u7b49\u4ef7\u65e0\u7a77\u5c0f\u66ff\u6362\uff09<\/li>\n<li>\u5229\u7528\u82e5 $\\mathop{\\lim }\\limits_{{x \\rightarrow  0}}\\frac{f\\left( x\\right) }{g\\left( x\\right) } = 1  ,\u5219  {\\int }_{0}^{\\varphi \\left( x\\right) }f\\left( t\\right) \\mathrm{d}t \\sim  {\\int }_{0}^{\\varphi \\left( x\\right) }g\\left( t\\right) \\mathrm{d}t  ,\u5176\u4e2d  \\mathop{\\lim }\\limits_{{x \\rightarrow  0}}\\varphi \\left( x\\right)  = 0$(\u4e0d\u5b9a\u79ef\u5206\u6362\u88ab\u79ef\u51fd\u6570)<\/li>\n<li>\u7ed3\u8bba: \u82e5 $f\\left( x\\right)$ \u5728 x=0 \u7684\u67d0\u90bb\u57df\u5185\u8fde\u7eed,\u4e14\u5f53 $x \\rightarrow  0$ \u65f6 $f\\left( x\\right)$ \u662f $x$ \u7684 $m$ \u9636\u65e0\u7a77\u5c0f,$\\varphi \\left( x\\right)$ \u662f $x$ \u7684 $n$ \u9636\u65e0\u7a77\u5c0f,\u5219\u5f53 $x\\rightarrow0$ \u65f6 $F\\left( x\\right)={\\int }_{0}^{\\varphi \\left( x\\right) }f\\left( t\\right) \\mathrm{d}t$ \u662f $x$ \u7684 $n\\left( {m + 1}\\right)$ \u9636\u65e0\u7a77\u5c0f) <strong>\u4e5f\u5c31\u662f\u4e0a\u9650\u51fd\u6570\u9636\u6570*\uff08\u88ab\u79ef\u51fd\u6570\u9636\u6570 +1\uff09<\/strong><\/li>\n<\/ol>\n<h3>\u95f4\u65ad\u70b9\u7684\u7c7b\u578b\u5224\u65ad<\/h3>\n<ul>\n<li>\u5f53 f(x)\u6ca1\u6709\u663e\u6027\u7ed9\u51fa\u65f6\u5019\uff0c\u8981\u6ce8\u610f\u5148\u6c42\u51fa\u8868\u8fbe\u5f0f\uff0c\u5728\u6c42\u8868\u8fbe\u5f0f\u7684\u8fc7\u7a0b\u4e2d\u53ef\u80fd\u4f1a\u5148\u9047\u5230\u4e0d\u6ee1\u8db3\u5b9a\u4e49\u57df\u7684\u60c5\u51b5\uff0c\u4e5f\u5c31\u662f\u6709\u4e9b\u60c5\u51b5\u4e0b\u8868\u8fbe\u5f0f\u4e0d\u5b58\u5728\uff0c\u8981\u5728\u8868\u8fbe\u5f0f\u5b58\u5728\u7684\u6761\u4ef6\u4e0b\u8fdb\u884c\u8ba8\u8bba\uff0c\u5148\u6c42\u51fa\u8868\u8fbe\u5f0f<\/li>\n<li>\u8868\u8fbe\u5f0f\u5728\u53d8\u578b\u8fc7\u7a0b\u4e2d\uff0c\u53ef\u80fd\u7ed9\u4e00\u4e2a\u5206\u6bcd\u4e58\u4e86\u4e00\u4e2a\u5f0f\u5b50\u8ba9\u5206\u5f0f\u6d88\u53bb\uff0c\u6ce8\u610f\u8fd9\u4e2a\u8fc7\u7a0b\u5b9a\u4e49\u57df\u7684\u53d8\u5316\uff0c\u53ef\u80fd\u5bfc\u81f4\u6709\u95f4\u65ad\u70b9\u7684\u51fd\u6570\u88ab\u5316\u7b80\u4e3a\u6ca1\u6709\u95f4\u65ad\u70b9\u6216\u8005\u95f4\u65ad\u70b9\u51cf\u5c11<\/li>\n<\/ul>\n<h3>\u4ecb\u503c\u5b9a\u7406\u3001\u6700\u503c\u5b9a\u7406\u53ca\u96f6\u70b9\u5b9a\u7406\u7684\u8bc1\u660e<\/h3>\n<p>\u5982\u679c\u9047\u5230\u51fd\u6570\u503c$f(x_0)$\u548c\u6781\u9650\u503c$lim_{x-&gt;x_0}f(x)$\uff0c\u8981\u80fd\u60f3\u5230\u6781\u9650\u7684\u4fdd\u53f7\u6027\u6216\u8005\u4e2d\u503c\u5b9a\u7406<\/p>\n<p>\u8bc1\u660e\u51fd\u6570\u503c\u7b49\u4e8e 