清華大學(xué)微積分(高等數(shù)學(xué))第6講_導(dǎo)數(shù)與微分(二)ppt課件

1、 作作 業(yè)業(yè) 6(3) (6) (9) (11) (14) (17). 9(4) (8) (15) (21). 10(8). 11(2). 12(2). P67 習(xí)題習(xí)題3.2 二、高階導(dǎo)數(shù)二、高階導(dǎo)數(shù)第六講第六講 導(dǎo)數(shù)與微分導(dǎo)數(shù)與微分( (二二) )一、導(dǎo)數(shù)與微分的運(yùn)算法那一、導(dǎo)數(shù)與微分的運(yùn)算法那么么一、導(dǎo)數(shù)與微分的運(yùn)算法那一、導(dǎo)數(shù)與微分的運(yùn)算法那么么1. 四那么運(yùn)算求導(dǎo)法四那么運(yùn)算求導(dǎo)法那么那么則則可可導(dǎo)導(dǎo)在在設(shè)設(shè)函函數(shù)數(shù),)(),(xxvxu且且可可導(dǎo)導(dǎo)在在函函數(shù)數(shù),)()()1(xxvxu )()( )()(xvxuxvxu 且且為為常常數(shù)數(shù)可可導(dǎo)導(dǎo)在在函函數(shù)數(shù)),()()2(Cxxu

2、C)( )(xuCxuC 且且可可導(dǎo)導(dǎo)在在函函數(shù)數(shù),)()()3(xxvxu )()()()( )()(xvxuxvxuxvxu 且且可可導(dǎo)導(dǎo)在在函函數(shù)數(shù),)()()4(xxvxu2)()()()()()()(xvxvxuxvxuxvxu )0)( xv)()(xvxuy 設(shè)設(shè))()()()(xxvxuxxvxxu vxuxxvu )()()()()()(xvxuxxvxu 證證 (3)xvxuxxvxuxy )()()()()()()()(limlim00 xvxuxvxuxvxuxxvxuxyyxx 可導(dǎo)必延續(xù)可導(dǎo)必延續(xù))()()()(xvxuxxvxxuy 的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例2

3、sinlncos24735 xxxxyxxxx1sin212524 解解)2(sin)(ln)(cos2)( 4)(35 xxxx)2sinlncos24(35 xxxxy)cossin()(tan xxxxxxxx2cos)(cossincos)(sin .seccos1cos)sin(sincoscos222xxxxxxx 的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例xxftan)(8 解解xxx22cos1sec)(tan 2、復(fù)合函數(shù)導(dǎo)數(shù)公式、復(fù)合函數(shù)導(dǎo)數(shù)公式1復(fù)合函數(shù)微分法鏈?zhǔn)椒敲磸?fù)合函數(shù)微分法鏈?zhǔn)椒敲辞仪乙惨部煽蓪?dǎo)導(dǎo)在在點(diǎn)點(diǎn)則則復(fù)復(fù)合合函函數(shù)數(shù)可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)函函數(shù)數(shù)可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)設(shè)設(shè)函函數(shù)

4、數(shù),)(,)(,)(xxgfyxxguuufy dxdududydxdy 或或)()()(xgxgfdxxgdf 證證 xyx 0lim xuuyxuuyxux 000limlimlimdxdududy )()(xgxgf 0,0 xx 時(shí)時(shí)當(dāng)當(dāng)不能保證中間變量的增量不能保證中間變量的增量)()(xgxxgu 總不等于零總不等于零上面的證法有沒(méi)有問(wèn)題？上面的證法有沒(méi)有問(wèn)題？ 證證 可可導(dǎo)導(dǎo))(ufy )(ufuy)0lim(0 u)(lim0ufuyu 上上式式化化為為時(shí)時(shí)當(dāng)當(dāng),0 u )1()(uuufy 0)()(,0 ufuufyu 時(shí)時(shí)當(dāng)當(dāng)(1) 式依然成立！式依然成立！xuxuufx

