A2-2函數(shù)的求導(dǎo)法則ppt課件

1、2.2 導(dǎo)數(shù)的運(yùn)算法則導(dǎo)數(shù)的運(yùn)算法則 0 和、差、積、商的求導(dǎo)法則和、差、積、商的求導(dǎo)法則0 反函數(shù)、復(fù)合函數(shù)的求導(dǎo)法則反函數(shù)、復(fù)合函數(shù)的求導(dǎo)法則一、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證證(3)(3),0)( ,)()()( xvx

2、vxuxf設(shè)設(shè)hxfhxfxfh)()(lim)(0 hxvhxvhxvxuxvhxuh)()()()()()(lim0 hxvxuhxvhxuh)()()()(lim0 hxvhxvxvhxvxuxvxuhxuh)()()()()()()()(lim0 )()()()()()()()(lim0 xvhxvhxvhxvxuxvhxuhxuh hxvhxvxvhxvxuxvxuhxuh)()()()()()()()(lim0 )()()()()()()()(lim0 xvhxvhxvhxvxuxvhxuhxuh 2)()()()()(xvxvxuxvxu .)(處處可可導(dǎo)導(dǎo)在在xxf推論推論;

3、)( )()1(11 niiniixfxf);( )()2(xfCxCf ; )()()()()()()()( )()3(1121211 ninikkkinnniixfxfxfxfxfxfxfxfxf例題分析例題分析例例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)(

4、cossincos)(sin xxx222cossincos xx22seccos1 .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.sinh的導(dǎo)數(shù)的導(dǎo)數(shù)求求xy 解解 )(21)(sinh xxeexy)(21xxee .cosh x 同理可得同理可得xxsinh)(cosh xx2cosh1)(tanh .,1csc2. 12yxxy 求求補(bǔ)充題補(bǔ)充題).(,0),1ln

5、(0,)(. 2xfxxxxxf 求求設(shè)設(shè)3. 設(shè)函數(shù)設(shè)函數(shù) f(x)在在 x=0的某鄰域內(nèi)可導(dǎo)的某鄰域內(nèi)可導(dǎo),且且).0(f 求求,3)(lim0 xxfx4. 求證求證:雙曲線雙曲線 x y = a2 (a0)上任一點(diǎn)處切線與坐標(biāo)軸上任一點(diǎn)處切線與坐標(biāo)軸構(gòu)成的三角形面積為常數(shù)構(gòu)成的三角形面積為常數(shù).解解2., 1)( xf,0時(shí)時(shí)當(dāng)當(dāng) x,0時(shí)時(shí)當(dāng)當(dāng) x,11)(xxf ,0時(shí)時(shí)當(dāng)當(dāng) xhhfh)01ln()0(lim)0(0 , 1 hhfh)01ln()0(1lnlim)0(0 , 1 . 1)0( f.0,110, 1)( xxxxf222)1(csc2)1(cotcsc2xxxxx

6、xy 222)1(2)1(cotcsc2xxxxx 解解1.解解3. . 3)(lim)0()(lim)0(, 0)0()(lim),(3)(:,3)(lim0000 xxfxfxfffxfxoxxfxxfxxxx由由極極限限與與無(wú)無(wú)窮窮小小的的關(guān)關(guān)系系解解4. 證明證明: 在曲線上任曲一點(diǎn)在曲線上任曲一點(diǎn)(x,y), )(:),(22xXxayYyx 的的切切線線方方程程為為過(guò)過(guò)點(diǎn)點(diǎn).222121)0 ,(), 0(:22222222222aayxxaxaayxxxaySayxxxay 切切線線與與坐坐標(biāo)標(biāo)軸軸的的交交點(diǎn)點(diǎn)為為222,xayxay 二、反函數(shù)的導(dǎo)數(shù)二、反函數(shù)的導(dǎo)數(shù)定理定理.)

