高等数学课件：2-2函数的求导法则

1、1第二节第二节 求导法则求导法则2xxuxxux )()(lim0 xxvxxvx )()(lim0 )()(xvxu . 设设)(xuu , ,)(xvv 可可导导, ,则则vu , ,uv, ,vu )0( v均均可可导导, ,且且有有 )( vuxxvxuxxvxxux )()()()(lim0 证证.)()1(vuvu 注注: : 可推广到有限多个函数的和与差。可推广到有限多个函数的和与差。 一、导数的四则运算法则一、导数的四则运算法则3证证xxvxuxxvxxuxy )()()()()()()()(xvxuxxvxu )()()()(xxvxuxxvxxuy .)()2(vuvuuv

2、 xxxvxuxxvxxuxy /)()()()(/xxvxuxxvxu /)()()()(4因因为为)(xv可可导导, ,必必连连续续, , )()()()(xvxuxvxu . 故故)()(lim0 xvxxvx , ,于于是是 )()()(xxvxxuxxuxy xxvxxvxu )()()(5hexehxxhxh000)(00sin)sin(lim 0)(sin xxxxe hexexexehxxhxhxhxh00000)(0)(0)(00sinsinsin)sin(lim hexexhexehxxhxhhxhxh00000)(00)(0)(00sinsinlimsin)sin(lim

3、 heexhexhxxhxhhxh)(sinlim)sin)(sin(lim000)(00)(000 heexhxhxexhxhhxh000)(0000)(0limsinsin)sin(lim 000)(sin)(sin0 xxxxxxexex 61 1. . uCCu )(； 2 2. .可可推推广广到到有有限限多多个个函函数数的的乘乘积积，如如 推论推论wuvwvuvwuuvw )(.)()2(vuvuuv 证证wuvwuvuvw )()()(wuvwvuvu )(.wuvwvuvwu 注意注意vuuv )(7haxahxxhxh0020)(200)(lim 0)(2xxxax hxahx

4、aaahxxxhxh20)(20)(00000)(1lim hxaxaxahxaahxxxxhx20)(2020200200000)(lim)(1 )(lim)(lim)(120)(02020020000hxaahxhxaaxhxhxhx 2202)()()(0000 xxxxxxxaaxax 8例例.tan的导数的导数求求xy 解解)(tan xyxxxxx2cos)(cossincos)(sin xxx222cossincos x2cos1 xx2sec)(tan xx2csc)(cot 类似可得类似可得即即)cossin( xx，x2sec .)()3(2vvuvuvu )(vuvu 注

5、注意意：9例例.sec的导数的导数求求xy 解解)(sec xyxx2cos)(cos .tansecxx xx2cossin xxxcotcsc)(csc 类似可得类似可得即即xxxtansec)(sec )cos1( x特别特别,.)()3(2vvuvuvu .)1(2vvv 10求求下下列列函函数数的的导导数数： 例例xxy3 . 12 xxyxcose . 22 xycot3 xycsc. 4 xxycos1sin5. 5 3lnsectan. 6 xxxy11求求下下列列函函数数的的导导数数： 例例xxy3 . 12 ;3ln3322xxxx )3(3)()3( 222 xxxxxx

6、yxxyxcose . 22 xxxcose2 xxxcose2 ;sine2xxx )(cosecos)(ecose)(222 xxxxxxyxxx12xycot. 3 解解)(cot xyxxxxx2sin)(sincossin)(cos xxx222sincossin x2sin1 xx2sec)(tan xx2csc)(cot 即即)sincos( xx，x2csc 13.csc. 4xy 解解)(csc xyxx2sin)(sin .cotcscxx xx2sincos xxxcotcsc)(csc 即即xxxtansec)(sec )sin1( xxxx sincossin1 14

7、xxycos1sin5. 5 解解3lnsectan. 6 xxxy解解2)cos1(sinsin5)cos1(cos5xxxxxy 2)cos1()1(cos5xx .cos15x .tansecsectan212xxxxxxy 152/16/arcsinlim2/1 xxx 2/1sin6/lim arcsin6/ x6/sinsin6/lim 6/ 6/6/sinsin1lim 6/ 6/6/sinsinlim1 6/ 6/)(sin1 2/1)(arcsin xx16二二 反函数的求导法则反函数的求导法则定理定理且且有有内内也也可可导导在在对对应应区区间间那那末末它它的的反反函函数数且

8、且内内单单调调、可可导导在在某某区区间间如如果果函函数数,)(,0)()(1xyIxfyyfIyfx 00)(1 )(1yyxxyfxf 其中其中).(010 xfy 170011)()(lim 0 xxxfxfxx )()(lim )(0010yfyfyyxfyyy 00)()(1lim 0yyyfyfyy 0)(1 yyyf 0) )( 1xxxf 00)()(lim1 0yyyfyfyy 证明证明 18例例.arcsin的导数的导数求函数求函数xy 解解,)2,2(sin内内单单调调、可可导导在在 yIyx, 0cos)(sin yy且且内内有有在在)1 , 1( xI00)(sin1)

9、(arcsinyyxxyx 0cos1y 02sin11y .1120 x )sin(00yx .11)(arcsin2xx 19；211)(arccosxx 类似可得类似可得211)(arctanxx 211)cotarc(xx 211)(arcsinxx 20三、复合函数求导法则三、复合函数求导法则21且其导数为且其导数为可导可导在点在点则复合函数则复合函数可导可导在点在点而而可导可导在点在点如果函数如果函数,)(,)()(,)(0000 xxfyxuufyxxu 定理定理. )()(dd000 xufxyxx 22证证)()(),(0000 xxxuxu 设设记记xxfxxfxy )()

10、(00 uufuufxy )()(00 xxxxxxxxfxxf )()()()()()(000000 )()(00 xfxxfy xxxx )()(00 . )()(dd000 xufxyxx 0 23推广推广),(),(),(xhvvguufy 设设的导数为的导数为则复合函数则复合函数)(xhgfy . )()()(ddddddddxhvgufxvvuuyxy 24求求下下列列函函数数的的导导数数： 例例102)1(. 1 xy21. 2xy xy1sine. 3 )(sec. 4122 xey25102)1(. 1 xy解解)1()1(10dd292 xxxyxx2)1(1092 .)1

11、(2092 xx解解xxy21212 .12xx 21. 2xy 26解解xy1sine. 3 )1(sine1sin xyx)1(1cose1sin xxx.1cose11sin2xxx 27)(sec. 4122 xey解解 )sec()sec(21122 xxeey)()tan()sec()sec(211112222 xxxxeeee) 1()(tan()(sec221112222 xeeexxx)(tan()(sec41112222 xxxeeex28基本初等函数的导数公式基本初等函数的导数公式)( C,0 )( x,1 x)( xa)e ( x)(log xa,lnaax )(ln

12、x,ex ,ln1ax ,1x ,21)(xx ,1)1(2xx 1. 1. 常数函数常数函数2. 2. 幂函数幂函数3. 3. 指数函数指数函数4. 4. 对数函数对数函数29)(sin x,cos x )(cos x,sin x )(tan x,sec2x )(cot x,csc2x )(sec x,tansecxx )(csc x,cotcscxx 5. 5. 三角函数三角函数30)(arcsin x)(arctan x,112x )(arccos x,112x ,112x )cotarc( x.112x 6. 6. 反三角函数反三角函数31xyxsin3. 1arctan 解解cos3sin113ln3sin1arctan2arctan2xxxxyxx