32,3,4導(dǎo)數(shù)和微分運(yùn)算法則74819751018100901

1、導(dǎo)數(shù)與微分1函數(shù)的四則運(yùn)算求導(dǎo)法則函數(shù)的四則運(yùn)算求導(dǎo)法則小結(jié)小結(jié) 思考題思考題 作業(yè)作業(yè)函數(shù)的求導(dǎo)法則函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則基本求導(dǎo)法則與導(dǎo)數(shù)公式基本求導(dǎo)法則與導(dǎo)數(shù)公式復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的求導(dǎo)法則2定理定理1,)(),(處可導(dǎo)處可導(dǎo)在點(diǎn)在點(diǎn)如果函數(shù)如果函數(shù)xxvxu )()()1(xvxu 并且并且則則它們的線性組合、積、商它們的線性組合、積、商在點(diǎn)在點(diǎn) x處也可導(dǎo)處也可導(dǎo),);()(xvxu )()()2(xvxu);()()()(xvxuxvxu )()()()()(2xvxvxuxvxu ).0)( xv.,r 一、函數(shù)的四則運(yùn)算求導(dǎo)法則一、函數(shù)的四則運(yùn)算

2、求導(dǎo)法則 )()()3(xvxu3證證則由則由導(dǎo)數(shù)的定義導(dǎo)數(shù)的定義有有hxfhxfxfh)()(lim)(0 hxuhxuh)()(lim0 hxvhxvh)()(lim0 ).()(xvxu )()()1(xvxu );()(xvxu .,r 0lim hh ),()()(xvxuxf 設(shè)設(shè)hxvxuhxvhxuh)()()()(lim0 )()(xuhxu )()(xvhxv 4 )()()3(xvxu).0)()()()()()(2 xvxvxvxuxvxu,vuy 設(shè)設(shè)證證.yvu 則則 )()(xvxu);()()()(xvxuxvxu 由乘積的導(dǎo)數(shù)由乘積的導(dǎo)數(shù): u得得故故vvyu

3、y vvvuu )0(2 vvvuvuvy ,vy y 特別特別 )(1xv)()(2xvxv .2vvuvuvu 即即5推論推論,處均可導(dǎo)處均可導(dǎo)在點(diǎn)在點(diǎn)、若若xwvu wvu uvw,wvu 則則,處也可導(dǎo)處也可導(dǎo)在同一點(diǎn)在同一點(diǎn)x且且uvw vw wvuwuv u vwu你能給出相應(yīng)的四則運(yùn)算你能給出相應(yīng)的四則運(yùn)算微分微分法則嗎？法則嗎？d uvdudvd cucdud u vvduudv2uvduudvdvv6例例.sin223的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxxy 解解23xy x4 例例.ln2sin的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxy 解解xxxylncossin2 xxxylncoscos2 xxxln)

4、sin(sin2 xxx1cossin2 .cos x .2sin1ln2cos2xxxx 7例例.tan的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解解)(tan xyx2cos xxx222cossincos xx22seccos1 .sec)(tan2xx )(cot x同理可得同理可得 xxcossin即即.csc2x 2vvuvuvu )(cossin xxxx cos)(sin 8例例.sec的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解解)cos1()(sec xxyxx2cos)(cos .tansecxx xx2cossin 同理可得同理可得 )(1xv)()(2xvxv 即即xxxtansec)(sec xxxcot

5、csc)(csc 9.11的導(dǎo)數(shù)的導(dǎo)數(shù)求求 xxy解解 法一法一2)1()1)(1()1()1( xxxxxy2)1(2 x法二法二11 xxy121 x)12()1( xy2)1(2 x注注在進(jìn)行求導(dǎo)運(yùn)算中在進(jìn)行求導(dǎo)運(yùn)算中,且也能提高結(jié)果的準(zhǔn)且也能提高結(jié)果的準(zhǔn)這樣使求導(dǎo)過程簡單這樣使求導(dǎo)過程簡單,盡量先化簡再求導(dǎo)盡量先化簡再求導(dǎo),確性確性.2)1(12x )(1xv)()(2xvxv 10用求導(dǎo)法則與用定義求導(dǎo)數(shù)時(shí)用求導(dǎo)法則與用定義求導(dǎo)數(shù)時(shí), 結(jié)果有時(shí)不一致結(jié)果有時(shí)不一致,這是為什么這是為什么? 如已知如已知).0(,sin)(3fxxxf 求求無意義無意義,解解.cossin31)(313

