導(dǎo)數(shù)運(yùn)算法則[1]

1、導(dǎo)數(shù)運(yùn)算法則最新一、和、差、積、商的求導(dǎo)法則定理定理并且并且可導(dǎo)可導(dǎo)處也處也在點(diǎn)在點(diǎn)分母不為零分母不為零們的和、差、積、商們的和、差、積、商則它則它處可導(dǎo)處可導(dǎo)在點(diǎn)在點(diǎn)如果函數(shù)如果函數(shù),)(,)(),(xxxvxu).0)()()()()()()()()3();()()()( )()()2();()( )()()1(2 xvxvxvxuxvxuxvxuxvxuxvxuxvxuxvxuxvxu導(dǎo)數(shù)運(yùn)算法則最新vuvu )(vuvuuv )(2)(vvuvuvu 導(dǎo)數(shù)運(yùn)算法則最新證證 (2)(2),()()(xvxuxf 設(shè)設(shè)xxfxxfxfx)()(lim)(0 xxvxuxxvxxux)()(

2、)()(lim0 xxvxxvxuxxvxuxxux)()()()()()(lim0)()()()(xvxuxvxu );()()()( )()(xvxuxvxuxvxu 導(dǎo)數(shù)運(yùn)算法則最新 n n2 21 1n n2 21 1n n2 21 1u uu uu uu uu uu u) )u uu u( (u u2推論uC)Cu( 1推論 n21uuu3推論2)1(VVV導(dǎo)數(shù)運(yùn)算法則最新例例1 1.sin223的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxxy 解解23xy x4 .4cos 例例2 2.4sin36的導(dǎo)數(shù)的導(dǎo)數(shù)求求 xxy解解56xy 3 .cos x . 0 導(dǎo)數(shù)運(yùn)算法則最新例例4 4.tan的的導(dǎo)導(dǎo)數(shù)

3、數(shù)求求xy 解解)cossin()(tan xxxyxxxxx2cos)(cossincos)(sin xxx222cossincos xx22seccos1 .sec)(tan2xx 即即.csc)(cot2xx 同理可得同理可得導(dǎo)數(shù)運(yùn)算法則最新例例5 5.sec的的導(dǎo)導(dǎo)數(shù)數(shù)求求xy 解解)cos1()(sec xxyxx2cos)(cos0 .tansecxx xx2cossin .cotcsc)(cscxxx 同理可得同理可得 )(secx.tansecxx導(dǎo)數(shù)運(yùn)算法則最新基本初等函數(shù)的導(dǎo)數(shù)有：基本初等函數(shù)的導(dǎo)數(shù)有：. 0)(C)(.)(1Rxx .cos)(sinxx .)(cossi

4、mxx.ln1)(logaxxa.1)(lnxx .sec)(tan2xx .csc)(cot2xx )(secx.tansecxx.cotcsc)(cscxxx 導(dǎo)數(shù)運(yùn)算法則最新122|11xyyxxy及求設(shè)12yxxxy求設(shè)cosln導(dǎo)數(shù)運(yùn)算法則最新二、反函數(shù)的導(dǎo)數(shù)定理定理)(1 )( )()()()(111xfyf，yf，yfxxfxxfy且有存在則在相應(yīng)點(diǎn)連續(xù)且反函數(shù)，處有不等于零的導(dǎo)數(shù)在點(diǎn)如果函數(shù)即即 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).導(dǎo)數(shù)運(yùn)算法則最新例例7 7.arcsin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xy )(sin1)(arcsin yxycos1

5、 y2sin11.112x)(arcsin x211x導(dǎo)數(shù)運(yùn)算法則最新.112x .11)(arccos2xx 同理可得同理可得;11)(arctan2xx )(arcsin x.11)cot(2xx arc導(dǎo)數(shù)運(yùn)算法則最新yaaayx求且已知)10(例：的反函數(shù)是解yxay：axlog)(xa)(log1yaaylnaaxln：即 )(xaaaxln特別地： xxee )(導(dǎo)數(shù)運(yùn)算法則最新三、復(fù)合函數(shù)的求導(dǎo)法則定理定理000,)(,)()(,)(0000 xxuuxxdxdududydxdyxxfyxuufyxxu且其導(dǎo)數(shù)為可導(dǎo)在點(diǎn)則復(fù)合函數(shù)可導(dǎo)在點(diǎn)而可導(dǎo)在點(diǎn)如果函數(shù)即即 函數(shù)函數(shù)對(duì)自變量求