0 \u8981\u80fd\u60f3\u5230\u96f6\u70b9\u5b9a\u7406<\/p>\n<p>\u82e5\u7ed9\u5b9a\u503c\uff0c\u8981\u80fd\u60f3\u5230\u6700\u503c\u5b9a\u7406<\/p>\n<h3>\u590d\u5408\u51fd\u6570\u6c42\u5bfc<\/h3>\n<p>\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219\uff1a<\/p>\n<ol>\n<li>\u8bfb\u8005\u4e0d\u96be\u9a8c\u8bc1\uff1a\u82e5 ( $y = g(f(x))$ ) \uff0c\u5219 ( $\\frac{\\mathrm{d}y}{\\mathrm{d}x} \\Bigg|_{x=0}$ ) \u4e0d\u5b58\u5728\u3002<\/li>\n<li>\u8bbe$y = f(u), u = g(x), u_0 = g(x_0)$\uff0c\u5982\u679c $g&#039;(x_0)$\u548c$f&#039;(u_0)$\u90fd\u5b58\u5728\uff0c\u5219$y = f(g(x))$\u5728 ($x_0$ ) \u5904\u53ef\u5bfc\uff0c\u4e14$\\frac{\\mathrm{d}y}{\\mathrm{d}x} \\Bigg|_{x=x_0} = f&#039;(u_0) \\cdot g&#039;(x_0)$ \uff0c\u5982\u679c$g&#039;(x_0)$\u548c$f&#039;(u_0)$\u81f3\u5c11\u6709\u4e00\u4e2a\u4e0d\u5b58\u5728\uff0c\u5219$y=f(g(x))\ufeff$\u5728$x_0$\u5904\u5e76\u975e\u4e00\u5b9a\u4e0d\u53ef\u5bfc\uff0c\u6b64\u65f6\uff0c\u5148\u6c42\u51fa\u590d\u5408\u51fd\u6570$y = f(g(x))$\u7684\u8868\u8fbe\u5f0f\uff0c\u7136\u540e\u518d\u8fdb\u4e00\u6b65\u8003\u67e5$y = f(g(x))$\u5728$x_0$\u5904\u7684\u53ef\u5bfc\u6027\u3002<\/li>\n<\/ol>\n<h3><a href=\"assets\/25\u6b66\u5fe0\u7965\u300a\u9ad8\u6570\u8f85\u5bfc\u8bb2\u4e49\u300b-20240731091616-x19jvdx.pdf#page=72\">25\u6b66\u5fe0\u7965\u300a\u9ad8\u6570\u8f85\u5bfc\u8bb2\u4e49\u300b.pdf - p72 - \u9690\u51fd\u6570\u6c42\u5bfc\u6cd5<\/a><\/h3>\n<p>\u4e00\u9636\u5bfc\u6570\u4ee3\u516c\u5f0f \uff0c\u4e8c \u9636\u5bfc\u6570\u5229\u7528$\\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2}=\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Big(\\frac{y^{&#039;}(t)}{x^{&#039;}(t)}\\Big)\\frac{1}{x^{&#039;}(t)}$\uff0c\u6ce8\u610f\uff0c\u4e8c\u9636\u5bfc\u662f\u5bf9x\u6c42\u5bfc\uff0c\u4f46\u662f\u4e0a\u9762\u662f\u5173\u4e8et\u7684\u5bfc\u6570\uff0c\u5bf9x\u4e0d\u80fd\u76f4\u63a5\u6c42\uff0c\u6240\u4ee5\u8981\u5148\u5bf9t\u6c42\u5bfc\uff0c\u7136\u540e\u518d$\\frac{dt}{dx}=\\frac{1}{\\frac{dx}{dt}}$<\/p>\n<h3><a href=\"assets\/25\u6b66\u5fe0\u7965\u300a\u9ad8\u6570\u8f85\u5bfc\u8bb2\u4e49\u300b-20240731091616-x19jvdx.pdf#page=73\">25\u6b66\u5fe0\u7965\u300a\u9ad8\u6570\u8f85\u5bfc\u8bb2\u4e49\u300b.pdf - p73 - \u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\u6cd5<\/a><\/h3>\n<p>\u5728\u53c2\u6570\u65b9\u7a0b\u6d89\u53ca\u5230\u6c42\u659c\u7387\u7684\u65b9\u7a0b\u4e2d\uff0c\u56e0\u4e3a\u659c\u7387\u662f$\\frac{dy}{dx}$,\u53c2\u6570\u65b9\u7a0b\u662f\u5173\u4e8et\u7684\u65b9\u7a0b\uff0c\u6240\u4ee5\u8981\u6ce8\u610f\u5c06 <strong>\u5bf9t\u7684\u5bfc\u6570\u6362\u6210\u5bf9x\u7684<\/strong> \uff0c\u6ce8\u610f\uff0c\u4e8c\u9636\u5bfc\u4e2d\uff0c$\\frac{dy}{dx}$\u4e5f\u5c31\u662f$\\frac{y&#039;(t)}{x&#039;(t)}$,$\\frac{d{\\frac{y(t)}{x(t)}}}{dx^2}$\u5c31\u662f$\\frac{y&#039;(t)}{x&#039;(t)}$\u518d\u6b21\u5bf9x\u6c42\u5bfc\uff0c\u7136\u540e\u518d$\\frac{dt}{dx}$\u4e5f\u5c31\u662f$\\frac{1}{\\frac{dt}{dx}}$\uff0c\u603b\u7684\u6765\u8bf4\uff0c\u516c\u5f0f\u6574\u7406\u540e\u5c31\u662f$\\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2}=\\frac{y^{\\prime\\prime}(t)x^{\\prime}(t)-x^{\\prime\\prime}(t)y^{\\prime}(t)}{x^{\\prime3}(t)}$<\/p>\n<h3>\u53cd\u51fd\u6570\u7684\u5bfc\u6570<\/h3>\n<ul>\n<li>\n<p>\u53cd\u51fd\u6570\u7684\u5bfc\u6570\uff0c\u56e0\u4e3a$\\varphi^{&#039;}(y)=\\frac{\\mathrm{d}x}{\\mathrm{d}y}=\\frac{1}{\\frac{\\mathrm{d}y}{\\mathrm{d}x}}=\\frac{1}{f^{&#039;}(x)}$,$\\varphi&#039;&#039;(y)=\\frac{\\mathrm{d}}{\\mathrm{d}x}\\biggl[\\frac{1}{f&#039;(x)}\\biggr]\\bullet\\frac{\\mathrm{d}x}{\\mathrm{d}y}=-\\frac{f&#039;&#039;(x)}{\\bigl[f&#039;(x)\\bigr]^2}\\bullet\\frac{1}{f&#039;(x)}$<\/p>\n<p>\u6ce8\u610f\uff1a$\\frac{dx}{dy}=\\frac{1}{\\frac{dy}{dx}},\u4f46\\frac{d^{2}x}{dy^{2}}\\neq\\frac{1}{\\frac{d^{2}y}{dx^{2}}}$<\/p>\n<\/li>\n<\/ul>\n<h3>\u5e38\u7528\u7684\u9ad8\u9636\u5bfc\u6570<\/h3>\n<ol>\n<li>$(e^{ax+b})^{(n)}=a^ne^{ax+b}$<\/li>\n<li>$\\sin(ax+b)^{(n)}=a^{n}\\sin(ax+b+\\frac{n\\pi}2)$<\/li>\n<li>$\\cos(ax+b)^{(n)}=a^{n}\\cos(ax+b+\\frac{n\\pi}2)$<\/li>\n<li>$\\ln(ax+b)^{(n)}=(-1)^{n-1}(n-1)!\\frac{a^n}{(ax+b)^n}$<\/li>\n<li>$(\\frac1{ax+b})^{(n)}=(-1)^nn!\\frac{a^n}{(ax+b)^{n+1}}$<\/li>\n<li>$(\\frac{c}{ax+b})^{(n)}=(-1)^{n}cn!\\frac{a^{n}}{(ax+b)^{n+1}}$<\/li>\n<\/ol>\n<h3>\u5e38\u89c1\u7684\u8f6c\u6362\uff1a\u6b63\u5f26\u51fd\u6570\u548c\u4f59\u5f26\u51fd\u6570\u4e09\u89d2\u6052\u7b49\u5f0f\u8f6c\u6362<\/h3>\n<ol>\n<li><strong>\u548c\u89d2\u516c\u5f0f<\/strong>:<br \/>$\\sin(x+y) = \\sin x \\cos y + \\cos x \\sin y$<br \/>$\\cos(x+y) = \\cos x \\cos y - \\sin x \\sin y$<\/li>\n<li><strong>\u5dee\u89d2\u516c\u5f0f<\/strong>:<br \/>$\\sin(x-y) = \\sin x \\cos y - \\cos x \\sin y$<br \/>$\\cos(x-y) = \\cos x \\cos y + \\sin x \\sin y$<\/li>\n<li><strong>\u500d\u89d2\u516c\u5f0f<\/strong>:<br \/>$\\sin 2x = 2 \\sin x \\cos x$<br \/>$\\cos 2x = \\cos^2 x - \\sin^2 x = 2 \\cos^2 x - 1 = 1 - 2 \\sin^2 