liu2-2函數(shù)的求導(dǎo)法則2ppt課件

1、第二節(jié)二、反函數(shù)的求導(dǎo)法則二、反函數(shù)的求導(dǎo)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 四、初等函數(shù)的求導(dǎo)問題四、初等函數(shù)的求導(dǎo)問題 一、函數(shù)的和、差、積、商求導(dǎo)法則一、函數(shù)的和、差、積、商求導(dǎo)法則 函數(shù)的求導(dǎo)法則 第二章 )(xf xyx 0limxxfxxfx )()(lim0;)(vuvu ;)(uCCu ;)(vuvuuv ，2)(vvuvuvu ) 0( v復(fù)習(xí)復(fù)習(xí)2)1(vvv )(xfy )(00 xfx，（x，)(yx ,0)( y )(xfy )(1)(yxf 請熟記以下公式請熟記以下公式(tan)x (cot )x (sec )x (csc ) x ()C (sin )x

2、 (cos )x ()()xR (ln)x (arcsin )x (arccos )x (arctan )x (arccot)x ()xa (e )x 211x 211x 211x 211x (log)ax 01.x ex1lnxa1xcos xsin x 2sec x2csc x sectanxxcsccotxx lnxaa)(xfy xyydd ,lim0 xyx )(ufy uyydd ,lim0uyu )(tfy tyydd ,lim0tyt xy uy ty tuutdd ,lim0tut txxtdd ,lim0txt yxxydd ,lim0yxy )(tfu )(tfx )(y

3、fx xy2sin xy2cos xx2cos2)2(sin xyddxy2sin xuuy2,sin ,cos)(sindduuuy , 2)2(dd xxu.ddddxuuy )2(sin xx2cos2 )(sin)2( ux)2()(sin xuucos )2( xxuuyxydddddd 三、三、 復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的求導(dǎo)法則定理定理3 3：d( ) ( )dyf u g xx 假如假如( )ug x 在點在點x可導(dǎo)，可導(dǎo)，)(ufy 而而在點在點( )ug x 可導(dǎo)，可導(dǎo)，則復(fù)合函數(shù)則復(fù)合函數(shù) ( )yf g x 在點在點x可導(dǎo)，可導(dǎo)，且其導(dǎo)數(shù)為且其導(dǎo)數(shù)為因變量對自變量求導(dǎo)

4、因變量對自變量求導(dǎo), ,等于因變量對中間變量等于因變量對中間變量xuuyxydddddd xuxuyy 即即或或即即求導(dǎo)求導(dǎo), ,乘以中間變量對自變量求導(dǎo)乘以中間變量對自變量求導(dǎo). . ( (鏈式法則鏈式法則) )在點在點x x可導(dǎo)可導(dǎo), , lim0 x( )uufuxx xyxyx0limdd定理定理3.3.( )ug x ( )yf u 在點在點( )ug x 可導(dǎo)可導(dǎo)復(fù)合函數(shù)復(fù)合函數(shù) yf ( )g x且且d( )( )dyfu g xx 在點在點 x x 可導(dǎo)可導(dǎo), ,證證: :)(ufy 在點在點 u u 可導(dǎo)可導(dǎo), , 故故)(lim0ufuyuuuufy)(（當（當 時時 )

5、)0u 0 故有故有( )( )fu g x uy)(uf( )(0)yuufuxxxx xu2cos2cos2 ，xy2sin )2()(sin)2(sin xuxu),(),(),(xvvuufy .ddddddddxvvuuyxy )(xfy uysin xu2 .)542(63的的導(dǎo)導(dǎo)數(shù)數(shù)求求 xxy)46(625 xu, 5423 xxu,6uy xuxxuy)542()(36 .)542)(23(12532 xxx,tanlnxy .ddxy,tanlnxy ，uyln xutan xydd xuuyddddxu2sec1 xxxxcossin1seccot2 ,2tanlnxy

6、.ddxy，uyln 2,tanxvvu 2tanlnxy xydd xvvuuyddddddxvuxvu)21()(tan)(ln 21sec12vu.cscxxxysin xxelnsin xydd xuuydddd)sinln(cosxxxxeu ).sinln(cossinxxxxxx ，uey xxulnsin xxysin .ddxy，) 0( x.1 xxxxex ln ).(R 1)( xx xy xeln ，uey xuln xydd xuuyddddxeu xuuyydddd,),(2xuufy )2)(xuf).(22xfx xwvuxwvuyy 例例7 7，xysinl

