函數(shù)的求導(dǎo)法則88870PPT學(xué)習(xí)教案

1、會(huì)計(jì)學(xué)1函數(shù)的求導(dǎo)法則函數(shù)的求導(dǎo)法則88870定理定理1.并且并且處可導(dǎo)處可導(dǎo)都在點(diǎn)都在點(diǎn)點(diǎn)外點(diǎn)外除分母為零的除分母為零的、商、商那么它們的和、差、積那么它們的和、差、積可導(dǎo)可導(dǎo)處處都在點(diǎn)都在點(diǎn)和和如果函數(shù)如果函數(shù), )( , )( )( xxxvvxuu );()( )()()1(xvxuxvxu );()()()( )()( )2(xvxuxvxuxvxu ).0)()()()()()()()()3(2 xvxvxvxuxvxuxvxu第1頁(yè)/共28頁(yè)證證: :則則令令 ),()()( xvxuxf xxvxuxxvxxux )()()()(lim0 xxvxxvxxuxxuxx )()(

2、lim)()(lim00);()(xvxu vuvu )( 1)(xxfxxfxfx )()(lim)(0此法則可此法則可推廣到任意有限項(xiàng)推廣到任意有限項(xiàng)的情形的情形. 例如例如,.)(wvuwvu 第2頁(yè)/共28頁(yè)證證: 設(shè)設(shè), )()()(xvxuxf 則則xxfxxfxfx )()(lim)(0 xxvxuxxvxxux )()()()(lim0故結(jié)論成立故結(jié)論成立.);()()()(xvxuxvxu xxxux )(lim0)(xu)(xxv xxv)( )(xu)(xxv 推論推論: : )()1uC )()2wvuuC wvuwvuwvu ( C為常數(shù)為常數(shù) )vuvuuv )(

3、2)()()()()(xxvxuxxvxu 第3頁(yè)/共28頁(yè))()()()(xvxuxvxu )()( lim0 xvxxvxxxvxxvxxvxuxvxxu )()()()()()(證證: 設(shè)設(shè)則有則有xxfxxfxfx )()(lim)(0 xx lim0,)()()(xvxuxf )()(xxvxxu )()(xvxu xxxu )()(xu )(xvxxxv )()(xu )(xv 故結(jié)論成立故結(jié)論成立.)()()()()(2xvxvxuxvxu 特殊地特殊地: : v1( C為常數(shù)為常數(shù) )2)( 3)(vvuvuvu .2vv 第4頁(yè)/共28頁(yè)例例1.735223的導(dǎo)數(shù)的導(dǎo)數(shù)求求

4、xxxy解解: :)7()3()5()2(23 xxxyx25 例例2. )2( )( ,2sincos4)(3 fxfxxxf 及及求求已知已知解解: :)2(sin)cos4()()(3 xxxf3 23x 232x 0 . 31062 xx2)()2( xxff. 4432 2sin4)2(32 xsin4 0 ,sin432xx uCCuuuuuuu )()(321321第5頁(yè)/共28頁(yè)例例3.ln的的導(dǎo)導(dǎo)數(shù)數(shù)求求xxy 解：解：xxyln)( )(ln xxxxxx1ln21 例例4. ),cos(sineyxxyx 求求).1ln21(1 xx解：解：)cos(sine)cos(s

5、in)e ( xxxxyxx)sin(cose)cos(sinexxxxxx .cose2xx vuvuuv )(第6頁(yè)/共28頁(yè)例例5.tan的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解：解： xxxycossin)(tanxxxxx2cos)(cossincos)(sin xxx222cossincos x2cos1 .sec)(tan2xx 即即同同理理.sec2x .csc)(cot2xx 2vvuvuvu 例例6.sec的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解：解： xxycos1)(secxxx2cos)(cos1cos)1( .tansecxx xx2cossin 同理同理.tansec)(sec xxx 即即.c