x$<\/li>\n<li><strong>\u534a\u89d2\u516c\u5f0f<\/strong>:<br \/>$\\sin \\frac{x}{2} = \\pm \\sqrt{\\frac{1 - \\cos x}{2}}$<br \/>$\\cos \\frac{x}{2} = \\pm \\sqrt{\\frac{1 + \\cos x}{2}}$<\/li>\n<li><strong>\u4f59\u5f26\u5230\u6b63\u5f26\u7684\u8f6c\u6362<\/strong>:<br \/>$\\sin x = \\cos \\left(\\frac{\\pi}{2} - x \\right)$<\/li>\n<li><strong>\u6b63\u5f26\u5230\u4f59\u5f26\u7684\u8f6c\u6362<\/strong>:<br \/>$\\cos x = \\sin \\left(\\frac{\\pi}{2} - x \\right)$<\/li>\n<li><strong>\u6b63\u5f26\u548c\u4f59\u5f26\u7684\u5468\u671f\u6027<\/strong>:<br \/>$\\sin(x + 2\\pi k) = \\sin x$<br \/>$\\cos(x + 2\\pi k) = \\cos x$<br \/>\u5176\u4e2d ( k ) \u662f\u4efb\u610f\u6574\u6570\u3002<\/li>\n<li><strong>\u6b63\u5f26\u548c\u4f59\u5f26\u7684\u5947\u5076\u6027<\/strong>:<br \/>$\\sin(-x) = -\\sin x$\uff08\u5947\u51fd\u6570\uff09<br \/>$\\cos(-x) = \\cos x$\uff08\u5076\u51fd\u6570\uff09<\/li>\n<\/ol>\n<h3>\u9ad8\u9636\u5bfc\u6570<\/h3>\n<p>\u524d\u4e24\u79cd\u4e3b\u8981\u7528\u6765\u6c42n\u9636\u5bfc\u51fd\u6570\uff0c\u6cf0\u52d2\u516c\u5f0f\u7528\u6765\u6c42\u67d0\u70b9\u7684\u5bfc\u6570<\/p>\n<ol>\n<li>\u516c\u5f0f\u6cd5<\/li>\n<li>\u5f52\u7eb3\u6cd5<\/li>\n<li>\u6cf0\u52d2\u516c\u5f0f<\/li>\n<\/ol>\n<p>\u200d<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5e38\u7528\u516c\u5f0f \u57fa\u672c\u6c42\u5bfc\u516c\u5f0f ${\\left( {x}^{\\alpha }\\right) }^{\\prime } = \\alpha {x}^{\\alpha - 1}\\left( {\\alpha \\text{ \u4e3a\u5e38\u6570 }}\\right)$ ${\\left( {a}^{x}\\right) }^{\\prime } = {a}^{x}\\ln a\\left( {a &gt; 0,a \\neq 1}\\right)$ ${\\left( {\\mathrm{e}}^{x}\\right) }^{\\prime } = {\\mathrm{e}}^{x}$ ${\\left( {\\log }_{a}x\\right) }^{\\prime } = \\frac{1}{x\\ln a}\\left( {a &gt; 0,a \\neq 1}\\right)$ ${\\left( \\ln \\left| x\\right| \\right) }^{\\prime } = [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[68],"tags":[],"class_list":["post-292","post","type-post","status-publish","format-standard","hentry","category-68"],"_links":{"self":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts\/292","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=292"}],"version-history":[{"count":0,"href":"https:\/\/solost.cn\/index.php?rest_route=\/wp\/v2\/posts\/292\/revisions"}],"wp:attachment":[{"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=292"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=292"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/solost.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}