數(shù)學(xué)分析課件：3-3 導(dǎo)數(shù)的運(yùn)算法則

1、3 導(dǎo)數(shù)的運(yùn)算法則 一、求導(dǎo)的四則運(yùn)算定理3.1并且并且也可導(dǎo)也可導(dǎo)們的和、差、積、商們的和、差、積、商則它則它可導(dǎo)可導(dǎo)在區(qū)間在區(qū)間如果函數(shù)如果函數(shù),)(),(Ixvxu).0)()()()()()()()( )3(2 xvxvxvxuxvxuxvxu);()()()( )()( )2(xvxuxvxuxvxu );()( )()( )1(xvxuxvxu ),0)( ,)()()( xvxvxuxf設(shè)設(shè)hxfhxfxfh)()(lim)(0 hxvhxvhxvxuxvhxuh)()()()()()(lim0 hxvxuhxvhxuh)()()()(lim0 證 僅證僅證(3).(3).hxv

2、hxvxvhxvxuxvxuhxuh)()()()()()()()(lim0 )()()()()()()()(lim0 xvhxvhxvhxvxuxvhxuhxuh 2)()()()()(xvxvxuxvxu .)(處可導(dǎo)處可導(dǎo)在在xxf推廣：四則運(yùn)算性質(zhì)可推廣到有限項(xiàng); )( )()1(11 niiniixfxf);( )()2(xfCxCf )()()()()()( )()3(21211xfxfxfxfxfxfxfnnnii ; )()(11 ninikkkixfxf例1.sin223的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxxy 解23xy x4 例2.ln2sin的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxy 解xxxylncos

3、sin2 xxxylncoscos2 xxxln)sin(sin2 xxx1cossin2 .cos x .2sin1ln2cos2xxxx 例3.tan的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解)cossin()(tan xxxyxxxxx2cos)(cossincos)(sin xxx222cossincos xx22seccos1 .sec)(tan2xx 即即.csc)(cot2xx 同理可得例4.sec的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解)cos1()(sec xxyxx2cos)(cos .tansecxx xx2cossin .cotcsc)(cscxxx 同理可得同理可得二、復(fù)合函數(shù)的求導(dǎo)法則定理3.2 鏈

4、式法則: 對(duì)中間變量的導(dǎo)數(shù)乘以中間變量對(duì) 自變量的導(dǎo)數(shù).)(,)(ufvvu而函數(shù)而函數(shù)可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn)設(shè)函數(shù)設(shè)函數(shù) 在點(diǎn)在點(diǎn)則復(fù)合函數(shù)則復(fù)合函數(shù)可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn))(,)(xfyvu 且且可導(dǎo)可導(dǎo),v).()(vufdvdududydvdy 證,)(可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)由由uufy )(lim0ufuyu )0lim()(0 uufuy故故uuufy )(則則vyv 0lim)(lim00vuvuufv vuvuufvvv 0000limlimlim)( ).()(00 xuf 0, 0 uv時(shí)時(shí)不不妨妨設(shè)設(shè)可推廣到有限次復(fù)合，例如),(),(),(xvvuufy 設(shè)設(shè)的導(dǎo)數(shù)為的導(dǎo)數(shù)為則復(fù)合函數(shù)則

5、復(fù)合函數(shù))(xfy 例5.sinln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解.sin,lnxuuy dxdududydxdy xucos1 xxsincos xcot .dxdvdvdududydxdy 例6.)1(102的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xy解)1()1(10292 xxdxdyxx2)1(1092 .)1(2092 xx例7.arcsin22222的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)axaxaxy 解)arcsin2()2(222 axaxaxy2222222222121xaaxaxxa .22xa )0( a例8.)(sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)nnnxfy 解)(sin)(sin1nnnnnxf

6、xnfy )(sin)(sin1nnnxxn 1cos nnnxx )(sincos113nnnnnxfxxn ).(sin)(sin)(sin1nnnnnxxfx 三、反函數(shù)的導(dǎo)數(shù)定理3.3存在并且可導(dǎo)，滿足存在并且可導(dǎo)，滿足則它的反函數(shù)則它的反函數(shù))(1yfx .)( 1)(1xyf ,0)( )( xfIxf內(nèi)嚴(yán)格單調(diào)，且內(nèi)嚴(yán)格單調(diào)，且區(qū)間區(qū)間設(shè)設(shè)yxyyfyyfyfyy 01101lim)()(lim)(證, 0, 0 xy時(shí)有時(shí)有由嚴(yán)格單調(diào)，由嚴(yán)格單調(diào)，.)( 11lim0 xfxyy 上式上式例9.arcsin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解,)2,2(sin內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可

7、導(dǎo)在在 yIyx, 0cos)(sin yy且且內(nèi)有內(nèi)有在在)1 , 1( xI)(sin1)(arcsin yxycos1 y2sin11 .112x .11)(arccos2xx 同理可得同理可得;11)(arctan2xx .11)cot(2xx arc 例10.log的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xya , 0ln)( aaayy且且,), 0(內(nèi)有內(nèi)有在在 xI)(1)(log yaaxaayln1 .ln1ax 解,),(內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(dǎo)在在 yyIax 四、小結(jié)axxaaaaxxln1)(logln)( xxeexx1)(ln)( xxxxxxxCtansec)(secsec)(tancos)(sin0)(2 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)xxxxxxxxxcotcsc)(csccsc)(cotsin)(cos)(