《大学文科数学》课程PPT教学课件（微积分）第三章 变量变化速度与局部改变量估值问题——导数与微分 3-2 求导数的方法——法则与公式

u'(x)v(x) -u(x)v'(x)X(3) [(v(x) ± 0).v(x)v(x)当u(x)=1时,=0(1)'v(x)-1· v'(x)-v(x)v2(x)v(x)v(x-媚不可以为

= 0 2 1 (1) ( ) 1 ( ) [ ] ( ) ( ) v x v x v x v x   −   = 2 ( ) ( ) ( ) ( ) ( ) (3) [ ] ( ( ) 0). ( ) ( ) u x u x v x u x v x v x v x v x   −  =  ( ) ( ) [ ] ( ) ( ) . u x u x v x v x   =  不可以为 当u x( ) 1 , = 时 2 ( ) . ( ) v x v x −  =

(tan x)' = sec" x,例3已知y=tanx,求y(cot x)' = -csc2 x.sin x解y= (tanx)=cosx(sin x)' cos x - sin x(cos x)cos? xcos? x + sin? x12=sec-xcos? x2cos" xu(x)=v(x)

解 sin (tan ) ( ) cos x y x x    = = x x x x x 2 cos (sin ) cos − sin (cos ) = 2 2 1 sec . cos x x = = 例3 tan , . 已知y x y = 求  2 2 (tan ) sec (cot ) csc . x x x x  =  = − ， x x x 2 2 2 cos cos + sin = 2 ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) u x u x v x u x v x v x v x   −  = sin cos x x ( )==

例4已知y = sec x,求y'解y=(secx)=(cos x-(cos x)"cos? x1-v(x)sinx福Ocosx= secx tanx.(sec x)'= secx tan x,(cscx)'= -cscxcot x

解 1 (sec ) ( ) cos y x x    = = 2 (cos ) cos x x −  = 2 sin cos x x = = sec tan . x x 例4 sec , . 已知y x y = 求  (sec ) sec tan , (csc ) csc cot . x x x x x x   = = − 2 1 ( ) [ ] ( ) ( ) v x v x v x −   =