[數(shù)學(xué)]35_導(dǎo)數(shù)與微分小結(jié)ppt課件

1、求求 導(dǎo)導(dǎo) 法法 那么那么根本公式根本公式導(dǎo)導(dǎo) 數(shù)數(shù)xyx 0lim微微 分分xydy 關(guān)關(guān) 系系)( xodyydxydyydxdy 高階導(dǎo)數(shù)高階導(dǎo)數(shù)高階微分高階微分一、主要內(nèi)容1 1、導(dǎo)數(shù)的定義、導(dǎo)數(shù)的定義(1)定義定義.)()(limlim00000 xxfxxfxyyxxxx 導(dǎo)數(shù)定義的等價(jià)方式導(dǎo)數(shù)定義的等價(jià)方式: ;)()(lim)(0000 xxxfxfxfxx ;)()(lim)(0000hxfhxfxfh .)()(lim)(0000 xxfxxfxfx (2)幾何意義幾何意義.)(,()()(000處處切切線線的的斜斜率率在在點(diǎn)點(diǎn)是是曲曲線線導(dǎo)導(dǎo)數(shù)數(shù)xfxxfyxf ).)(

2、)(000 xxxfxfy 切切線線方方程程為為. 0)( )()(1)(0000 xfxxxfxfy法法線線方方程程為為右導(dǎo)數(shù)右導(dǎo)數(shù):(3)單側(cè)導(dǎo)數(shù)單側(cè)導(dǎo)數(shù)左導(dǎo)數(shù)左導(dǎo)數(shù):;)()(lim)()(lim)(0000000 xxfxxfxxxfxfxfxxx ;)()(lim)()(lim)(0000000 xxfxxfxxxfxfxfxxx 函函數(shù)數(shù))(xf在在點(diǎn)點(diǎn)0 x處處可可導(dǎo)導(dǎo)左左導(dǎo)導(dǎo)數(shù)數(shù))(0 xf 和和右右導(dǎo)導(dǎo)數(shù)數(shù))(0 xf 都都存存在在且且相相等等.2 2、根本導(dǎo)數(shù)公式、根本導(dǎo)數(shù)公式22211)(arctan11)(arcsinln1)(logln)(sec)(secsec)(t

3、ancos)(sin0)(xxxxaxxaaaxtgxxxxxxCaxx 常數(shù)和根本初等函數(shù)的導(dǎo)數(shù)公式常數(shù)和根本初等函數(shù)的導(dǎo)數(shù)公式222111)cot(11)(arccos1)(ln)(csc)(csccsc)(cotsin)(cos)(xxxxxxeexctgxxxxxxxxxx arc3 3、求導(dǎo)法那么、求導(dǎo)法那么設(shè)設(shè))(),(xvvxuu 可可導(dǎo)導(dǎo)，則則（1）vuvu )(, （2）uccu )(c是是常常數(shù)數(shù)),（3）vuvuuv )(, （4）)0()(2 vvvuvuvu.(1) 函數(shù)的和、差、積、商的求導(dǎo)法那么函數(shù)的和、差、積、商的求導(dǎo)法那么(2) 反函數(shù)的求導(dǎo)法那么反函數(shù)的求導(dǎo)

4、法那么.)(1)(),()(xxfxfyyx 則則有有的的反反函函數(shù)數(shù)為為如如果果函函數(shù)數(shù)(3) 復(fù)合函數(shù)的求導(dǎo)法那么復(fù)合函數(shù)的求導(dǎo)法那么).()()()()(),(xufxydxdududydxdyxfyxuufy 或或的的導(dǎo)導(dǎo)數(shù)數(shù)為為則則復(fù)復(fù)合合函函數(shù)數(shù)而而設(shè)設(shè)(4) 對(duì)數(shù)求導(dǎo)法對(duì)數(shù)求導(dǎo)法先在方程兩邊取對(duì)數(shù)先在方程兩邊取對(duì)數(shù),然后利用隱函數(shù)的求導(dǎo)方法然后利用隱函數(shù)的求導(dǎo)方法求出導(dǎo)數(shù)求出導(dǎo)數(shù).適用范圍適用范圍: :.)()(的情形的情形數(shù)數(shù)多個(gè)函數(shù)相乘和冪指函多個(gè)函數(shù)相乘和冪指函xvxu(5) (5) 隱函數(shù)求導(dǎo)法那么隱函數(shù)求導(dǎo)法那么用復(fù)合函數(shù)求導(dǎo)法那么直接對(duì)方程兩邊求導(dǎo)用復(fù)合函數(shù)求導(dǎo)法那么

