定理,则可导,且设严格单调函数0)()(yyx也可导,并且其反函数)(xfy.1,)(1)(dydxdxdyyxf或法则:反函数的导数是原来函数的导数的倒数.2.4.3反函数求导法则证明:xyyx1yyyy)()(1.0,0)(yxxfy可知,的严格单调性和连续性由因此xyxfx0lim)(yyyyy)()(lim10.)(1y.证毕).22(sinarcsinyyxxy的反函数是解:).22(,0cosyydydxdydxdxdy1.arcsin),1,1(的导数求函数设例:xyxycos1211xy2sin11.,arctandxdyxy求设例:解:).22(tanarctanyyxxy的反函数是函数.0sec2yyydydx)(tandydxdxdy1y2sec1y2tan11.112x,2cotarctan2arccosarcsinxarcxxx以及利用得211)(arccosxx.11)cot(2xxarc.)1,0(的导数求函数例:aaayx解:).0(logyyxayax的反函数是aydydxln1.ln1aaxdydxdxdy1.lnaax.)(xxee特别的,1.10xx)(aaaxxln)(.11基本求导公式2.)(.12xxee.11)(arcsin.132xx.11)(arccos.142xx.11)(arctan.152xx.11)cot(.162xxarc.)(.17chxshx.)(.18shxchx.11))1(ln()(.1922xxxarshx定理1(链式法则)且处也可导,在点则也可导处在对应点处可导,在点若xxfyxuufyxxu)(,)()()(xuuyxydddddd且)()(xufdxdy即2.4.4复合函数求导法则证明:)()()(xufyxxu在对应于且处可导,在点由故处也可导,的点u)1(,)()(uuuufy)2(,)()(xxxxu即时的无穷小,和分别是和其中00)()(xuxu.0)(lim,0)(lim00xuxu得式式代入将,)1()2(.)()()()()(uuxxufxxufy得将等式两边同除以,x.)()()()()(xuuxufxufxy续的性质处可导且函数可导必连在点根据xxu)(从而有,时有当知道,00ux,0)(lim)(lim00uuux.lim)(lim)(lim)()()(lim0000xuuxufxufxyxuxx于是.)()(xufdxdy因此)())(()('''xxfxf或可写成.),1sin(2dxdyxy求设例:)1cos(22cos2xxxudxdududydxdy.1,sin2xuuy设解:).1cos(2)1)(1cos())1(sin(2222xxxxx)()(lnxex例:xexln)ln(lnxex.1x)()(lnaxxea)ln(lnaxeaxaeaxlnln.lnaax))tan(sin(xx)tan(sin)tan(sinxxxx))(sin(sinsec)tan(sin2xxxx.cos)(sinsec)tan(sin2xxxx)1(22xx22222112)1(12xxxxxx222222)1()1(1)(xxxxx.)1(22323xxx推广均可导,设)(),(),(.1xvvuufy.)]}([{dxdvdvdududydxdyxfy的导数为则复合函数.)),12(ln(tan33yxy求解))12(ln(tan332xy))12(ln(sec32x236121xx)).12(ln(sec))12(ln(tan1218323232xxxx例.,3)sin(cos22yyxx求设函数解:3ln3)sin(cos22xxy))sinsin(()sincos(222xxxx).2cos(sin22xxxx例求下列函数的导数:,ln)1(xy)cos(sin)3(2xxy322)2()4(xxy)1ln()5(2xxy.2)6(lnxxy课内练习xxy22tan)2(.1)(lnxx.0,1)(lnxxx.0)ln(,0lnlnxxxxxy.0,1))(ln(xxxxxxxxx4222tansectan2tan2.tan22xx,ln)1(xyxxy22tan)2()cos(sin)3(2xxyy)cos)(cossin(2xxxx)sin)(coscossin(2xxxxx322)2()4(xxyy6222232)2(2)2(3)2(2xxxxxx.)2()1(4422xxxy221)1(1xx2211xxx)1ln()5(2xxy.2)6(lnxxy221)1(xxxxy)ln(2ln2lnxxxx)ln1ln1(2ln22lnxxxx.)0()),1ln(sin()(fxxf求设函数解)1ln(sin)0(f0))1(ln(sin)0(f)1cot