NAN DA XUE JING PIN KE CHENG 例3.设y=x",n为正整数,求y)和ym+) 解 n(n-1)xy2,y3)=n(n-1)(n-2) -1)3·21xn=nl! 而y+=(m)=0 易见,若f(x),g(x)均存在n阶导数,则 ((x)+g(x))(n=f(n(x)+g n(x) 类似,设f(x)=ax+a1xn1a2x-2+.+an1xn+an, 为n次多项式,则f(n(x)=aol,而(+x)=0 OD 高等數粤例3. 解: y' = nxn–1 , , , . ( ) ( +1) = n n n 设y x n为正整数 求y 和y y'' = n(n–1)x n–2 , y (3)= n(n–1)(n–2) x n–3 , …, y (n)= n(n–1)… 3 ·2 ·1x n–n = n! 而 y (n+1)= (n!)' = 0 易见, 若f (x), g(x)均存在n阶导数, 则 ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) f x g x f x g x n n n  =  类似, 设f (x)=a0x n +a1x n–1 +a2x n–2 +…+an–1 x n +an , 为n次多项式, 则f (n) (x)=a0n!, 而f (n+1)(x)= 0