5、直接對(duì)方程兩邊求導(dǎo).,)()(間間的的函函數(shù)數(shù)關(guān)關(guān)系系與與確確定定若若參參數(shù)數(shù)方方程程xytytx ;)()(ttdtdxdtdydxdy .)()()()()(322tttttdxyd (6) (6) 參變量函數(shù)的求導(dǎo)法那么參變量函數(shù)的求導(dǎo)法那么(7) 分段函數(shù)的導(dǎo)數(shù)分段函數(shù)的導(dǎo)數(shù) ).(, )()( 00 xGxxaxxxfxG 求求設(shè)設(shè), 0時(shí)時(shí)當(dāng)當(dāng)xx ),()(xfxG , 0時(shí)時(shí)當(dāng)當(dāng)xx 000)()(lim)(0 xxxGxGxGxx ,)(lim00 xxaxfxx 留意留意: )()(0 axG).(, )( )()( 00 xHxxxgxxxfxH 求求設(shè)設(shè), 0時(shí)時(shí)當(dāng)當(dāng)x

6、x )()(xfxH , 0時(shí)時(shí)當(dāng)當(dāng)xx )()(xgxH , 0時(shí)時(shí)當(dāng)當(dāng)xx 000)()(lim)(0 xxxHxHxHxx ,)()(lim000 xxxfxfxx 000)()(lim)(0 xxxHxHxHxx ,)()(lim000 xxxfxgxx ),()( 00 xHxH 若若. )( 0存在存在則則xH ),()( 00 xHxH 若若. )( 0不不存存在在則則xH 留意留意: 0)()(0 xxxgxH 0)()(0 xxxfxH 4 4、高階導(dǎo)數(shù)、高階導(dǎo)數(shù),)()(lim) )(0 xxfxxfxfx 二階導(dǎo)數(shù)二階導(dǎo)數(shù)記作記作.)(,),(2222dxxfddxydy

7、xf或或 .,),(33dxydyxf 二階導(dǎo)數(shù)的導(dǎo)數(shù)稱為三階導(dǎo)數(shù)二階導(dǎo)數(shù)的導(dǎo)數(shù)稱為三階導(dǎo)數(shù),記作記作階導(dǎo)數(shù)階導(dǎo)數(shù)的的函數(shù)函數(shù)階導(dǎo)數(shù)的導(dǎo)數(shù)稱為階導(dǎo)數(shù)的導(dǎo)數(shù)稱為的的函數(shù)函數(shù)一般地一般地,)( 1)(,nxfnxf .)(,),()()(nnnnnndxxfddxydyxf或或(二階和二階以上的導(dǎo)數(shù)統(tǒng)稱為高階導(dǎo)數(shù)二階和二階以上的導(dǎo)數(shù)統(tǒng)稱為高階導(dǎo)數(shù))高階導(dǎo)數(shù)公式高階導(dǎo)數(shù)公式: nmnmmnmxnmmmxnmnm , 0 , ! ,)1()1()( )1()(xnxnxnxeeaaaa )()()( );0( ln)( )2()2sin()(sin )3()( nkxkkxnn )2cos()(cos

8、 )4()( nkxkkxnn 1)()(!)1(1 )5( nnnaxnaxnnnxnx)!1()1()ln( )6(1)( )()()()( )7(nnnvuvu )()(0)()( )8(knknkknnvuCuv 5、微分的定義、微分的定義定義定義.),(,)(,)(),()()()(,)(000000000 xAdyxdfdyxxxfyxAxxfyxAxoxAxfxxfyxxxxfyxxxx 即即或或記作記作的微分的微分于自變量增量于自變量增量相應(yīng)相應(yīng)在點(diǎn)在點(diǎn)為函數(shù)為函數(shù)并且稱并且稱可微可微在點(diǎn)在點(diǎn)則稱函數(shù)則稱函數(shù)無關(guān)的常數(shù)無關(guān)的常數(shù)是與是與其中其中成立成立如果如果在這區(qū)間內(nèi)在這區(qū)間