()1sin()1cos()(xxxxf.1cot)0(f例?),(sin)(2dxdyxfyxf求可导,设解dxdyxxxfcossin2)(sin2.2sin)(sin2xxf例0.1)(Cxxcos)(sin.52.4.5基本求导数公式1)(.2xxxxee)(aaaxxln)(.3xx1)(lnaxxaln1)(log.4xxsin)(cosxx2sec)(tanxx2csc)(cotxxxtansec)(secxxxcotcsc)(csc211)(arcsin.6xx211)(arccosxx211)(arctanxx211)cot(xxarcxxch)sh('xxsh)ch('xx2ch1)th('11)arsh(2'xx11)arch(2'xx211)arth('xx双曲双曲与反双曲函数的导数公式xx2sh1)cth('211)arcth('xxxeeeexxxxxch2)2()sh(''xeeeexxxxxsh2)2()ch(''xxxxxxx2222ch1)ch()sh()ch()chsh()th(''xxxxxxx2222sh1)sh()ch()sh()shch()cth(''反函数及其导数。例:求函数2shxxeex2shyxxeex解:令0122xxyee2212442yyyyex0xe21yyex)1ln(2yyx)1ln(:2xxyyx得与互换)1ln(arsh2xxx即22222222111111)1221(111)1()arsh(xxxxxxxxxxxxxxx111sh1chy1)sh(1)arsh(22xyyx或者,反函数及其导数。例:求函数2chxxeex2chyxxeex解:令0122xxyee(减号舍去)1244222yyyyex)1ln(2yyx)1ln(:2xxyyx得与互换)1ln(arch2xxx即111111)1221(111)1()arch(22222222xxxxxxxxxxxxxxx111ch1shy1)ch(1)arch(22xyyx或者,yyyx22sh1ch)th1)arth('(yy22chth1)th1(12y211xyyycx22ch-1sh)th1)arcth('(yy22shcth1)cth1(12y211x.)1(102的导数求函数xy解)1()1(10292xxdxdyxx2)1(1092.)1(2092xx例2.arcsin22222的导数求函数axaxaxy解)arcsin2()2(222axaxaxy2222222222121xaaxaxxa.22xa)0(a例3.)2(21ln32的导数求函数xxxy解),2ln(31)1ln(212xxy)2(31211212xxxy)2(3112xxx例4.1sin的导数求函数xey解)1(sin1sinxeyx)1(1cos1sinxxex.1cos11sin2xexx例5小结1.反函数的求导法则(注意成立条件)2.复合函数的求导法则(注意函数的复合过程,合理分解正确使用链式法则)一、填空题:1、设4)52(xy,则y=.2、设xy2sin,则y=.3、设)arctan(2xy,则y=.4、设xycosln,则y=.5、设xxy2tan10,则y=.6、设)(xf可导,且)(2xfy,则dxdy=.练习题3)52(8xx2sin412xxxtan)2sec22(tan10ln1022tanxxxxx)(22xfx7.设xkexftan)(,则)(xf=,若ef4,则k.xxkekxk21tansectan21二、-求下列函数的导数:1.xy1arccos2.xxy2sin3.)ln(22xaxy4.)cotln(cscxxy;5.2)2(arcsinxy6.xeyarctan122xxx22sin2cos2xxxx221xaxcsc2arcsin422xxxexxarctan)1(217.xxyarccosarcsin8.xxy11arcsin三.设)(xf,)(xg可导,且0)()(22xgxf,求函数)()(22xgxfy的导数.22)(arccos12xx)1(2)1(1xxx)()()()()()(22xgxfxgxgxfxfy.三、四、设)(xf在0x处可导,且0)0(f,0)0(f,又)(xF在0x处可导,证明)(xfF在0x处也可导.0)0()()0()()]0([)]([lim0)]0([)]([lim00xfxffxffFxfFxfFxfFxx证明:)0(0)0()(lim0)(0fxfxfxxfx处可导,在)0(0)0()(lim)0()()]0([)]([lim,)(0)(00FuFuFfxffFxfFuxfxxFux则处可导,令在)0()0(0)]0([)]([lim0fFxfFxfFx可导。在即0)]([xxfF