函數(shù)的求導(dǎo)法則及基本導(dǎo)數(shù)公式

1、 2.3 函數(shù)的求導(dǎo)法則及基函數(shù)的求導(dǎo)法則及基本導(dǎo)數(shù)公式本導(dǎo)數(shù)公式(Operational rules of derivative)一、和、差、積、商的求導(dǎo)法則一、和、差、積、商的求導(dǎo)法則二、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)的求導(dǎo)法則三、復(fù)合函數(shù)的求導(dǎo)法則四、基本求導(dǎo)法則與導(dǎo)數(shù)公式四、基本求導(dǎo)法則與導(dǎo)數(shù)公式五、小結(jié)五、小結(jié) 并并且且可可導(dǎo)導(dǎo)處處也也在在點(diǎn)點(diǎn)分分母母不不為為零零們們的的和和、差差、積積、商商則則它它處處可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)如如果果函函數(shù)數(shù),)(,)(),(xxxvxu一、和、差、積、商的求導(dǎo)法則定理定理1);()( )()()(xvxuxvxu1).)()()()()

2、()()()()(032xvxvxvxuxvxuxvxu);()()()( )()()(xvxuxvxuxvxu2幻燈片 5證證(3)(3),0)( ,)()()( xvxvxuxf設(shè)設(shè)hxfhxfxfh)()(lim)(0 hxvhxvhxvxuxvhxuh)()()()()()(lim0 hxvxuhxvhxuh)()()()(lim0 證證(1)(1)、(2)(2)略略. .hxvhxvxvhxvxuxvxuhxuh)()()()()()()()(lim0 hxvhxvxvhxvxuxvxuhxuh)()()()()()()()(lim0 )()()()()()()()(lim0 xvh

3、xvhxvhxvxuxvhxuhxuh 2)()()()()(xvxvxuxvxu .)(處可導(dǎo)處可導(dǎo)在在xxf上面的法則可分別簡(jiǎn)寫(xiě)為：上面的法則可分別簡(jiǎn)寫(xiě)為：;)()(vuvu1).()()(032vvvuvuvu;)()(vuvuvu2可推廣到有限多項(xiàng)可推廣到有限多項(xiàng)特別地特別地.)(uccu如如.)(wvuwvuwvuwvu例例1 1.,cossin02342xyyxxxy及及求求解解23xy x4 .cos x . 10 xy例例2 2解解.),cos(sinyxxeyx求求)cos(sin)cos(sin)(xxexxeyxx)sin(cos)cos(sinxxexxexx.cos

4、xex2vuvuvu )(例例3 3.tan的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解解)cossin()(tan xxxyxxxxx2cos)(cossincos)(sin xxx222cossincos xx22seccos1 .sec)(tan2xx 即即.csc)(cot2xx 同理可得同理可得2vvuvuvu )(例例4 4.sec 的的導(dǎo)導(dǎo)數(shù)數(shù)求求xy 解解)cos1()(sec xxyxx2cos)(cos .tansecxx xx2cossin .cotcsc)(cscxxx 同理可得同理可得.tansec)(secxxx即即.)( )(,),(|)(,)()(dydxdxdyyfxfIyyfx

5、xIxfyyfIyfxyxy11011或或且且內(nèi)內(nèi)也也可可導(dǎo)導(dǎo)在在區(qū)區(qū)間間那那末末它它的的反反函函數(shù)數(shù)可可導(dǎo)導(dǎo)且且內(nèi)內(nèi)單單調(diào)調(diào)、在在某某區(qū)區(qū)間間如如果果函函數(shù)數(shù)二、反函數(shù)的導(dǎo)數(shù)定理定理2簡(jiǎn)單地說(shuō)：簡(jiǎn)單地說(shuō)：反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù)反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù). .幻燈片 11證證,xIx 任取任取xx 以增量以增量給給的單調(diào)性可知的單調(diào)性可知由由)(xfy1, 0 y于是有于是有,1yxxy ,)(連連續(xù)續(xù)xf1),0(0 xy0)(yf又知又知xyxfx01lim )(yxy 1lim0.)(yf 1), 0(xIxxx 例例5 5.arcsin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xy

