$2.3復(fù)合函數(shù)及反函數(shù)的求導(dǎo)法

1、2-3復(fù)合函數(shù)及反函數(shù)的求導(dǎo)法 指出下列復(fù)合函數(shù)的復(fù)合過(guò)程：2(1)(32)yx2(2)ln(1)yx2(3)cosyx2.3.12.3.1復(fù)合函數(shù)的導(dǎo)數(shù)復(fù)合函數(shù)的導(dǎo)數(shù)設(shè)函數(shù) ，且 在點(diǎn) 處可導(dǎo)， 在對(duì)應(yīng)點(diǎn) 處也可導(dǎo)，則復(fù)合函數(shù) 在點(diǎn) 處可導(dǎo)，且 上式也可寫成)(),(xuufy)(xux)(ufy )(xfyxdxdududydxdy)( )( )( xufxyuyyxux或u復(fù)合函數(shù)求導(dǎo)法則（又稱為鎖鏈法則復(fù)合函數(shù)求導(dǎo)法則（又稱為鎖鏈法則）復(fù)合函數(shù)的求導(dǎo)法則： 復(fù)合函數(shù)求導(dǎo)步驟： xuxyyu分分解解求求導(dǎo)導(dǎo)相相乘乘回回代代dxdyxxuuy求已知例,sin,. 133,. 34yeyx求

2、已知例.,sinln. 2yxy求已知例2.(sinln ) ,yxy例5已知求210023.(1)sin(2)(2 +8)(3)(4)ln(65)xyxyxyeyx例4求下列函數(shù)的導(dǎo)數(shù)：.sinsin,.nynxxy例6已知求練習(xí)2(1)tan 3(2)lntan(21)(3)sin2xyxyxyex求下列函數(shù)的導(dǎo)數(shù)求下列函數(shù)的導(dǎo)數(shù):xxyxxycos1sin)5()sin()4(24反函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則處可導(dǎo)，且：點(diǎn)在，則它的反函數(shù)點(diǎn)可導(dǎo)且在續(xù)，點(diǎn)的某鄰域內(nèi)單調(diào)且連在如果函數(shù))()(0)( )(yxxxfyyyyyx2.3.22.3.2反函數(shù)的導(dǎo)數(shù)反函數(shù)的導(dǎo)數(shù))( 1)( yx

3、f.) 1, 0(. 7的導(dǎo)數(shù)求指數(shù)函數(shù)例aaayx.arctan. 9的導(dǎo)數(shù)求例xy .arcsin. 8的導(dǎo)數(shù)求例xy 一、基本初等函數(shù)的導(dǎo)數(shù)公式一、基本初等函數(shù)的導(dǎo)數(shù)公式(2)()x(1) () c(4)(ln )x 5 (sin )x(6)(cos )x(3)(log)ax0()c為常數(shù)1()x為常數(shù)1(0,1)lnaaxa且1xcosxsin x)(7(xa) 1, 0(lnaaaax)(8(xexe)(tan9(xx2sec)(cot10(xx2csc)(sec11(xxxtansec)(arcsin13(x211x)(csc12(xxxcotcsc)(arccos14(x211x

4、)(arctan15(x211x)cot)(16(xarc211x2.3.32.3.3初等函數(shù)的導(dǎo)數(shù)初等函數(shù)的導(dǎo)數(shù)二、函數(shù)的和、差、積、商的求導(dǎo)法則導(dǎo)法則為：的和、差、積、商的求為常數(shù)，則函數(shù)都是可導(dǎo)函數(shù)，和設(shè)Cvu)0()(4( )(3()( )(2( )(1 (2vvuvvuvuuvvuuvCCuCuvuvu是常數(shù)三、復(fù)合函數(shù)的求導(dǎo)法則的導(dǎo)數(shù)為：則復(fù)合函數(shù)都可導(dǎo)及設(shè)函數(shù))(,)()(xfyxuufyxuxuyydxdududydxdy或課堂小結(jié)課堂小結(jié)一、導(dǎo)數(shù)的概念一、導(dǎo)數(shù)的概念1.導(dǎo)數(shù)的定義2.導(dǎo)數(shù)的幾何意義3.左、右導(dǎo)數(shù)二、導(dǎo)數(shù)的基本公式和四則運(yùn)算二、導(dǎo)數(shù)的基本公式和四則運(yùn)算1.導(dǎo)數(shù)的基本公式2.導(dǎo)數(shù)的四則運(yùn)算法則三、復(fù)合函數(shù)求導(dǎo)法則三、復(fù)合函數(shù)求導(dǎo)法則練習(xí)求下列函數(shù)在給定點(diǎn)的導(dǎo)數(shù)值求下列函數(shù)在給定點(diǎn)的導(dǎo)數(shù)值:25021(1) ( )2,(0),( );4(2) ( )sin cos ,(),();64(3)