1.2求導(dǎo)法則復(fù)合函數(shù)求導(dǎo)1ppt課件

1、求求 導(dǎo)導(dǎo) 法法 那那么么目的與要求目的與要求掌握導(dǎo)數(shù)運(yùn)算法則和基本初等函數(shù)的求導(dǎo)公掌握導(dǎo)數(shù)運(yùn)算法則和基本初等函數(shù)的求導(dǎo)公 式式, 能熟練的求初等函數(shù)的一階能熟練的求初等函數(shù)的一階,二階導(dǎo)數(shù)二階導(dǎo)數(shù)掌握復(fù)合函數(shù)的求導(dǎo)掌握復(fù)合函數(shù)的求導(dǎo)掌握隱函數(shù)所確定的函數(shù)的一、二階導(dǎo)數(shù)掌握隱函數(shù)所確定的函數(shù)的一、二階導(dǎo)數(shù)理解二階導(dǎo)數(shù)的物理意義理解二階導(dǎo)數(shù)的物理意義12222221.02.()3.ln114. (log). (ln )ln5. sincos6. cossin117. tan8. cotcossin119. arcsin10.s111111. arctan12.cot11nnxxaCxnxaaax

2、espxxaxxxxxxxxxxarccoxxxxarcxxx 基基本本初初等等函函數(shù)數(shù)的的導(dǎo)導(dǎo)數(shù)數(shù)公公式式一、和、差、積、商的求導(dǎo)法則一、和、差、積、商的求導(dǎo)法則定理定理并并且且可可導(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).0)()()()()()()()( )3();()()()( )()( )2();()( )()( )1(2 xvxvxvxuxvxuxvxuxvxuxvxuxvxuxvxuxvxu推論推論; )( )()1(11 niiniixfxf);( )()2(xfCxCf

3、; )()()()()()()()( )()3(1121211 ninikkkinnniixfxfxfxfxfxfxfxfxf二、例題分析二、例題分析例例1 1.sin223的導(dǎo)數(shù)的導(dǎo)數(shù)求求xxxy 解解23xy x4 例例2 2.ln2sin的的導(dǎo)導(dǎo)數(shù)數(shù)求求xxy 解解xxxylncossin2 xxxylncoscos2 xxxln)sin(sin2 xxx1cossin2 .cos x .2sin1ln2cos2xxxx 例例3 3.tan的的導(dǎo)導(dǎo)數(shù)數(shù)求求xy 解解)cossin()(tan xxxyxxxxx2cos)(cossincos)(sin xxx222cossincos xx

4、22seccos1 .sec)(tan2xx 即即.csc)(cot2xx 同理可得同理可得例例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 同理可得同理可得例例5 5).(,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) xhxhxxfh)1ln()1ln(lim)(0 )11ln(1lim0 xhhh ,11x ,0時(shí)時(shí)當(dāng)當(dāng) xhhfh)01ln()0(lim)0(0 , 1 hhfh)01ln()0(1lnlim)0(

5、0 , 1 . 1)0( f.0,110, 1)( xxxxf二、復(fù)合函數(shù)的求導(dǎo)法則二、復(fù)合函數(shù)的求導(dǎo)法則定理定理).()(,)(,)()(,)(0000000 xufdxdyxxfyxuufyxxuxx 且且其其導(dǎo)導(dǎo)數(shù)數(shù)為為可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)則則復(fù)復(fù)合合函函數(shù)數(shù)可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)而而可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)如如果果函函數(shù)數(shù)即即 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)中間變等于因變量對(duì)中間變量求導(dǎo)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).(鏈?zhǔn)椒▌t鏈?zhǔn)椒▌t)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

6、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.sin,lnxuuy dxdududydxdy xucos1 xxsincos xcot 例例7 7.)1(102的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù) xy解解)1()1(10292 xxdxdyxx2)1(1092 .)1(2092 xx例例8 8.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 axaxaxy2222222222121xaaxaxxa .22xa )0( a例例9 9.)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(3

