例3证明函数=√2x-x2满足yy"+1=0 2-2x 证 2y2x-x2√2x-x 2 X-x 2x-x 2 2x-x2 2 2x+x2-(1-x) (2x-x2)2x-x2(2x-x 23/2 3 +1=0
例 3 2 1 0 2 3 证明函数y = x − x 满 足 y y + = 证 , 21 2 22 2 2 2 x xx x xx y −− = − − = 2 2 2 2 21 2 ( 1 ) x x x xx x x x y − −− − − − − = 2 2 2 2 ( 2 ) 2 2 ( 1 ) x x x x x x x − − − + − − = , 1 ( 2 ) 1 2 3 2 3 x x y− = −− = 1 0 . 3 y y + =
例4设y=ex,求y 解 J≡e";y≡e",y=e e e,即(e) 类似地,y=a,y=alma,y"=a(nd)2, a In a a"In a
例4 , . x (n) 设y = e 求y 解 y = e x , y = e x , y = e x , y (4) = e x , n x x n x y = e e = e ( ) ( ) , 即 ( ) , ln , (ln ) , 2 y a y a a y a a x x x 类似地, = = = ( ) (ln ) . n x n y = a a ( ) ( ) (ln ) . x n x n a = a a
例5求正弦函数及余弦函数的n阶导数 yd A y=sinx, y'=coS x=sin(x y"=cos(x+)=si(x+2·), J”=coS(x+2.2 sin(x+3 (n)=sin(x+n 即(inx)=sin(x+n·) 类似地,(cosx)=cos(x+n·“)
), 2 sin , cos sin( y = x y = x = x + ), 2 ) sin( 2 2 cos( = + y = x + x π π cos( 2 ) sin( 3 ), , 2 2 y x x = + = + ( ) ). 2 sin( y = x + n n 例5 求正弦函数及余弦函数的n阶导数. ( ) ). 2 (sin ) sin( x = x + n 即 n 类似地, ( ) π (cos ) cos( ). 2 n x x n = + 解