6、2xxxxxf )0(f 所以所以,)0(f 不存在不存在.上述解法有問題嗎上述解法有問題嗎?注意問題出在注意問題出在)(0 xfx 處處不連續(xù)不連續(xù).因此因此)(xf 可能在不連續(xù)點(diǎn)處不代表該點(diǎn)處的導(dǎo)數(shù)值可能在不連續(xù)點(diǎn)處不代表該點(diǎn)處的導(dǎo)數(shù)值.,0時(shí)時(shí)當(dāng)當(dāng) x )(xf,0時(shí)時(shí)當(dāng)當(dāng) x, 0用定義用定義!,cossin31332xxxx 例例解解.),ln(2dyexyx求求設(shè)設(shè) ,2122xxexxey .2122dxexxedyxx 例例解解.,cos31dyxeyx求求設(shè)設(shè) )(cos)(cos3131xdeedxdyxx .sin)(cos,3)(3131xxeexx dxxedxex

7、dyxx)sin()3(cos3131 .)sincos3(31dxxxex 12)(1 )(1yfxf 或或.dd1ddyxxy 第一章定理第一章定理: 單調(diào)的連續(xù)函數(shù)必有單調(diào)的連續(xù)函數(shù)必有單調(diào)的連續(xù)反函數(shù)單調(diào)的連續(xù)反函數(shù). .定理定理2內(nèi)單調(diào)、內(nèi)單調(diào)、在某區(qū)間在某區(qū)間如果函數(shù)如果函數(shù)yiyfx)( ,0)( yf且且在在那末它的反函數(shù)那末它的反函數(shù))(1xfy ,內(nèi)也可導(dǎo)內(nèi)也可導(dǎo)對(duì)應(yīng)區(qū)間對(duì)應(yīng)區(qū)間xi且且可導(dǎo)可導(dǎo)二、反函數(shù)的求導(dǎo)法則二、反函數(shù)的求導(dǎo)法則證證,xix 任取任取xx 以以增增量量給給)()(11xfxxfy ), 0(xixxx , 0 .1yx xy 有有130lim x )(

8、1xfyxxy 10lim y.)(1yf 反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù)反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù). .連續(xù)連續(xù), 0lim0 yx)(1xfy 故故從而從而因因14.112x 例例.arcsin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解解yxsin yycos)(sin 且且內(nèi)有內(nèi)有在在)1 , 1( xi)(arcsin xycos1 y2sin11 .112x .11)(arccos2xx 同理可得同理可得;11)(arctan2xx , 0 )(sin1 y)(arcsin x)(1 )(1yfxf 單調(diào)、可導(dǎo)單調(diào)、可導(dǎo),直接函數(shù)直接函數(shù) 反函數(shù)反函數(shù) 內(nèi)內(nèi)在在 2,2 yi.11

9、)cotarc(2xx 15注注如果利用三角學(xué)中的公式如果利用三角學(xué)中的公式:,arcsin2arccosxx ,11)(arccos2xx .11)cot(2xx arc,arctan2cotarcxx 也可得公式也可得公式也可得公式也可得公式16例例.log的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xya , 0ln)( aaayy且且內(nèi)有內(nèi)有在在), 0( xi)(1)(log yaaxaayln1 .ln1ax 解解,),(內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(dǎo)在在 yyiax特別地特別地.1)(lnxx 17定理定理3 鏈導(dǎo)法則鏈導(dǎo)法則)(ufy 而而 xydd)(xg )(uf 三、復(fù)合函數(shù)的求導(dǎo)法則三、復(fù)合函數(shù)

10、的求導(dǎo)法則可導(dǎo)可導(dǎo), ,且其導(dǎo)數(shù)為且其導(dǎo)數(shù)為或或 uyxydddd.ddxu因變量對(duì)自變量求導(dǎo)因變量對(duì)自變量求導(dǎo), ,等于因變量對(duì)中間等于因變量對(duì)中間變量求導(dǎo)變量求導(dǎo), ,乘以中間變量對(duì)自變量求導(dǎo)乘以中間變量對(duì)自變量求導(dǎo). .,)(可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn)如果函數(shù)如果函數(shù)xxgu ,)(可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn)xgu xxgfy在在點(diǎn)點(diǎn)則則復(fù)復(fù)合合函函數(shù)數(shù))( 18證證,)(可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn)由由uufy )(lim0ufuyu )(ufuy故故uuufy )(則則xy xuufx0lim)( xydd0lim xxuxuuf )( 0lim xxuxx 00limlim , 0,0 ux時(shí)時(shí)當(dāng)當(dāng) )(uf).