7、n 求求.ddxy解解 xydd)sin(ln x)(sinsin1 xxxxsincos .cotx 解解 xydd 21xxx知知例例8 8,1arcsin2xy 求求.ddxy知知 21arcsinx)1()1 (1122 xx . 01 ,11, 10 ,1122xxxx),cos(lnxey xyddxydd)cos( xe )cos()sin(xxee).tan(xxee )( xe )cos(xe四、初等函數(shù)的求導(dǎo)問題四、初等函數(shù)的求導(dǎo)問題 1. 1. 常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)常數(shù)和基本初等函數(shù)的導(dǎo)數(shù) (P95)(P95) )(C0 )(x1x )(sin xxcos )(cos

8、xxsin )(tan xx2sec )(cot xx2csc )(secxxxtansec )(cscxxxcotcsc )(xaaaxln )(exxe )(log xaaxln1 )(ln xx1 )(arcsin x211x )(arccosx211x )(arctanx211x )cot(arcx211x2. 2. 有限次四則運算的求導(dǎo)法則有限次四則運算的求導(dǎo)法則 )(vuvu )( uCuC )( vuvuvuvu2vvuvu( C為常數(shù) )0( v3. 3. 復(fù)合函數(shù)求導(dǎo)法則復(fù)合函數(shù)求導(dǎo)法則)(, )(xuufyxydd)()(xuf4. 4. 初等函數(shù)在定義區(qū)間內(nèi)可導(dǎo)初等函數(shù)在

9、定義區(qū)間內(nèi)可導(dǎo), , )(C0 )(sin xxcos )(ln xx1由定義證 ,說明說明: : 最基本的公式最基本的公式uyddxudd其他公式用求導(dǎo)法則推出.且導(dǎo)數(shù)仍為初等函數(shù)且導(dǎo)數(shù)仍為初等函數(shù)例例10. 10. 求求解解: :11,11xxyxx .y22212xxy 21xx 1y 2121x (2 )x 211xx 例例11.11.設(shè)設(shè)(0),aaxaxayxaaa解解: :1aaaya x lnaxaa 1aax lnxaaa 求求.y lnxaa 先化簡后求導(dǎo)先化簡后求導(dǎo)例例12. 12. 求求解解: :2sin2earctan1 ,xyx.y2( )arctan1yx 2si

10、ne( )x 2sinex2cos x 2x 21x2121x 2x 2x 2arctan1x 2sinex2cos x2sinex211xx 關(guān)鍵關(guān)鍵: : 搞清復(fù)合函數(shù)結(jié)構(gòu)搞清復(fù)合函數(shù)結(jié)構(gòu) 由外向內(nèi)逐層求導(dǎo)由外向內(nèi)逐層求導(dǎo)例例13. 設(shè)設(shè)求,1111ln411arctan21222xxxy.y解解: y22)1(1121x21xx) 11ln() 11ln(22xx111412x21xx1112x21xx2121xx221x21x231)2(1xxx，xeyx2cos2 yy )(coscos)(2222 xexexx)sin(cos2cos)2(222xxexxexx ).sincos(

11、cos22xxxxex 解解例例1515y 求求知知5( )(log)arcsin2 ,xf xexx xxexfx2arcsin)log()(5)2)(arcsinlog(5 xxex xxex2arcsin)5ln1()2()2(1log25 xxxex xxex2arcsin)5ln1(.41)log( 225xxex xyln 解解，2ln x y22)(xx 2222)(xxx 2xx x1 故故xx1ln ）（例例1616xyln xln）（ x ln0 x0 xy 求求知知12 xxy y12212 xx112 xx，87x xxxy y.8781 x,112 xxyy ,xxx

12、y y 例例1919，xeyarcsin .y 解解 y)(arcsin xe求求知知xearcsin 2arcsin)(11xex .)1(2arcsinxxex )(arcsin x)( x.1sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey )1(sin1sinxeyx )1(1cos1sinxxex.1cos11sin2xexx )(12222 axxaxxy）的導(dǎo)數(shù)）的導(dǎo)數(shù)（求求22lnaxxy )()(12222 axxaxx221ax 2222221axaxxaxx xvvuuyydddddd )(sin3xfy )(xf),(ufy ,3vu .sinxv )(sin)()(3 xvufx

13、vufcos3)(2 xxxfcossin3)(sin23 )(sincossin332xf xx )(sin )(sin33xfxf .)(sin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)nnnxfy )(sin)(sin1nnnnnxfxnfy )(sin)(sin1nnnxxn )(sin)(sin)(sin1nnnnnnnxxfxnf )(sin)(sin1nnnnnxfxnf )(sin)(sin12nnnnnxfxfn nnnnxxxsin)(sin)(sin1 nnnnnxxxxcos)(sin)(sin1 )(sin)(sin12nnnnnxfxfn ).(sin)(sin)(sin1nnnnnxxxf )(sincos113nnnnnxfxxn ，)()(ddxufxy xwvuxwvuyy xuxuyy vuvu )(uCCu )(vuvuuv )(2)(vvuvuv