积分微分点积差积

xxfxxfxydxdyxfyxx )()(limlim)(00tan)(xf yydtdyytdyddydydtddtydy22dxdyy tytytxxydtdytydtdy (16)(arctan x)211x。(1)(C)0，(2)(xm)m xm1，(3)(sin x)cos x，(4)(cos x)sin x，(5)(tan x)sec2x，(6)(cot x)csc2x，(7)(sec x)sec x tan x，(8)(csc x)csc x cot x，(9)(ax)ax ln a，(10)(ex)ex，(12)(ln x)x1，(13)(arcsin x)211x，(14)(arccos x)211x，(15)(arctan x)211x，(11)(log a x)axln1(a0,a1)，函数的和、差、积、商的求导法则：函数的和、差、积、商的求导法则：(1)(u v)=u v，(2)(Cu)=Cu(C是常数)，(3)(uv)=uv+u v，(4)2)(vvuvuvu(v0)。复合函数的求导法则：复合函数的求导法则：f 1(y)(1xf (f(x)0)。反函数求导法：反函数求导法：求导法则求导法则 dxdududydxdy，或 yyuux，其中 y=f(u)，u=(x)。dxdududydxdy，或 yyuux。解：解：函数y=lntan x是由y=ln u，u=tan x复合而成，dxdududydxdyxxxu22seccotsec1 xxcossin1。dxdududydxdyxxxu22seccotsec1dxdududydxdyxxxu22seccotsec1 例例1 1y=lntan x，求dxdy。dxdududydxdy3332xuxexedxdududydxdy3332xuxexedxdududydxdy3332xuxexe。解解：函数3xey 是由 yeu，ux3 复合而成，例例2 2y=3xe，求dxdy。dxdududydxdy2222)1()2()1(2cosxxxu 222212cos)1()1(2xxxx。解解：212sinxxy是由 ysin u，212xxu复合而成，dxdududydxdy2222)1()2()1(2cosxxxu 例例3 3212sinxxy，求dxdy。)()(dxdxxAy0 dxxAdy)(dxxfdy)(dxdyxfyxG)()(CxydydxdxdydxyxG)()()()()()(aFbFxFdxxfbabak x C(k是常数)，arctan x C，arcsin x C，ln|x|C，sin x C，cos x C，(1)k dx (2)x mdx (3)x1dx (4)211xdx (5)211xdx (7)sin x dx11mx m1C，(6)cos x dx(10)sec x tan x dxsec x C，(11)csc x cot x dx csc x C，(12)e x dx e 2C，(13)a x dx(14)sh x dx ch x C，(15)ch x dx sh x C(8)x2cos1dx(9)x2sin1dx dx sec 2x dxtan xC，dx csc 2x dxcot x C，sec x tan x dxsec x C，csc x cot x dx csc x C，e x dx e 2C，a x dx aaxlnC，sh x dx ch x C，ch x dx sh x C f(x)g(x)dxf(x)dxg(x)dxkf(x)dxkf(x)dx(k 是常数，k 0)25xdx 521xdx dx 25xdx 521xdx7227x53223xC72x 3x310 xxC23)1(xx dx 223133xxxxdx(x 3 x3 21x)dx x dx 3dx 3x1dx 21xdx21x 23x 3 ln|x|x1 C dxxxdxxx)()(2125255dxxx231)(2x ex dx(2e)x dx)2ln()2(eexC)2ln()2(eexC 2ln12xxe C)1(122xxxxdx)1()(122xxxxdx(211x x1)dx211xdx x1dx arctan x ln|x|C 24111xxdx22211)1)(1(xxxdx(x2 1 211x)dx x 2dx dx 211xdx31x 3x arctan x C dxexx2dxxxxx)(2211dxxx241AkBAjBAiBAkBjBiBkAjAiAAzzyyxxzyxzyx)()()()()(BkBAjBAiBAkBjBiBkAjAiAAzzyyxxzyxzyx)()()()()(BABBABAA AkAjAiAkAjAiAAzyxzyx )(A 0 AA0 AB),(BA),cos(BABABABABABAzzyyxxkBABAjBABAiBABABBBAAAkjiBAxyyxzxyzyzzyzyxzyx)()()(),sin(BABABABA)(tfttfttft)()(lim0)(tfdttfdtf)()()(tf