n+ 1)(x )= 0 例4.. ),1,0(,)2(,)1( )(n xxyaaayey求设?????解: (1) y' = e ?x??, y'' = e ?x?? 2,y (3) =e ?x?? 3,…, 故y (n ) = e ?x?? n.特别, 取?= 1, 得(e x) (n ) = e x(a x) (n ) = ( e x lna) (n)取?= –1, 得(e –x) (n ) =(–1) (n)e x. (2) 由于 a x =e x lna, 由(1) 得= a x ( lna) n = e x lna ( lna) n 例5.求 y = sin x的n阶导数 y (n).解:我们知道 y' = cos x, y'' = – sin x, y (3) = – cos x, y (4)= sin x,…但y (n)的通项公式难写, 并且不好记. ).2 sin( cos ???xx由于从而) (sin ???xy ).2 sin( ???x = cos x???????????)2 sin( ?xy )2 cos( ???x ).2 2 sin( ????x ??????????)2 2 sin( )3(?xy )2 2 cos( ????x ).2 3 sin( ????x )()() (sin nnxy?故).2 sin( ????nx ).2 cos( ) (cos , )(????nxx n类似例6.设 y = sin 2x, 求y (n).解: y' = ( sin 2 x)' y'' = (sin2 x)' = sin2 x. = 2 sin x cox2)2 2 sin( ????x,2)2 22 sin( 2)3(?????xy …….22 )1(2 sin 1 )(???????????? n nnxy ?
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