6、導(dǎo)對(duì)自變量求導(dǎo), ,等于等于函數(shù)函數(shù)對(duì)中間變量求對(duì)中間變量求導(dǎo)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).(.(鏈?zhǔn)椒▌t鏈?zhǔn)椒▌t) )導(dǎo)數(shù)運(yùn)算法則最新證證,)(0可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)由由uufy )(lim00ufuyu )0lim()(00 uufuy故故uuufy )(0則則xyx 0lim)(lim00 xuxuufx xuxuufxxx 0000limlimlim)().()(00 xuf 導(dǎo)數(shù)運(yùn)算法則最新推廣推廣),(),(),(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) dxdududydxdy 導(dǎo)數(shù)運(yùn)算法則最新

7、例例1010.sinln的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy 解解.sin,lnxuuy dxdududydxdy xucos1 xxsincos xcot 9例例.3的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xey 解解,3xueyu dxdududydxdy .33223xexexu 導(dǎo)數(shù)運(yùn)算法則最新例例1111.)1(102的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù) xy解解xu2109 xx2)1(1092 .)1(2092 xxdxdududydxdy 1,210 xuuy導(dǎo)數(shù)運(yùn)算法則最新例例1212.arcsin22222的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)axaxaxy 解解)arcsin2()2(222 axaxaxy 22122x

8、xa.22xa )0( a2222222222121xaaxaxxa 222xa x20 22a2)(1ax a1xx21)( 211)(arcsinxx 導(dǎo)數(shù)運(yùn)算法則最新例例1313.)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 xxy11212 xy)2(3112 xxx例例1414.1sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey 解解xey1sin xe1sin .1cos11sin2xexx x2 )2(31 x)1(sin xx1cos )1( x導(dǎo)數(shù)運(yùn)算法則最新 只需在方程只需在方程F(x,y)=0的兩邊同時(shí)對(duì)的兩邊同時(shí)對(duì)x求導(dǎo)。而在求求導(dǎo)。而在

9、求導(dǎo)過程中，把導(dǎo)過程中，把y看成看成x的函數(shù)。（導(dǎo)數(shù)結(jié)果中可含有的函數(shù)。（導(dǎo)數(shù)結(jié)果中可含有y)四、隱函數(shù)求導(dǎo)法：四、隱函數(shù)求導(dǎo)法：隱函數(shù)：隱函數(shù)：若若x與與y的函數(shù)關(guān)系由方程的函數(shù)關(guān)系由方程F(x,y)=0確定，確定，則稱這種函數(shù)關(guān)系為隱函數(shù)。則稱這種函數(shù)關(guān)系為隱函數(shù)。.)(形形式式稱稱為為顯顯函函數(shù)數(shù)xfy 0),( yxF)(xfy 隱函數(shù)的顯化隱函數(shù)的顯化問題問題:隱函數(shù)不易顯化或不能顯化如何求導(dǎo)隱函數(shù)不易顯化或不能顯化如何求導(dǎo)?導(dǎo)數(shù)運(yùn)算法則最新例例1 1.,00 xyxdxdydxdyyeexy的的導(dǎo)導(dǎo)數(shù)數(shù)所所確確定定的的隱隱函函數(shù)數(shù)求求由由方方程程解解,求導(dǎo)求導(dǎo)方程兩邊對(duì)方程兩邊對(duì)x

10、0 dxdyeedxdyxyyx解得解得,yxexyedxdy , 0, 0 yx由原方程知由原方程知000 yxyxxexyedxdy. 1 導(dǎo)數(shù)運(yùn)算法則最新五、對(duì)數(shù)求導(dǎo)法五、對(duì)數(shù)求導(dǎo)法觀察函數(shù)觀察函數(shù).,)4(1)1(sin23xxxyexxxy 方法方法: :先在方程兩邊取對(duì)數(shù)先在方程兩邊取對(duì)數(shù), 然后利用隱函數(shù)的求導(dǎo)然后利用隱函數(shù)的求導(dǎo)方法求出導(dǎo)數(shù)方法求出導(dǎo)數(shù).-對(duì)數(shù)求導(dǎo)法對(duì)數(shù)求導(dǎo)法適用范圍適用范圍: :.)()(的情形的情形數(shù)數(shù)多個(gè)函數(shù)相乘和冪指函多個(gè)函數(shù)相乘和冪指函xvxu導(dǎo)數(shù)運(yùn)算法則最新例例1 1解解 142)1(3111)4(1)1(23 xxxexxxyx等式兩邊取對(duì)數(shù)得等式

11、兩邊取對(duì)數(shù)得xxxxy )4ln(2)1ln(31)1ln(ln求導(dǎo)得求導(dǎo)得上式兩邊對(duì)上式兩邊對(duì) x142)1(3111 xxxyy.,)4(1)1(23yexxxyx 求求設(shè)設(shè)導(dǎo)數(shù)運(yùn)算法則最新例例2 2解解.),0(sinyxxyx 求求設(shè)設(shè)等式兩邊取對(duì)數(shù)得等式兩邊取對(duì)數(shù)得xxylnsinln 求求導(dǎo)導(dǎo)得得上上式式兩兩邊邊對(duì)對(duì)xxxxxyy1sinlncos1 )1sinln(cosxxxxyy )sinln(cossinxxxxxx 導(dǎo)數(shù)運(yùn)算法則最新六、由參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù)六、由參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù).,)()(定定的的函函數(shù)數(shù)稱稱此此為為由由參參數(shù)數(shù)方方程程所所確確間間的的函