9、內(nèi)及及在某區(qū)間內(nèi)有定義在某區(qū)間內(nèi)有定義設(shè)函數(shù)設(shè)函數(shù).的線性主部的線性主部叫做函數(shù)增量叫做函數(shù)增量微分微分ydy ( (微分的本質(zhì)微分的本質(zhì)) )6 6、導(dǎo)數(shù)與微分的關(guān)系、導(dǎo)數(shù)與微分的關(guān)系).(, )( )(000 xfAxxfxxf 且且處處可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)可可微微的的充充要要條條件件是是函函數(shù)數(shù)在在點(diǎn)點(diǎn)函函數(shù)數(shù)定理定理7 7、 微分的求法微分的求法dxxfdy)( 求法求法: :計(jì)算函數(shù)的導(dǎo)數(shù)計(jì)算函數(shù)的導(dǎo)數(shù), ,乘以自變量的微分乘以自變量的微分. .根本初等函數(shù)的微分公式根本初等函數(shù)的微分公式xdxxxdxdxxxdxdxxdxdxxdxdxxdxdxxddxxxdCdcotcsc)(csc

10、tansec)(seccsc)(cotsec)(tansin)(coscos)(sin)(0)(221 dxxxddxxxddxxxddxxxddxxxddxaxxddxeedadxaadaxxxx222211)cot(11)(arctan11)(arccos11)(arcsin1)(lnln1)(log)(ln)( arc 函數(shù)和、差、積、商的微分法那么函數(shù)和、差、積、商的微分法那么2)()()()(vudvvduvududvvduuvdCduCuddvduvud 8 8、 微分的根本法那么微分的根本法那么 微分方式的不變性微分方式的不變性的的微微分分形形式式總總是是函函數(shù)數(shù)是是自自變變量量

11、還還是是中中間間變變量量無無論論)(,xfyx dxxfdy)( 二、主要題型1.用定義求導(dǎo)數(shù)用定義求導(dǎo)數(shù) .0 )()(,)0(,), 0( )( 1.處的導(dǎo)數(shù)處的導(dǎo)數(shù)在在求求內(nèi)連續(xù)內(nèi)連續(xù)在在已知已知 xxxgxfagUxgex Solution. xfxffx)0()(lim)0(0 xxxgx0)(lim0 )(lim0 xgx .)0(ag .)()(lim,0)(. 20 xxfsxfxxfexx 求求處處可可導(dǎo)導(dǎo)在在設(shè)設(shè)Solution.xfxffsxfxxfsxfxx)0()()0()(lim)()(lim00 xfxfxfsxfxx)0()(lim)0()(lim00 xfxf

12、sxfsxfsxx)0()(lim)0()(lim00 )0()0(ff s ).0()1(fs ).( ,cos)( ),()()()()( , 1)0(, 0)0(,),()(. 322122121xfexxxxxfxxfxxfffxfexx 求求其其中中又又且且內(nèi)內(nèi)有有定定義義在在設(shè)設(shè) Solution. , 1)0( ,22sin)(222xxexxexx , 0)0( xxfxxfxfx )()(lim)(0 xxfxxfxxfx )()()()()(lim0 xxxfxxfxxfxx )()(lim)()()(lim00 xxfxxxxfxx 0)(lim)(1)(lim)(00

13、xfxfxxxxfxx )0()(lim)()0()(lim)(00 )0()()0()(fxxf )(x 2.復(fù)合函數(shù)求導(dǎo)數(shù)復(fù)合函數(shù)求導(dǎo)數(shù) .,. 41sin2yyexex 求求設(shè)設(shè)Solution. exy1sin2 xex1sin21sin2 xxex1sin1sin21sin2 xxxex11cos1sin21sin2 21sin11cos1sin22xxxexxxex2sin11sin22 .,cossin11arctan. 5ynxxxxyexn 求求設(shè)設(shè)Solution. )cos(sin11arctan nxxxxyn 1111112xxxx)(cossincos)(sin n

14、xxnxxnn222)1()1()1()1(2)1( xxxxxnxxxnncoscossin1 nnxxn)sin(sin .sinsincoscossin1112nxxnnxxxnxnn .,),1(. 633dxydfxfyex求求具有三階導(dǎo)數(shù)具有三階導(dǎo)數(shù)其中其中設(shè)設(shè) Solution. )1(xfdxdy)1()1( xxf)1(12xfx )1(1222xfxdxyd )1(1)1(122xfxxfx)1(1)1(2224xfxxfxx )1(1)1(243xfxxfx )1(1)1(24333xfxxfxdxyd )1(1)1(1)1(2)1(24433xfxxfxxfxxfxy3