6、otcsc)(cscxxx 第7頁(yè)/共28頁(yè)即即 反函數(shù)的導(dǎo)數(shù)等于反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù)直接函數(shù)導(dǎo)數(shù)的倒數(shù). .)(1 )(1yfxf 定理定理2. 且由反函數(shù)的連續(xù)性知且由反函數(shù)的連續(xù)性知 內(nèi)內(nèi)單單調(diào)調(diào)、可可導(dǎo)導(dǎo)且且在在區(qū)區(qū)間間若若函函數(shù)數(shù)yIyfx)( , 0)( yfyxxydd1dd 或或,00 yx時(shí)必有時(shí)必有xyxfx 01lim )( lim0 yyx .)(1yf 1),(|)( 1yxIyyfxxIxfy 在在則其反函數(shù)則其反函數(shù)可導(dǎo)且可導(dǎo)且證證:在在 x 處給增量處給增量由反函數(shù)的單調(diào)性知由反函數(shù)的單調(diào)性知,0 x)()(11xfxxfy ,0 xyyx 1第8

7、頁(yè)/共28頁(yè)例例7.arcsin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xy 解：解： , 2,2 sin內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(dǎo)在在因?yàn)橐驗(yàn)?yIyx, 0cos)(sin yy且且 )1 , 1(內(nèi)有內(nèi)有所以在所以在 xI)(sin1)(arcsin yxycos1 y2sin11 .112x 同理同理;11)(arctan2xx .11)(arcsin 2xx 即即.11)(arccos2xx .11)cotarc(2xx 第9頁(yè)/共28頁(yè)在點(diǎn)在點(diǎn) x 可導(dǎo)可導(dǎo), lim0 xxuxuuf )(xyxyx 0limdd定理定理3.)(xgu )(ufy 在點(diǎn)在點(diǎn))(xgu 可導(dǎo)可導(dǎo)復(fù)合函數(shù)復(fù)合函數(shù) fy

8、 )(xg且且)()(ddxgufxy 在點(diǎn)在點(diǎn) x 可導(dǎo)可導(dǎo),證證:)(ufy 在點(diǎn)在點(diǎn) u 可導(dǎo)可導(dǎo), 故故)(lim0ufuyu uuufy )(（當(dāng)（當(dāng) 時(shí)時(shí) )0u0).()(xguf uy)(uf)0()( xxuxuufxy .dddddd xuuyxy 或或第10頁(yè)/共28頁(yè)例例8.tanln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解：解： tan ,lntanln 復(fù)合而成復(fù)合而成可由可由xuuyxy xuuyxydddddd u1.seccot2xx x2sec.e .93的的導(dǎo)導(dǎo)數(shù)數(shù)求求例例xy 解：解：, ,e e 33復(fù)復(fù)合合而而成成可可由由xuyyux xuuyxydd dd

9、 dd 23exu .e332 xx 分解復(fù)合函數(shù)是關(guān)鍵分解復(fù)合函數(shù)是關(guān)鍵! !第11頁(yè)/共28頁(yè)例例10.12sin2的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xxy 解：解：xuuyxydd dd dd 2222)1()1(2)1()2(cosxxxxxu .12cos)1()1(22222xxxx 212cosxxu復(fù)合而成復(fù)合而成由由2212 ,sin 12sin xxuuyxxy 第12頁(yè)/共28頁(yè)例例11.)1(102的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xy解：解： xyddxx2)1(1092 .)1(2092 xx)1(2 x 92)1(10 x例例12.21 32的的導(dǎo)導(dǎo)數(shù)數(shù)求求xy 解解：)21(dd

10、312 xxy1312)21 (31 x)4()21(31322xx .)21(34322xx )21(2 x由外到里逐層求導(dǎo)由外到里逐層求導(dǎo)! !第13頁(yè)/共28頁(yè)例如例如,)(, )(, )(xvvuufy xydd)()()(xvuf yuvx uydd vuddxvdd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).( (鏈?zhǔn)椒▌t鏈?zhǔn)椒▌t) )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 第14頁(yè)/共28頁(yè)例例13.e )2( ;)ecos(ln )1( 1sinxxyy 求下列函數(shù)的導(dǎo)數(shù)：求下列函數(shù)的導(dǎo)數(shù)：解解: : (1) )ecos(1

11、x )esin(xxe).etan(exx xvvuuye ,cos ,ln xvvuuyxydd dd dd dd xvue)sin(1 解解: : (2)xvvuuyydddddd xvu1cose.1cose11sin2xxx xvvuyu1 ,sin ,e 第15頁(yè)/共28頁(yè)例例14. )(sin,)( 2的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)可導(dǎo)可導(dǎo)設(shè)設(shè)xfyxf 解解: :xvvuufysin , ),( 2 xvvuuyxydddddddd )(uf ).(sin 2sin2xfx v2 xcos 第16頁(yè)/共28頁(yè).)()2( ;)()1( xxx 解解: (1)e ()(ln xx xln