11、(xg xuuyxydddddd 即即19推廣推廣),(ufy 設(shè)設(shè)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))(xfy 例例.2sinln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解解,lnuy uyxydddd u1xx2sin2cos2 x2cot2 xydd),(vu ),(xv ,sinvu vcos uydd vudd.ddxv.2xv vuddxvdd 2 20例例.)1(102的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xy解解 92)1(10 xy 92)1(10 x.)1(2092 xx例例.arcsin22222的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)axaxaxy 解解 )2(22xaxy2221xa 2221xa )0

12、( ax2)arcsin2(2 axa22112 axa)1(2 x)(22 xa ax22122xax2222xax2222aax21例例.)2(21ln32的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xxxy解解),2ln(31)1ln(212 xxyxxy211212 )2(3112 xxx例例.1sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey 解解)1(sin x xe1sin.1cos11sin2xexx )2(31 x xey1sin x1cos)1( x220 x,ln xex )()(ln xex 因?yàn)橐驗(yàn)樗运?xeln x)ln( x .1 xx1 的情形證明冪函數(shù)的導(dǎo)數(shù)公式的情形證明冪函數(shù)的導(dǎo)數(shù)公

13、式1)( xx一階微分形式的不變性;)(,)1(dxxfdyx 是自變量時(shí)是自變量時(shí)若若則則微函數(shù)微函數(shù)的可的可即另一變量即另一變量是中間變量時(shí)是中間變量時(shí)若若),(,)2(txtx ),()(xfxfy 有導(dǎo)數(shù)有導(dǎo)數(shù)設(shè)函數(shù)設(shè)函數(shù)dttxfdy)()( ,)(dxdtt .)(dxxfdy 結(jié)論結(jié)論：的微分形式總是的微分形式總是函數(shù)函數(shù)是自變量還是中間變量是自變量還是中間變量無論無論)(,xfyx 微分形式的不變性微分形式的不變性dxxfdy)( 例例解解.,sindybxeyax求求設(shè)設(shè) )(sin)(cosaxdebxbxbxdedyaxax dxaebxbdxbxeaxax)(sinco

14、s .)sincos(dxbxabxbeax 例例解解.),12sin(dyxy求求設(shè)設(shè) . 12,sin xuuyududycos )12()12cos( xdxdxx2)12cos( .)12cos(2dxx 25xxxxxxxctansec)(secsec)(tancos)(sin0)(2 1. 常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)公式常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)公式xxxxxxxxxcotcsc)(csccsc)(cotsin)(cos)(21 axxaaaaxxln1)(logln)( xxeexx1)(ln)( 四、基本求導(dǎo)法則與導(dǎo)數(shù)公式四、基本求導(dǎo)法則與導(dǎo)數(shù)公式211)(arcsinxx 211

15、)(arccosxx 211)(arctanxx 211)cotarc(xx 262. 函數(shù)的線性組合、積、商的求導(dǎo)法則函數(shù)的線性組合、積、商的求導(dǎo)法則)(),(xvvxuu 設(shè)設(shè)都可導(dǎo)都可導(dǎo), 則則3. 反函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則)(1 )(1yfxf 或或.dd1ddyxxy 內(nèi)單調(diào)、內(nèi)單調(diào)、在某區(qū)間在某區(qū)間如果函數(shù)如果函數(shù)yiyfx)( ,0)( yf且且在對(duì)應(yīng)區(qū)間在對(duì)應(yīng)區(qū)間則它的反函數(shù)則它的反函數(shù))(1xfy 且且可導(dǎo)可導(dǎo).,r ,)()1(vuvu .)()2(vuvuvu ).0()3(2 vvvuvuvu,內(nèi)也可導(dǎo)內(nèi)也可導(dǎo)xi274. 復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的求導(dǎo)法則,

16、)()()(),(都可導(dǎo)都可導(dǎo)及及且且而而設(shè)設(shè)xgufxguufy 初等函數(shù)的導(dǎo)數(shù)仍為初等函數(shù)初等函數(shù)的導(dǎo)數(shù)仍為初等函數(shù).注注的導(dǎo)數(shù)為的導(dǎo)數(shù)為則復(fù)合函數(shù)則復(fù)合函數(shù))(xgfy ).()()(ddddddxgufxyxuuyxy 或或 利用上述公式及法則初等函數(shù)求導(dǎo)問題利用上述公式及法則初等函數(shù)求導(dǎo)問題可完全解決可完全解決.28,ch)sh(xx ,sh)ch(xx )th(x證明下列雙曲函數(shù)與反雙曲函數(shù)的導(dǎo)數(shù)公式證明下列雙曲函數(shù)與反雙曲函數(shù)的導(dǎo)數(shù)公式:例例證證 1shch22 xx xxchsh,ch1)th(2xx ,11)arsh(2xx ,11)arch(2 xx.11)arth(2xx