7、1211212 xxxy)2(3112 xxx例例1010.1sin的導(dǎo)數(shù)的導(dǎo)數(shù)求函數(shù)求函數(shù)xey 解解)1(sin1sin xeyx)1(1cos1sin xxex.1cos11sin2xexx 三、隱函數(shù)的導(dǎo)數(shù)三、隱函數(shù)的導(dǎo)數(shù)定義定義: :.)(稱(chēng)稱(chēng)為為隱隱函函數(shù)數(shù)由由方方程程所所確確定定的的函函數(shù)數(shù)xyy .)(形形式式稱(chēng)稱(chēng)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(wèn)題問(wèn)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)?隱函數(shù)求導(dǎo)法則隱函數(shù)求導(dǎo)法則: :用復(fù)合函數(shù)求導(dǎo)法則直接對(duì)方程兩邊求導(dǎo)用復(fù)合函數(shù)求導(dǎo)法則直接對(duì)方程兩邊求導(dǎo).例例11

8、11.,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)x0 dxdyeedxdyxyyx解得解得,yxexyedxdy , 0, 0 yx由原方程知由原方程知000 yxyxxexyedxdy. 1 例例1212.,)23,23(,333線(xiàn)線(xiàn)通通過(guò)過(guò)原原點(diǎn)點(diǎn)在在該該點(diǎn)點(diǎn)的的法法并并證證明明曲曲線(xiàn)線(xiàn)的的切切線(xiàn)線(xiàn)方方程程點(diǎn)點(diǎn)上上求求過(guò)過(guò)的的方方程程為為設(shè)設(shè)曲曲線(xiàn)線(xiàn)CCxyyxC 解解,求導(dǎo)求導(dǎo)方程兩邊對(duì)方程兩邊對(duì)xyxyyyx 333322)23,23(22)23,23(xyxyy . 1 所求切線(xiàn)方程為所求切線(xiàn)方程

9、為)23(23 xy. 03 yx即即2323 xy法線(xiàn)方程為法線(xiàn)方程為,xy 即即顯然通過(guò)原點(diǎn)顯然通過(guò)原點(diǎn).例例1313.)1 , 0(, 144處處的的值值在在點(diǎn)點(diǎn)求求設(shè)設(shè)yyxyx 解解求求導(dǎo)導(dǎo)得得方方程程兩兩邊邊對(duì)對(duì)x)1(04433 yyyxyx得得代入代入1, 0 yx;4110 yxy求求導(dǎo)導(dǎo)得得兩兩邊邊再再對(duì)對(duì)將將方方程程x)1(04)(122123222 yyyyyxyx得得4110 yxy, 1, 0 yx代代入入.16110 yxy四、反函數(shù)的導(dǎo)數(shù)四、反函數(shù)的導(dǎo)數(shù)定理定理.)(1)(,)(,0)()(xxfIxfyyIyxxy 且且有有內(nèi)內(nèi)也也可可導(dǎo)導(dǎo)在在對(duì)對(duì)應(yīng)應(yīng)區(qū)區(qū)間間

10、那那末末它它的的反反函函數(shù)數(shù)且且內(nèi)內(nèi)單單調(diào)調(diào)、可可導(dǎo)導(dǎo)在在某某區(qū)區(qū)間間如如果果函函數(shù)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1414.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 yIyx, 0cos)(sin yy且且內(nèi)內(nèi)有有在在)1 , 1( xI)(sin1)(arcsin yxycos1 y2sin11 .112x .11)(arccos2xx 同理可得同理可得;11)(arctan2xx )(arcsin x.11)cot(2xx arc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)

11、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例例1515解解 142)1(3111)4(1)1(23 xxxexxxyx等式兩邊取對(duì)數(shù)得等式兩邊取對(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)例例1616解解.),0(sinyxxyx 求求設(shè)設(shè)等

12、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 一般地一般地)0)()()()( xuxuxfxv)()(1)(lnxfdxdxfxfdxd 又又)(ln)()(xfdxdxfxf )()()()(ln)()()()(xuxuxvxuxvxuxfxv )(ln)()(lnxuxvxf 六、高階導(dǎo)數(shù)的定義六、高階導(dǎo)數(shù)的定義問(wèn)題問(wèn)題: :變速直線(xiàn)運(yùn)動(dòng)的加速度變速直線(xiàn)運(yùn)動(dòng)的加速度. .),(tfs 設(shè)設(shè))()(tftv 則瞬時(shí)速度為則瞬時(shí)速度為的的變變化