12、e )ln( x x x .1 x)e ()(ln xxxxxx lne )ln( xxxx ).1ln( x(2)uvvulne 第17頁(yè)/共28頁(yè),tansec)(sec,sec)(tan,cos)(sin, 0)(2xxxxxxxC 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,cotcsc)(csc,csc)(cot,sin)(cos,)(21xxxxxxxxx ,ln1)(log,ln)(axxaaaaxx ,1)(ln,e)e (xxxx ,11)(arctan,11)(arcsin22xxxx .11)cotarc(,11)(arccos22xxxx 第18頁(yè)/

13、共28頁(yè)2. 函數(shù)的和、差、積、商的求導(dǎo)法則函數(shù)的和、差、積、商的求導(dǎo)法則 , )( ),( 則則都可導(dǎo)都可導(dǎo)設(shè)設(shè)xvvxuu ),()()2(,)()1(是常數(shù)是常數(shù)CuCCuvuvu ).0()4(,)( (3)2 vvvuvuvuvuvuuv3. 反函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則.dd1dd )(1 )(,),(|)( ,0)( )(11yxxyyfxfIyyfxxIxfyyfIyfxyxy 或或且有且有內(nèi)也可導(dǎo)內(nèi)也可導(dǎo)在區(qū)間在區(qū)間則它的反函數(shù)則它的反函數(shù)且且內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(dǎo)在區(qū)間在區(qū)間如果函數(shù)如果函數(shù)第19頁(yè)/共28頁(yè)4. 復(fù)合函數(shù)求導(dǎo)法則復(fù)合函數(shù)求導(dǎo)法則的導(dǎo)數(shù)為的導(dǎo)數(shù)為，則復(fù)

14、合函數(shù)，則復(fù)合函數(shù)設(shè)設(shè))( )(, )( xfyxuufy xydd)()(xuf 5. 初等函數(shù)在定義區(qū)間內(nèi)可導(dǎo)初等函數(shù)在定義區(qū)間內(nèi)可導(dǎo),uyddxudd 且導(dǎo)數(shù)仍為初等函數(shù)且導(dǎo)數(shù)仍為初等函數(shù).第20頁(yè)/共28頁(yè). )( sinsinynxnxyn ，求求為為常常數(shù)數(shù)解解: :)(sinsinsin)(sin xnxxnxynnxnxnnsincos xxnnxncossinsin1 )cossinsin(cossin1xnxxnxxnn .)1sin(sin1xnxnn 第21頁(yè)/共28頁(yè)例例17.)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(21

15、2 xxy)2(3121121 2 xxxy.)2(3112 xxx第22頁(yè)/共28頁(yè)解解:. ,1arctane2sin2yxyx 求求1arctan) (2 xy) (e2sinx 2sinex2cos x x2 21x1212 xx2 x2 1arctan2 x2sinex2cos x.e2sinx112 xx第23頁(yè)/共28頁(yè)例例19.).(,0),1ln(0,)(xfxxxxxf 求求設(shè)設(shè)解解: :; 1)()( xxf,0時(shí)時(shí)當(dāng)當(dāng) x,0時(shí)時(shí)當(dāng)當(dāng) x)1(11)( xxxf;11x ,0時(shí)時(shí)當(dāng)當(dāng) xhhfh)01ln()0(lim)0(0 , 1 hhfh)01ln()0(1lnlim)0(0 , 1 . 1)0( f.0,110, 1)( xxxxf總之總之第24頁(yè)/共28頁(yè),ch)sh(xx ,sh)ch(xx ,ch1)th(2xx *6. 雙曲函數(shù)與反雙曲函數(shù)的導(dǎo)數(shù)雙曲函數(shù)與反雙曲函數(shù)的導(dǎo)數(shù),11)arsh(2xx ,11)arch(2 xx.11)arth(2xx 證證 )ee (21)sh(xxx)ee (21xx .chx )ee (21)ch(xxx)ee (21xx .shx 第25頁(yè)/共