5、y )(xuxuufxyxxxx 0000limlimlim)(lim )()(lim)(0 xgufxydxxgdfx 0limlim00 ux)()()(xgufdxxgfd 連連續(xù)續(xù)可可導(dǎo)導(dǎo))()(xguxgu 00ux 2微分的方式不變性微分的方式不變性(復(fù)合函數(shù)微分法那么復(fù)合函數(shù)微分法那么)且且其其微微分分為為也也可可微微則則復(fù)復(fù)合合函函數(shù)數(shù)函函數(shù)數(shù)均均為為可可微微和和設(shè)設(shè)函函數(shù)數(shù),)(,)()(xgfyxguufy dxxuufduufdy)( )( )( xxgfdyx )(證證xxgxgf )()( duuf )(有有時(shí)時(shí)當(dāng)當(dāng),)(xxgu xxxgdu )(xdx 我我們們將

6、將微微分分寫(xiě)寫(xiě)成成因因此此對(duì)對(duì)于于自自變變量量,xdxxfxxfxdf)()()( dxxfxdf)()( uduxxgu ,)(時(shí)時(shí)當(dāng)當(dāng)不不能能將將微微分分寫(xiě)寫(xiě)成成對(duì)對(duì)于于中中間間變變量量),(xuu uufxudf )()( dxxuufduufxudf)( )( )()( 的的函函數(shù)數(shù)，微微分分形形式式不不變變還還是是中中間間變變量量是是自自變變量量不不論論uxy但有但有 微分的微分的方式不變性方式不變性.11123的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例 xxy解解 11112321xxdxdxxdxdy221)1(21123 xxx2521)1()1(3 xx42,ln xvtgvuuy設(shè)設(shè)xv

7、uxxtgvuy)42()()(ln .)42lntan(2的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例 xy解解21cos112 vu21)(cos1)(142242 xxtg)sin(12 xxcos1 xsec .)1ln(32的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例 xxyxxxxxxy)1(1122 1111112222 xxxxxx)1(11122xxxx 解解)1(121111222xxxxx .ln4的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例xy 時(shí)時(shí)當(dāng)當(dāng)時(shí)時(shí)當(dāng)當(dāng)0, )ln(0,lnlnxxxxxyxxyx1)(ln,0 時(shí)時(shí)當(dāng)當(dāng) )ln(,0 xyx時(shí)時(shí)當(dāng)當(dāng))0(1)(ln xxxxxx1)(1 解解.lnln同同

8、的的導(dǎo)導(dǎo)數(shù)數(shù)公公式式有有相相與與xx)()( )(ln)(lnxfxfxfxf .,1中中間間變變量量都都便便于于求求導(dǎo)導(dǎo)應(yīng)應(yīng)使使每每一一個(gè)個(gè)地地選選取取中中間間變變量量恰恰當(dāng)當(dāng)在在于于分分析析清清楚楚函函數(shù)數(shù)關(guān)關(guān)系系關(guān)關(guān)鍵鍵：復(fù)復(fù)合合函函數(shù)數(shù)求求導(dǎo)導(dǎo)數(shù)數(shù)時(shí)時(shí)注注意意.,2就就用用什什麼麼求求導(dǎo)導(dǎo)法法則則什什麼麼運(yùn)運(yùn)算算碰碰到到四四則則運(yùn)運(yùn)算算的的函函數(shù)數(shù)關(guān)關(guān)系系時(shí)時(shí)又又有有：當(dāng)當(dāng)遇遇到到既既有有復(fù)復(fù)合合運(yùn)運(yùn)算算注注意意3. 反函數(shù)求導(dǎo)法那反函數(shù)求導(dǎo)法那么么)(1 )(,)()(, 0)(,11xfyfxfyyyfxxfxfx 且且可可導(dǎo)導(dǎo)在在反反函函數(shù)數(shù)則則它它的的且且可可導(dǎo)導(dǎo)在在單單調(diào)調(diào)且

