例4. 设y= sin x, 求 (n)解: y'=cosx =sin(x+号)y"= cos(x+)= sin(x++)= sin(x + 2 ·)y"=cos(x+2·号)= sin(x+3·号2元一般地，(sinx)(n)= sin(x+n2类似可证：(cos x)(n) = cos(x + n ·元2上页目录下页返回结束机动
例4. 设 求 解: y = cos x sin( ) 2 = x + cos( ) 2 y = x + sin( ) 2 2 = x + + sin( 2 ) 2 = x + cos( 2 ) 2 y = x + sin( 3 ) 2 = x + 一般地 , x = x + n (sin ) sin( ( ) 类似可证: x = x + n (cos ) cos( ( ) ) 2 n ) 2 n
例5.设 =ea× sin bx (a,b为常数),求y(n)解: y'= aeax sin bx + beax cos bx= eax (asinbx +bcos bx)=eaxx/a? +b? sin(bx + )(@ = arctan 1一a[aeax sin(bx + p)+ beax cos(bx +)]y"=Va?+b?= Va? + b? eax a? + b? sin(bx + 2p)hy(n) =(a? +b?)2 eax sin(bx + np)@ = arctana目录上页下页返回结束机动
例5 . 设 y e bx ax = sin 解: y = a e b x + a x sin e (a sin bx b cos bx) a x = + (a , b为常数 ), 求 . (n) y be bx a x cos ( sin cos ) 2 2 2 2 2 2 bx a b b bx a b a a b + + + + cos sin ax = e sin( ) 2 2 a + b b x + ( arctan ) a b = 2 2 y = a + b ( ) 2 2 2 ( ) n n y = a + b a x a b e 2 2 = + ( arctan ) a b = sin( 2 ) 2 2 a + b b x + e sin(b x n ) a x +