13、化率率對(duì)對(duì)時(shí)時(shí)間間是是速速度度加加速速度度tva. )()()( tftvta定義定義.)() )(,)()(lim) )(,)()(0處的二階導(dǎo)數(shù)處的二階導(dǎo)數(shù)在點(diǎn)在點(diǎn)為函數(shù)為函數(shù)則稱(chēng)則稱(chēng)存在存在即即處可導(dǎo)處可導(dǎo)在點(diǎn)在點(diǎn)的導(dǎo)數(shù)的導(dǎo)數(shù)如果函數(shù)如果函數(shù)xxfxfxxfxxfxfxxfxfx 記作記作.)(,),(2222dxxfddxydyxf或或 記記作作階階導(dǎo)導(dǎo)數(shù)數(shù)的的函函數(shù)數(shù)階階導(dǎo)導(dǎo)數(shù)數(shù)的的導(dǎo)導(dǎo)數(shù)數(shù)稱(chēng)稱(chēng)為為的的函函數(shù)數(shù)一一般般地地,)(1)(,nxfnxf .)(,),()()(nnnnnndxxfddxydyxf或或三階導(dǎo)數(shù)的導(dǎo)數(shù)稱(chēng)為四階導(dǎo)數(shù)三階導(dǎo)數(shù)的導(dǎo)數(shù)稱(chēng)為四階導(dǎo)數(shù), 二階和二階以上的導(dǎo)

14、數(shù)統(tǒng)稱(chēng)為高階導(dǎo)數(shù)二階和二階以上的導(dǎo)數(shù)統(tǒng)稱(chēng)為高階導(dǎo)數(shù).)(;)(,稱(chēng)稱(chēng)為為一一階階導(dǎo)導(dǎo)數(shù)數(shù)稱(chēng)稱(chēng)為為零零階階導(dǎo)導(dǎo)數(shù)數(shù)相相應(yīng)應(yīng)地地xfxf .,),(33dxydyxf 二階導(dǎo)數(shù)的導(dǎo)數(shù)稱(chēng)為三階導(dǎo)數(shù)二階導(dǎo)數(shù)的導(dǎo)數(shù)稱(chēng)為三階導(dǎo)數(shù),.,),(44)4()4(dxydyxf七、七、 高階導(dǎo)數(shù)求法舉例高階導(dǎo)數(shù)求法舉例例例1 1).0(),0(,arctanffxy 求求設(shè)設(shè)解解211xy )11(2 xy22)1(2xx )1(2(22 xxy322)1()13(2xx 022)1(2)0( xxxf0322)1()13(2)0( xxxf; 0 . 2 直接法直接法: :由高階導(dǎo)數(shù)的定義逐步求高階導(dǎo)數(shù)由高階導(dǎo)

15、數(shù)的定義逐步求高階導(dǎo)數(shù).例例2 2.),()(nyRxy求求設(shè)設(shè) 解解1 xy)(1 xy2)1( x3)2)(1( x)1(2 xy)1()1()1()( nxnynn則則為自然數(shù)為自然數(shù)若若,n )()()(nnnxy , !n ) !()1( nyn. 0 例例3 3.),1ln()(nyxy求求設(shè)設(shè) 解解注意注意: :xy 112)1(1xy 3)1(! 2xy 4)4()1(! 3xy )1! 0, 1()1()!1()1(1)( nxnynnn 求求n n階導(dǎo)數(shù)時(shí)階導(dǎo)數(shù)時(shí), ,求出求出1-31-3或或4 4階后階后, ,不要急于合不要急于合并并, ,分析結(jié)果的規(guī)律性分析結(jié)果的規(guī)律性, ,寫(xiě)出寫(xiě)出n n階導(dǎo)數(shù)階導(dǎo)數(shù).(.(數(shù)學(xué)歸納數(shù)學(xué)歸納法證明法證明) )例例4 4.,sin)(nyxy求求設(shè)設(shè) 解解xycos )2sin( x)2c