9、且嚴(yán)嚴(yán)格格的的某某鄰鄰域域連連續(xù)續(xù)設(shè)設(shè)函函數(shù)數(shù)在在的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)例例xxfyarcsin)( 解解2211 yxyxxysin,)1, 1(arcsin 存存在在反反函函數(shù)數(shù)增增加加且且嚴(yán)嚴(yán)格格上上連連續(xù)續(xù)在在yyxcos1)(sin1)(arcsin 2211sin11xy 由反函數(shù)由反函數(shù)求導(dǎo)法那求導(dǎo)法那么么4. 隱函數(shù)求導(dǎo)法隱函數(shù)求導(dǎo)法定義：隱函數(shù)定義：隱函數(shù).0),()(,0),(,.,的的隱隱函函數(shù)數(shù)確確定定是是方方程程或或關(guān)關(guān)系系則則稱稱此此對(duì)對(duì)應(yīng)應(yīng)對(duì)對(duì)應(yīng)應(yīng)唯唯一一的的由由方方程程若若設(shè)設(shè)有有非非空空數(shù)數(shù)集集 yxFxfyfYyyxFXxRYX0)(, xfxFXx有有的

10、的解解必必是是方方程程確確定定的的隱隱函函數(shù)數(shù)由由方方程程注注意意0),()(0),( yxFxfyyxF.),(0),(可可導(dǎo)導(dǎo)并并且且函函數(shù)數(shù)隱隱函函數(shù)數(shù)能能夠夠確確定定假假定定方方程程fxfyyxF ?,)(如如何何求求出出導(dǎo)導(dǎo)數(shù)數(shù)的的情情況況下下問(wèn)問(wèn)題題：在在不不解解出出顯顯式式xfy 隱函數(shù)求導(dǎo)問(wèn)題的提法隱函數(shù)求導(dǎo)問(wèn)題的提法.,0)(,(),(,0),(xyxxyxFxxfyxyyxF 解解出出求求導(dǎo)導(dǎo)兩兩邊邊對(duì)對(duì)的的恒恒等等式式：關(guān)關(guān)于于于于是是方方程程可可看看成成的的函函數(shù)數(shù)：看看成成把把中中在在方方程程.,求求導(dǎo)導(dǎo)法法則則因因此此需需要要應(yīng)應(yīng)用用復(fù)復(fù)合合函函數(shù)數(shù)的的函函數(shù)數(shù)是是

11、要要注注意意注注意意：左左端端求求導(dǎo)導(dǎo)時(shí)時(shí)xy 隱函數(shù)求導(dǎo)法隱函數(shù)求導(dǎo)法得得求求導(dǎo)導(dǎo)方方程程兩兩邊邊對(duì)對(duì),x)2(02sincos3cos)(22223 xxxxxyxyyyexy.),(01cos 123xxyyxfyxxye 求求隱隱函函數(shù)數(shù)確確定定由由方方程程例例解解)1(0)1()cos()(23 xxeyyexyxy得得解解出出,y )1(sincos6cos2222223xyeeyxxxxyxyxy ?)0(: y問(wèn)問(wèn)0)0(1)0( yy5. 參數(shù)方程求導(dǎo)法參數(shù)方程求導(dǎo)法參參數(shù)數(shù)方方程程)1(2, 0sincos1 ttbytax橢橢圓圓：例例0,)cos1()sin(2 ata

12、yttax擺擺線線：例例aa 22, 0sincos333 ttaytax星星形形線線：例例a內(nèi)旋輪線內(nèi)旋輪線0,323232 aayx隱隱函函數(shù)數(shù)方方程程：0120(2) 參數(shù)方程求導(dǎo)法參數(shù)方程求導(dǎo)法?dxdy如如何何求求).()(, 0)(,)(),(1xttxttt 存存在在可可導(dǎo)導(dǎo)的的反反函函數(shù)數(shù)且且都都存存在在設(shè)設(shè)確確定定由由參參數(shù)數(shù)方方程程：設(shè)設(shè)函函數(shù)數(shù) )()()(tytxxfy 的的復(fù)復(fù)合合函函數(shù)數(shù)成成為為通通過(guò)過(guò)xty)(ty 分析函數(shù)關(guān)系分析函數(shù)關(guān)系:)()(1xttx )(1xy 利用復(fù)合函數(shù)和反函數(shù)微分法利用復(fù)合函數(shù)和反函數(shù)微分法, 得得)()(ttdtdxdtdydxdtdtdyd