6、解解,)2,2(sin內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(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 ;11)(arctan2xx )(arcsin x.11)cot(2xx arc.112x 而而可導(dǎo)可導(dǎo)在點(diǎn)在點(diǎn)如果函數(shù)如果函數(shù),)(xxgu 三、復(fù)合函數(shù)的求導(dǎo)法則定理定理3意義：意義：因變量對(duì)自變量求導(dǎo)因變量對(duì)自變量求導(dǎo), ,等于因變量對(duì)中間等于因變量對(duì)中間 變量求導(dǎo)變量求導(dǎo), ,乘以中間變量對(duì)自變量求導(dǎo)乘以中間變量對(duì)自變量求導(dǎo). . 則復(fù)合函數(shù)則復(fù)合函數(shù)可導(dǎo)可導(dǎo)在

7、點(diǎn)在點(diǎn),)(xgu 且且其其導(dǎo)導(dǎo)數(shù)數(shù)為為可可導(dǎo)導(dǎo),x.)()(dxdududydxdyxgufdxdy或或)(ufy 在點(diǎn)在點(diǎn))(xgfy 幻燈片 15證證,)(可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)由由uufy )(limufuyu0)lim()(00uufuy故故uuufy)(則則xyx 0lim)(limxuxuufx0 xuxuufxxx000limlimlim)().()(xguf證畢證畢.可推廣到多個(gè)中間變量可推廣到多個(gè)中間變量),(),(),(xvvuufy 設(shè)設(shè).)(dxdvdvdududydxdyxfy 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) 例例6 6.sinln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解解

8、.sin,lnxuuy dxdududydxdy xucos1 xxsincos xcot .xvuxvuyy 或或.)cos(ln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey .,cos,lnxevvuuy所所給給函函數(shù)數(shù)可可分分解解為為,sin,xedxdvvdvduududy1xevudxdy)sin(1例例7 7解解).tan()cos()sin(xxxxxeeeee熟練后熟練后, 不必再寫(xiě)出中間變量不必再寫(xiě)出中間變量.例例9 9.)1(102的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xy解解)1()1(10292 xxdxdyxx2)1(1092 .)1(2092 xx復(fù)合函數(shù)求導(dǎo)順序復(fù)合函數(shù)求導(dǎo)順序“由外往里，

9、逐層求導(dǎo)由外往里，逐層求導(dǎo)”. .例例8 8解解.sin的的導(dǎo)導(dǎo)數(shù)數(shù)求求xy25. )(sinxxy2254)(sinsinxxy2254.cossinxx22104)(cossinxxx22254例例1010.1sin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xey 解解)1(sin1sin xeyx)1(1cos1sin xxex.1cos11sin2xexx 例例1111.)2(21ln32的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù) xxxy解解),2ln(31)1ln(212 xxy)2(31211212 xxxy.)(23112xxx,0時(shí)時(shí)當(dāng)當(dāng) xxxfx)ln(lim)(0100, 1 xxfx)ln()ln(li

10、m)(01100, 1 . 1)0( f例例1212).(,0),1ln(0,)(xfxxxxxf 求求設(shè)設(shè)解解, 1)( xf,0時(shí)時(shí)當(dāng)當(dāng) x,0時(shí)時(shí)當(dāng)當(dāng) x,)(xxf11.)()(lim)(axafxfafax.)(10 f且且例例1212).(,0),1ln(0,)(xfxxxxxf 求求設(shè)設(shè)解解.,)(01101xxxxf, 1)( xf,0時(shí)時(shí)當(dāng)當(dāng) x,0時(shí)時(shí)當(dāng)當(dāng) x,)(xxf11例例1313.ln2sin的的導(dǎo)導(dǎo)數(shù)數(shù)求求xxy 解解.2sin1ln2cos2xxxx )(lnsinln)(sin xxxxy22四、基本求導(dǎo)法則與導(dǎo)數(shù)公式0 )(C1.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1xx)(axxaaaaxxln1)(logln)( xxeexx1)(ln)( xxxxxxxtansec)(secsec)(tancos)(sin2xxxxxxxcotcsc)(csccsc)(cotsin)(cos2221111xxxx)(arctan)(arcsin2. 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.3. 反函數(shù)的求導(dǎo)法則反函數(shù)的求導(dǎo)法則.4. 復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的求導(dǎo)法則.221111xxarcxx)cot()(arccos五、小結(jié)1