\u8bbe\u51fd\u6570$y=f(x)$\u5728\u70b9$x_0$\u7684\u67d0\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u82e5<\/p>\n
$$ <\/p>\n \u5b58\u5728\uff0c\u5219\u79f0\u51fd\u6570$f(x)$\u5728\u70b9$x_0$\u5904\u53ef\u5bfc\uff0c\u5e76\u79f0\u6b64\u6781\u9650\u4e3a$y=f(x)$\u5728\u70b9$x_0$\u7684\u5bfc\u6570\u3002\u8bb0\u4f5c<\/p>\n $$ \u8bbe$u=u(x),\\,v=v(x)$\u90fd\u53ef\u5bfc\uff0c\u5219<\/p>\n $ $ $ $ \u5982\u679c\u51fd\u6570$x=f(y)$\u5728\u533a\u95f4$I_y$\u5185\u5355\u8c03\u3001\u53ef\u5bfc\u4e14$f'(y) \\neq 0$\uff0c\u5219\u5b83\u7684\u53cd\u51fd\u6570$y=f'(x)$\u5728\u533a\u95f4$I_x=\\{x|x=f(y), \\, y \\in I_y\\}$\u5185\u4e5f\u53ef\u5bfc\uff0c\u4e14\u6709<\/p>\n $$ \u8bbe$y=f(u),u=\\varphi(x)$\uff0c\u5219\u590d\u5408\u51fd\u6570$y=f[\\varphi(x)]$\u7684\u5bfc\u6570\u4e3a<\/p>\n $$ \u82e5\u51fd\u6570$y=f(x)$\u7684\u5bfc\u6570$y’=f'(x)$\u53ef\u5bfc\uff0c\u5219\u79f0$f'(x)$\u7684\u5bfc\u6570\u4e3a$f(x)$\u7684\u4e8c\u9636\u5bfc\u6570\uff0c\u8bb0\u4f5c$y”$\u6216$\\frac{d^2y}{dx^2}$\uff0c\u5373<\/p>\n $$ \u7c7b\u4f3c\u7684\uff0c\u4e8c\u9636\u5bfc\u6570\u7684\u5bfc\u6570\u79f0\u4e3a\u4e09\u9636\u5bfc\u6570\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c$n-1$\u9636\u5bfc\u6570\u7684\u5bfc\u6570\u79f0\u4e3a$n$\u9636\u5bfc\u6570\uff0c\u5206\u522b\u8bb0\u4f5c<\/p>\n $$ $$ \u4e8c\u9636\u548c\u4e8c\u9636\u4ee5\u4e0a\u7684\u5bfc\u6570\u7edf\u79f0\u4e3a\u9ad8\u9636\u5bfc\u6570\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u5bfc\u6570\u7684\u5b9a\u4e49 \u8bbe\u51fd\u6570$y=f(x)$\u5728\u70b9$x_0$\u7684\u67d0\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u82e5 $$ \\lim \\limits_{x \\ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2131","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2131","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2131"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2131\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2131"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2131"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\lim \\limits_{x \\to x_0} \\frac{f(x)-f(x_0)}{x-x0} = \\lim \\limits<\/em>{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}
\n$$<\/p>\n
\ny’\\left|
\n\\right.
\nx=x_0 \\\\
\nf'(x_0) \\\\
\n\\frac{dy}{dx}\\left|
\n\\right.
\nx=x_0 \\\\
\n\\frac{df(x)}{dx}\\left|
\n\\right.
\nx=x_0
\n$$<\/p>\n\u5bfc\u6570\u516c\u5f0f\u4e0e\u57fa\u672c\u6c42\u5bfc\u6cd5\u5219<\/h2>\n
\u5e38\u6570\u548c\u57fa\u672c\u521d\u7b49\u51fd\u6570\u7684\u5bfc\u6570\u516c\u5f0f<\/h3>\n
\n\n
\n \n<\/th>\n <\/th>\n<\/tr>\n<\/thead>\n \n $(C)’=0$<\/td>\n $(x^{\\mu})’=\\mu x^{\\mu-1}$<\/td>\n<\/tr>\n \n $(\\sin x)’=\\cos x$<\/td>\n $(\\cos x)’=-\\sin x$<\/td>\n<\/tr>\n \n $(\\tan x)’=\\sec^2 x$<\/td>\n $(\\cot x)’=-\\csc^2x$<\/td>\n<\/tr>\n \n $(\\sec x)’=\\sec x \\tan x$<\/td>\n $(\\csc x)’=-\\csc x \\cot x$<\/td>\n<\/tr>\n \n $(a^x)’=a^x \\operatorname{ln}a$<\/td>\n $(e^x)’=e^x$<\/td>\n<\/tr>\n \n $(\\operatorname{log}_ax)’=\\frac{1}{x\\operatorname{ln}a}$<\/td>\n ($\\operatorname{ln}x)’=\\frac{1}{x}$<\/td>\n<\/tr>\n \n $(\\operatorname{arcsin}x)’=\\frac{1}{\\sqrt{1-x^2}}$<\/td>\n ($\\operatorname{arccos}x)’=-\\frac{1}{\\sqrt{1-x^2}}$<\/td>\n<\/tr>\n \n $(\\operatorname{arctan}x)’=\\frac{1}{\\sqrt{1+x^2}}$<\/td>\n ($\\operatorname{arccot}x)’=-\\frac{1}{\\sqrt{1+x^2}}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u51fd\u6570\u7684\u548c\u3001\u5dee\u3001\u79ef\u3001\u5546\u7684\u6c42\u5bfc\u6cd5\u5219<\/h3>\n
\n(1) \\, (u\u00b1v)’ = u’ \u00b1 v’
\n$<\/p>\n
\n(2) \\, (Cu)’ = Cu’
\n$<\/p>\n
\n(3) \\, (uv)’ = u’v \u00b1 uv’
\n$<\/p>\n
\n(4) \\, (\\frac{u}{v})’ = \\frac{u’v – uv’}{v^2} \\, (v \\neq 0)
\n$<\/p>\n\u53cd\u51fd\u6570\u7684\u6c42\u5bfc\u6cd5\u5219<\/h3>\n
\n[f^{-1}(x)]’ = \\frac{1}{f'(y)} \\, \u6216 \\, \\frac{dy}{dx}=\\frac{1}{\\frac{dx}{dy}}
\n$$<\/p>\n\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219<\/h3>\n
\n\\frac{dy}{dx} = \\frac{dy}{du}\u00b7\\frac{du}{dx} = f'(u)\u00b7\\varphi'(x)
\n$$<\/p>\n\u9ad8\u9636\u5bfc\u6570<\/h2>\n
\ny”=(y’)’ \\, \u6216 \\, \\frac{d^2y}{dx^2}=\\frac{d}{dx}(\\frac{dy}{dx})
\n$$<\/p>\n
\n\\frac{d^3y}{dx^3},\\,\\frac{d^4y}{dx^4},\u00b7\u00b7\u00b7,\\frac{d^ny}{dx^n}
\n$$<\/p>\n
\n\\frac{d^ny}{dx^n}=\\frac{d}{dx}(\\frac{d^{n-1}y}{dx^{n-1}})
\n$$<\/p>\n