推广(Ztu,(x)l'=Eu(x);(1)i=1i=1(2)[Cu(x)}' = Cu'(x);1(3) [/Iu,;(x)' = u'(x)u,(x)...u.,(x)i=1+...+u,(x)u,(x)...u,(x)u,(x)u (x);i=1 k=1kti
推广 1 1 (1) [ ( )] ( ); n n i i i i u x u x = = = (2) [ ( )] ( ); Cu x Cu x = 1 2 1 1 2 1 1 (3) [ ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ); n i n i n n n i k i k k i u x u x u x u x u x u x u x u x u x = = = = + + =
例1 求y=x3-2x2+sinx的导数解 y'= 3x2-4x+ cos x.例2 求y= sin2x·lnx的导数解y=2sinx·cosx·ln xy' = 2cos x .cosx.ln x+ 2sin x(-sin x).ln x+ 2sin x.cosx.x1=2cos2xlnx+=sin2x.x
例 1 2 sin . 求 y = x3 − x2 + x 的导数 解 2 y x x x = − + 3 4 cos . 例2 求 y = sin2x ln x 的导数 . 解 y = 2sin x cos x ln x y = 2cos x cos x ln x+ 2sin x (− sin x) ln x x x x 1 + 2sin cos sin 2 . 1 2cos 2 ln x x = x x +
例3求y=tanx的导数解 y'=(tanx)'=(sinz)cosx(sin x)'cos x - sin x(cos x)cos" x1cos x + sin' x=sec"x2cos"xcos"x即(tan x) = sec’ x.同理可得(cot x)' = -csc2 x
例3 求 y = tan x 的导数 . 解 ) cos sin = (tan ) = ( x x y x x x x x x 2 cos (sin ) cos − sin (cos ) = x x x 2 2 2 cos cos + sin = x x 2 2 sec cos 1 = = 2 即 (tan ) sec . x x = 2 同理可得 (cot ) csc . x x = −
例4 求 y= secx的导数,解'=(secx)=(cosx_ -(cosx)cosxsinx= secx tanx.2cosx同理可得(cscx)' = -csc xcot x
例4 求 y = sec x 的导数 . 解 ) cos 1 = (sec ) = ( x y x x x 2 cos − (cos ) = = sec x tan x. x x 2 cos sin = 同理可得 (csc ) csc cot . x x x = −
例5. y=/x(x3 -4cosx-sin1),求y'及j'|x=1 解: y' =(Vx)(x3 -4cosx -sin1)+/x(x3 - 4cos x -sin1)1(x3 -4cosx -sin1) + /x (3x + 4sin x)2Vx1(1-4cos1- sin1)+(3+4sin1)x=127.7sin1-2cos1+一一22
解: 3 1 5. ( 4cos sin1) , . x y x x x y y = 例 = − − 求 及 3 y x x x = − − ( ) ( 4cos sin1) 3 + − − x x x ( 4cos sin1) 1 3 ( 4cos sin1) 2 x x x = − − 1 1 (1 4cos1 sin1) ( 3 4sin1) 2 x y = = − − + + 7 7 sin1 2cos1 2 2 = + − 2 + + x x x ( 3 4sin )