15、.分段函數(shù)的導(dǎo)數(shù)分段函數(shù)的導(dǎo)數(shù) ).0(,0 , 00 ,sin)(. 72fxxxxxfex 求求設(shè)設(shè)Solution. 0)0()(lim)0(0 xfxffx00sinlim20 xxxx220sinlimxxx . 1 4.表達(dá)式中含有絕對(duì)值或最值的函數(shù)的導(dǎo)數(shù)表達(dá)式中含有絕對(duì)值或最值的函數(shù)的導(dǎo)數(shù) ).(,max)(. 82xFxxxFex 求求設(shè)設(shè)Solution. 21 ,10 ,max)(22xxxxxxxF; 1)(,10 xFx時(shí)時(shí)當(dāng)當(dāng);2)(,21xxFx 時(shí)時(shí)當(dāng)當(dāng),1時(shí)時(shí)當(dāng)當(dāng) x; 111lim1)1()(lim)1(11 xxxFxFFxx; 2)1(lim11lim1)

16、1()(lim)1(1211 xxxxFxFFxxx.)1(不存在不存在F 21 ,21 ,10 , 1)(xxxxxF不不存存在在5.參數(shù)方程的導(dǎo)數(shù)參數(shù)方程的導(dǎo)數(shù) .,)()(3)(2. 922dxydtf ttfytfxex求求設(shè)設(shè) Solution. )(2)()(3tftf ttfdxdyttftf ttftf )()()()( )(1)(222tftftdxyd .,arctan)1ln(.10332dxydttytxex求求設(shè)設(shè) Solution.dtdxdtdydxdy ,21211122tttt dtdxtdtddxyd 22221221tt dtdxttdtddxyd 412

17、3322212164)1(42tttttt ,412tt .8134tt 6.隱函數(shù)的導(dǎo)數(shù)隱函數(shù)的導(dǎo)數(shù) ., 11lnsin.110 xdxdyyxxyex求求設(shè)設(shè)Solution.,0 時(shí)時(shí)當(dāng)當(dāng) x.11lneyy , 1ln)1ln(sin yxxy),1(ln)1ln(sin dxdyxxydxd 則則, 0111)(cos yyxyxyxy01 0, eyeeyx代代入入得得將將).1(| 00eeydxdyxx .,0.1222dxydxyexyeexy求求的的可可微微函函數(shù)數(shù)為為確確定定了了設(shè)設(shè)方方程程 Solution. 方程兩邊對(duì)方程兩邊對(duì)x求導(dǎo)得求導(dǎo)得, , 0)( exye

18、dxdy. 0 yxyyey. yexyy 求二階導(dǎo)數(shù)有兩個(gè)方法求二階導(dǎo)數(shù)有兩個(gè)方法: 方法方法 1. yexyy2)()1)()(yyyexyeyexy .)(2232yyyexeyyexy 方法方法 2. , 0 yxyyey, 0)( yxyyey則則. 0)(2 yxyyyeyeyyyyexyyey 2)( 232)(22yyyexeyyexy 7.表達(dá)式中含有冪指函數(shù)的導(dǎo)數(shù)表達(dá)式中含有冪指函數(shù)的導(dǎo)數(shù) .,)1(.13sin2yxyexx 求求設(shè)設(shè)Solution.),1ln(sinln 2 xxy),1ln(sin)(ln2 xxdxdydxdyy 1,12sin)1ln(cos22 xxxxx.1sin2)1ln(cos)1( 22sin2 xxxxxxyx8.求高階導(dǎo)數(shù)求高階導(dǎo)數(shù) .,21.14)(2nyxxxyex求求設(shè)設(shè) Solution. 1211131)1)(12(212xxxxxxxxy 211211131xx 1121!)1(211!)1(31nnnnxnxn 11122113!)1(nnnnxxn. ,sin .15)(2nyxyex求求設(shè)設(shè) Solution. ,2cos212122cos1 xxy .)(2cos)(cos)(nnunuu nnnxy222cos21 )( .22cos21 nxn.),1ln(.162)sin(dyeyexxb