liu22函数的求导法则2

1、第二节二、反函数的求导法则二、反函数的求导法则 三、复合函数求导法则三、复合函数求导法则 四、初等函数的求导问题四、初等函数的求导问题 一、函数的和、差、积、商求导法则一、函数的和、差、积、商求导法则 函数的求导法则 第二章 )(xf xyx 0limxxfxxfx )()(lim0;)(vuvu ;)(uccu ;)(vuvuuv ，2)(vvuvuvu ) 0( v复习复习2)1(vvv )(xfy )(00 xfx，（x，)(yx ,0)( y )(xfy )(1)(yxf 请熟记以下公式请熟记以下公式(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、 复合函数的求导法则复合函数的求导法则定理定理3 3：d( ) ( )dyf u g xx 如果如果( )ug x 在点在点x可导，可导，)(ufy 而而在点在点( )ug x 可导，可导，则复合函数则复合函数 ( )yf g x 在点在点x可导，可导，且其导数为且其导数为因变量对自变量求导

4、因变量对自变量求导, ,等于因变量对中间变量等于因变量对中间变量xuuyxydddddd xuxuyy 即即或或即即求导求导, ,乘以中间变量对自变量求导乘以中间变量对自变量求导. . ( (链式法则链式法则) )在点在点x可导可导, , lim0 x( )uufuxx xyxyx0limdd定理定理3.3.( )ug x ( )yf u 在点在点( )ug x 可导可导复合函数复合函数 yf ( )g x且且d( )( )dyfu g xx 在点在点 x 可导可导, ,证证: :)(ufy 在点在点 u 可导可导, ,故故)(lim0ufuyuuuufy)(（当（当 时时 ) )0u 0 故

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

6、yln 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，xysinln 求求.dd

7、xy解解 xydd)sin(ln x)(sinsin1 xxxxsincos .cotx 解解 xydd 21xxx已知已知例例8 8,1arcsin2xy 求求.ddxy已知已知 21arcsinx)1()1 (1122 xx . 01 ,11, 10 ,1122xxxx),cos(lnxey xyddxydd)cos( xe )cos()sin(xxee).tan(xxee )( xe )cos(xe四、初等函数的求导问题四、初等函数的求导问题 1. 1. 常数和基本初等函数的导数常数和基本初等函数的导数 (p95)(p95) )(c0 )(x1x )(sin xxcos )(cosxxs

8、in )(tan xx2sec )(cot xx2csc )(secxxxtansec )(cscxxxcotcsc )(xaaaxln )(exxe )(log xaaxln1 )(ln xx1 )(arcsin x211x )(arccosx211x )(arctan x211x )cot(arcx211x2. 2. 有限次四则运算的求导法则有限次四则运算的求导法则 )(vuvu )( ucuc )( vuvuvuvu2vvuvu( c为常数为常数 )0( v3. 3. 复合函数求导法则复合函数求导法则)(, )(xuufyxydd)()(xuf4. 4. 初等函数在定义区间内可导初等函数

9、在定义区间内可导, , )(c0 )(sin xxcos )(ln xx1由定义证 ,说明说明: : 最基本的公式uyddxudd其他公式用求导法则推出.且导数仍为初等函数且导数仍为初等函数例例10. 10. 求求解解: :11,11xxyxx .y22212xxy 21xx1y 2121x (2 )x 211xx 例例11.11.设设(0),aaxaxayxaaa解解: :1aaaya x lnaxaa 1aax lnxaaa 求求.y lnxaa 先化简后求导先化简后求导例例12. 12. 求求解解: :2sin2earctan1 ,xyx.y2( )arctan1yx 2sine( )x

10、 2sinex2cos x 2x 21x2121x 2x 2x 2arctan1x 2sinex2cos x2sinex211xx 关键关键: : 搞清复合函数结构搞清复合函数结构 由外向内逐层求导由外向内逐层求导例例13. 设求,1111ln411arctan21222xxxy.y解解: y22)1(1121x21xx) 11ln() 11ln(22xx111412x21xx1112x21xx2121xx221x21x231)2(1xxx，xeyx2cos2 yy )(coscos)(2222 xexexx)sin(cos2cos)2(222xxexxexx ).sincos(cos22xx

11、xxex 解解例例1515y 求求已知已知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 求求已知已知12 xxy y12212 xx112 xx，87x xxxy y.8781 x,112 xxyy ,xxxy y

12、 例例1919，xeyarcsin .y 解解 y)(arcsin xe求求已知已知xearcsin 2arcsin)(11xex .)1(2arcsinxxex )(arcsin x)( x.1sin的导数的导数求函数求函数xey )1(sin1sinxeyx )1(1cos1sinxxex.1cos11sin2xexx )(12222 axxaxxy）的导数）的导数（求求22lnaxxy )()(12222 axxaxx221ax 2222221axaxxaxx xvvuuyydddddd )(sin3xfy )(xf),(ufy ,3vu .sinxv )(sin)()(3 xvufxv

13、ufcos3)(2 xxxfcossin3)(sin23 )(sincossin332xf xx )(sin )(sin33xfxf .)(sin的的导导数数求求函函数数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)(vvuvuvu