7、(1)(,)(,0)()(yxfIxfyyIyxxy 且有且有內(nèi)也可導(dǎo)內(nèi)也可導(dǎo)在對(duì)應(yīng)區(qū)間在對(duì)應(yīng)區(qū)間那末它的反函數(shù)那末它的反函數(shù)且且內(nèi)單調(diào)、可導(dǎo)內(nèi)單調(diào)、可導(dǎo)在某區(qū)間在某區(qū)間如果函數(shù)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).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)(xfy , 0 y于是有于是有,1yxxy ,)(連連續(xù)續(xù)xf),0(0 xy0)( y 又又知知xyxfx 0lim)(yxy 1lim0)(1y .)(1)(yxf 即即), 0(xIxxx 例例1 1.arcsin的的導(dǎo)導(dǎo)數(shù)數(shù)求求函函數(shù)數(shù)xy 解解,)2,

8、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 同理可得同理可得;11)(arctan2xx )(arcsin x.11)cot(2xx arc例例2 2.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 解解,),(內(nèi)內(nèi)單單調(diào)調(diào)、可可導(dǎo)導(dǎo)在在 yyIax特別地特別地.1)(lnxx hxhxyaahlog)(loglim0 x

9、xhxhah1)1(loglim0 hxahxhx)1(loglim10 .ln1log1axexa 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)證證,)(0可可導(dǎo)導(dǎo)在在點(diǎn)點(diǎn)由由uufy )(lim00ufuyu )0lim()

10、(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3 3.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4 4.)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5 5.arc

11、sin22222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6 6.)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)2(3112 xxx例例7 7.1sin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 , ,求求y y( (/2)./2).xxy22cos1cos1 2222)cos1(sincos2)cos1()c

12、os1(sincos2xxxxxxxy 22)cos1(2sin2xx , y, y( (/2)=0./2)=0.例例8 8例例9 9xnxynsinsin xxnnxxnxnynncossinsinsincos)1( xnxnxnxxnxxnnn)1sin(sin)cossinsin(cossin11 .,arctan1arctanyeeyxx 求求已知已知xtwxvutwyvuyy 21解解：222111)1(11xteeuwv 221arctan21111)1(11xxeeexxx xxxxtgxxxCtansec)(secsec)(cos)(sin0)(2 1.常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)

13、公式常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)公式P94）xxxxctgxxxxxcotcsc)(csccsc)(sin)(cos)(21 axxaaaaxxln1)(logln)( xxeexx1)(ln)( 2211)(arctan11)(arcsinxxxx 2211)(11)(arccosxarcctgxxx 2211)(arctan11)(arcsinxxxx 2211)cot(11)(arccosxxxx arc2.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)法則設(shè)設(shè))(),(xvvxuu 可導(dǎo)，那么可導(dǎo)，那么（1） vuvu )(, （2）uccu )(（3）vuvuuv )(, （4

14、）)0()(2 vvvuvuvu.( ( 是常數(shù)是常數(shù)) )C 3.反函數(shù)、復(fù)合函數(shù)的求導(dǎo)法則反函數(shù)、復(fù)合函數(shù)的求導(dǎo)法則.)(1)(,)(,0)()(xxfIxfyyIyxxy 且且有有內(nèi)內(nèi)也也可可導(dǎo)導(dǎo)區(qū)區(qū)間間在在對(duì)對(duì)應(yīng)應(yīng)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)反函數(shù)的求導(dǎo)法則注意成立條件）反函數(shù)的求導(dǎo)法則注意成立條件）;).()()()()(),(xufxydxdududydxdyxfyxuufy 或或?qū)?shù)為導(dǎo)數(shù)為的的則復(fù)合函數(shù)則復(fù)合函數(shù)而而設(shè)設(shè)利用上述公式及法則初等函數(shù)求導(dǎo)問(wèn)題可完全利用上述公式及法則初等函數(shù)求導(dǎo)問(wèn)題可完全處置處置.注意注意: :初等函數(shù)的導(dǎo)數(shù)仍為初等函數(shù)初等函數(shù)的導(dǎo)數(shù)仍為初等函數(shù). .復(fù)合函數(shù)的求導(dǎo)法則復(fù)合函數(shù)的