17、 x2ch xx ch)sh( xx sh)ch( xxx222chshch x2ch1 29 211xx211x xarsh)1( 證證 )1ln(2xx 2121x x2 211)arsh(xx )arsh(x211xx )1(2 xx30例例解解 y x2th11xxx222ch1chsh11 xx22shch1 .sh2112x x2th11x2ch1)th( x1shch22 xx.)tharctan(的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy xx2ch1)th( 31例例.的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xxxy 解解 y xxx21)211(211(21xxxxxx .812422xxxxxxxxx

18、x )( xxxxxx 21 1()(21 xxxx32例例.,可導(dǎo)可導(dǎo)其中函數(shù)其中函數(shù)的導(dǎo)數(shù)的導(dǎo)數(shù)求求g xgey1解解 xg1 xge1 2111xxgexg xgexxg121 y xge1 xg1 x133例例).(000sin)(2xfxxxxxf 求求設(shè)設(shè)解解,0時(shí)時(shí) x,0時(shí)時(shí) xxxxx0sinlim20 220sinlimxxx 22sin2sinxxxx xxxf2sin)(0)0()(lim)0(0 xfxffx1 所以所以 010sin2sin)(22xxxxxxxf34例例.)(sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)nnnxfy 解解 y)(sin1nnxn nxcos ).

19、(sin)(sin)(sin)(sincos1113nnnnnnnnnnxxfxxfxxn )(sin1nnnxnf )(sinnnxf )(sinnx 1 nnx35例例與兩坐標(biāo)軸的交點(diǎn)所與兩坐標(biāo)軸的交點(diǎn)所過曲線過曲線證明證明24: xxy, 0 x令令證證, 0 y令令);2 , 0(ay軸的交點(diǎn)為軸的交點(diǎn)為曲線與曲線與),0 , 4(bx軸的交點(diǎn)為軸的交點(diǎn)為曲線與曲線與. 2 y得得. 4 x得得 24xxy,210 xy,214 xy由于斜率相等由于斜率相等,知二切線平行知二切線平行.(1) 求交點(diǎn)求交點(diǎn),)2(22 x 221x分別為曲線在分別為曲線在a, b點(diǎn)點(diǎn)的切線斜率的切線斜率

20、.(2) 求導(dǎo)數(shù)求導(dǎo)數(shù)作的曲線的切線彼此平行作的曲線的切線彼此平行.36xexeexxxx22sin)1(sin)1(cos 解解 21sin11xex 1sinxex y.1sinarctan的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xexy37解解),(ufy 設(shè)設(shè) yxxf3cos)3(sin3 注注xu3sin uy )(ufx3cos3則則xu .的導(dǎo)數(shù)的導(dǎo)數(shù)對(duì)對(duì)不表示不表示xf上式中上式中是函數(shù)是函數(shù) f 對(duì)括號(hào)中的中間對(duì)括號(hào)中的中間變量求導(dǎo)變量求導(dǎo),? .,3sin可導(dǎo)可導(dǎo)其中函數(shù)其中函數(shù)的導(dǎo)數(shù)的導(dǎo)數(shù)求求fxfy )3(sin xf )3(sin)3(sin xfxf38.)(,)(的導(dǎo)數(shù)的導(dǎo)數(shù)

21、求求是可導(dǎo)函數(shù)是可導(dǎo)函數(shù)設(shè)設(shè)xfxeefyf 解解 分析分析 這是抽象函數(shù)與具體函數(shù)相結(jié)合的導(dǎo)數(shù)這是抽象函數(shù)與具體函數(shù)相結(jié)合的導(dǎo)數(shù), 綜合運(yùn)用函數(shù)線性組合、積、商求導(dǎo)法則以及綜合運(yùn)用函數(shù)線性組合、積、商求導(dǎo)法則以及 復(fù)合函數(shù)求導(dǎo)法則復(fù)合函數(shù)求導(dǎo)法則.)()( xfxeefy )( )(xfxeef)(xfe )()()()(xfefeefexxxxf )()( xfxeefxxeef )()()(xfexf )(xef39.的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xaaaxaaaxy 答案答案 1aaaxay axaaxaa1lnxaxaaa 2ln解解axafxfafax )()(lim)(axxaxax 0)()(lim )(limxax )(a ),()()(,)(xaxxfaxx 處連續(xù)處連續(xù)在在若若).(af 求求微分的求法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.1.基本初等函數(shù)的微分公式基本初等函數(shù)的微分公式xdxxxdxdxxxdxdxxdxdxxdxdxxdxdxxddxxxdcdcotcsc)(csctansec)(seccsc)(cotsec)(ta