導(dǎo)數(shù)運算法則

1、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)處也處也在點在點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)在點在點如果函數(shù)如果函數(shù),)(,)(),(xxxvxu).0)()()()()()()()()3();()()()( )()()2();()( )()()1(2 xvxvxvxuxvxuxvxuxvxuxvxuxvxuxvxuxvxuvuvu )(vuvuuv )(2)(vvuvuvu 證證 (2)(2),()()(xvxuxf 設(shè)設(shè)xxfxxfxfx)()(lim)(0 xxvxuxxvxxux)()()()(lim0 xxvxxvxuxxvxuxx

2、ux)()()()()()(lim0)()()()(xvxuxvxu );()()()( )()(xvxuxvxuxvxu 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 例例4 4.tan的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解解)cossin()(tan xxxyxxxxx2cos)(cossincos)(s

3、in xxx222cossincos xx22seccos1 .sec)(tan2xx 即即.csc)(cot2xx 同理可得同理可得例例5 5.sec的導(dǎo)數(shù)的導(dǎo)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 .)(cossimxx.ln1)(logaxxa.1)(lnxx .sec)(tan2xx .csc)(cot2xx )(secx.tansecx

4、x.cotcsc)(cscxxx 122|11xyyxxy及求設(shè)12yxxxy求設(shè)cosln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)點連續(xù)且反函數(shù)，處有不等于零的導(dǎo)數(shù)在點如果函數(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).例例7 7.arcsin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xy )(sin1)(arcsin yxycos1 y2sin11.112x)(arcsin x211x.112x .11)(arccos2xx 同理可得同理可得;11)(arctan2xx )(arcsin x.11)cot(

5、2xx arcyaaayx求且已知)10(例：的反函數(shù)是解yxay：axlog)(xa)(log1yaaylnaaxln：即 )(xaaaxln特別地： xxee )(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)在點則復(fù)合函數(shù)可導(dǎo)在點而可導(dǎo)在點如果函數(shù)即即 函數(shù)函數(shù)對自變量求導(dǎo)對自變量求導(dǎo), ,等于等于函數(shù)函數(shù)對中間變量求對中間變量求導(dǎo)導(dǎo), ,乘以中間變量對自變量求導(dǎo)乘以中間變量對自變量求導(dǎo).(.(鏈式法則鏈式法則) )證證,)(0可可導(dǎo)導(dǎo)在在點點由由uufy )(lim00ufuyu )0lim

6、()(00 uufuy故故uuufy )(0則則xyx 0lim)(lim00 xuxuufx xuxuufxxx 0000limlimlim)().()(00 xuf 推廣推廣),(),(),(xvvuufy 設(shè)設(shè).)(dxdvdvdududydxdyxfy 的導(dǎo)數(shù)為的導(dǎo)數(shù)為則復(fù)合函數(shù)則復(fù)合函數(shù) dxdududydxdy 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的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey 解解,3xueyu dxdududydxdy .33223xexexu 例例111

7、1.)1(102的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù) xy解解xu2109 xx2)1(1092 .)1(2092 xxdxdududydxdy 1,210 xuuy例例1212.arcsin22222的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)axaxaxy 解解)arcsin2()2(222 axaxaxy 22122xxa.22xa )0( a2222222222121xaaxaxxa 222xa x20 22a2)(1ax a1xx21)( 211)(arcsinxx 例例1313.)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(212 xxy11212 xy)2(3112 xxx例

8、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 只需在方程只需在方程f(x,y)=0的兩邊同時對的兩邊同時對x求導(dǎo)。而在求求導(dǎo)。而在求導(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)。.)(形式稱為顯函數(shù)形式稱為顯函數(shù)xfy 0),( yxf)(xfy

9、 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)?例例1 1.,00 xyxdxdydxdyyeexy的導(dǎo)數(shù)的導(dǎo)數(shù)所確定的隱函數(shù)所確定的隱函數(shù)求由方程求由方程解解,求導(dǎo)求導(dǎo)方程兩邊對方程兩邊對x0 dxdyeedxdyxyyx解得解得,yxexyedxdy , 0, 0 yx由原方程知由原方程知000 yxyxxexyedxdy. 1 五、對數(shù)求導(dǎo)法五、對數(shù)求導(dǎo)法觀察函數(shù)觀察函數(shù).,)4(1)1(sin23xxxyexxxy 方法方法: :先在方程兩邊取對數(shù)先在方程兩邊取對數(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)

10、.-對數(shù)求導(dǎo)法對數(shù)求導(dǎo)法適用范圍適用范圍: :.)()(的情形的情形數(shù)數(shù)多個函數(shù)相乘和冪指函多個函數(shù)相乘和冪指函xvxu例例1 1解解 142)1(3111)4(1)1(23 xxxexxxyx等式兩邊取對數(shù)得等式兩邊取對數(shù)得xxxxy )4ln(2)1ln(31)1ln(ln求導(dǎo)得求導(dǎo)得上式兩邊對上式兩邊對 x142)1(3111 xxxyy.,)4(1)1(23yexxxyx 求求設(shè)設(shè)例例2 2解解.),0(sinyxxyx 求求設(shè)設(shè)等式兩邊取對數(shù)得等式兩邊取對數(shù)得xxylnsinln 求導(dǎo)得求導(dǎo)得上式兩邊對上式兩邊對xxxxxyy1sinlncos1 )1sinln(cosxxxxyy

11、)sinln(cossinxxxxxx 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).,)()(定的函數(shù)定的函數(shù)稱此為由參數(shù)方程所確稱此為由參數(shù)方程所確間的函數(shù)關(guān)系間的函數(shù)關(guān)系與與確定確定若參數(shù)方程若參數(shù)方程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),()(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 且且都可導(dǎo)都可導(dǎo)再設(shè)函數(shù)再設(shè)函數(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)

12、的求導(dǎo)法則得dxdtdtdydxdy dtdxdtdy1 )()(tt dtdxdtdydxdy 即即,)()(中中在方程在方程 tytx 例例解解dtdxdtdydxdy ttcos1sin taatacossin 2cos12sin2 tdxdy. 1 .方方程程處的切線處的切線在在求擺線求擺線2)cos1()sin( ttayttax.),12(,2ayaxt 時時當當 所求切線方程為所求切線方程為)12( axay)22( axy即即 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)(c

13、os)(sin)()(xxxxxxxc )cot()(arctan)(arccos)(arcsin)(ln)(log)()(xarcxxxxxeaaxx01 xxcosxsin x2secx2csc xx tansec xx cotcsc xaxlnxeaxln/1x/121/1x 21/1x )1/(12x )1/(12x 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)可導(dǎo)，則可導(dǎo)，則、設(shè)設(shè))()(xvxu )( 10vuv vu u ) ( 20cuuc 03 )(uvvuvu 04 )(vu2vvu

14、vu 的導(dǎo)數(shù)為的導(dǎo)數(shù)為則復(fù)合函數(shù)則復(fù)合函數(shù))(xfy ).()()(xufxydxdududydxdy 或或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的兩邊同時對的兩邊同時對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、對數(shù)求導(dǎo)法、對數(shù)求導(dǎo)法：先對函數(shù)取對數(shù)再求導(dǎo)的方法。先對函數(shù)取對數(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：。例例1515.的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xxxy 解解xxxy 21xxx 21xxxxx 211(21.812422xxxxxxxxxx )( xxx 1(xx 21)( xx)211(x 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 xsi