12、函數(shù)數(shù)關(guān)關(guān)系系與與確確定定若若參參數(shù)數(shù)方方程程xytytx 例如例如 ,22tytx2xt 22)2(xty 42x xy21 消去參數(shù)消去參數(shù)問題問題: : 消參困難或無法消參如何求導(dǎo)消參困難或無法消參如何求導(dǎo)?t導(dǎo)數(shù)運(yùn)算法則最新),()(1xttx 具有單調(diào)連續(xù)的反函數(shù)具有單調(diào)連續(xù)的反函數(shù)設(shè)函數(shù)設(shè)函數(shù))(1xy , 0)(,)(),( ttytx 且且都都可可導(dǎo)導(dǎo)再再設(shè)設(shè)函函數(shù)數(shù)由復(fù)合函數(shù)及反函數(shù)的求導(dǎo)法則得由復(fù)合函數(shù)及反函數(shù)的求導(dǎo)法則得dxdtdtdydxdy dtdxdtdy1 )()(tt dtdxdtdydxdy 即即,)()(中中在方程在方程 tytx 導(dǎo)數(shù)運(yùn)算法則最新例例解解d

13、tdxdtdydxdy ttcos1sin taatacossin 2cos12sin2 tdxdy. 1 .方程方程處處的的切切線線在在求求擺擺線線2)cos1()sin( ttayttax導(dǎo)數(shù)運(yùn)算法則最新.),12(,2ayaxt 時(shí)時(shí)當(dāng)當(dāng) 所求切線方程為所求切線方程為)12( axay)22( axy即即導(dǎo)數(shù)運(yùn)算法則最新 七、初等函數(shù)的求導(dǎo)問題七、初等函數(shù)的求導(dǎo)問題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 )(csc)(sec)(cot)(tan)(cos)(sin)()(xxxxxxxC )cot()(arctan)(arccos)(arcsin)(ln)(log)

14、()(xarcxxxxxeaaxx01 xxcosxsin x2secx2csc xx tansec xx cotcsc xaxlnxeaxln/1x/121/1x 21/1x )1/(12x )1/(12x 導(dǎo)數(shù)運(yùn)算法則最新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3.復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的求導(dǎo)法則),(),(xuufy 而而設(shè)設(shè)可可導(dǎo)導(dǎo)，則則、設(shè)設(shè))()(xvxu )( 10vuv vu u ) ( 20CuuC 03 )(uvvuvu 04 )(vu2vvuvu 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 ).()()(xufxydxdududydxdy

15、或或?qū)?shù)運(yùn)算法則最新5、隱函數(shù)求導(dǎo)法、隱函數(shù)求導(dǎo)法：只需在方程只需在方程F(x,y)=0的兩邊同時(shí)對(duì)的兩邊同時(shí)對(duì)x求導(dǎo)。求導(dǎo)。而在求導(dǎo)過程中，把而在求導(dǎo)過程中，把y看成看成x的函數(shù)。（導(dǎo)數(shù)結(jié)果中可含有的函數(shù)。（導(dǎo)數(shù)結(jié)果中可含有y)4、反函數(shù)的求導(dǎo)法則、反函數(shù)的求導(dǎo)法則：反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)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)的倒數(shù).6、對(duì)數(shù)求導(dǎo)法、對(duì)數(shù)求導(dǎo)法：先對(duì)函數(shù)取對(duì)數(shù)再求導(dǎo)的方法。先對(duì)函數(shù)取對(duì)數(shù)再求導(dǎo)的方法。7、參數(shù)方程求導(dǎo)法、參數(shù)方程求導(dǎo)法：。導(dǎo)數(shù)運(yùn)算法則最新例例1515.的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xxxy 解解xxxy 21xxx 21xxxxx 211(21.812422xxxxxxxxxx )( xxx 1(xx 21)( xx)211(x 導(dǎo)數(shù)運(yùn)算法則最新yxyx 求求例例, 16sin1sinsin xxxy解解：xxyxlnsin )(xfa )(xf指數(shù)函數(shù)指數(shù)函數(shù)冪函數(shù)冪函數(shù)冪指函數(shù)冪指函數(shù))(sin xxy)(sinln xxe)(lnsin xxe )(lnsinxxe)ln(sin xx xxsinx(cosxln xsin