Question

y=x/(x^2-3x+2),求y的n阶导数，求详细过程

Answer 1

y=x/(x^2-3x+2)
=2/(x-2) -1/(x-1)
故y的n阶导数就等于2/(x-2)与1/(x-1)的n阶导数之差，
而
[2/(x-2)]′= -2(x-2)^(-2)
[2/(x-2)]′′=2*(-1)*(-2)*(x-2)^(-3)
[2/(x-2)]′′′=2*(-1)*(-2)*(-3)*(x-2)^(-4)
......
[2/(x-2)]^n=2*(-1)(-2)(-3)....(-n) *(x-2)^(-n-1)
同理
1/(x-1)的n阶导数= (-1)(-2)(-3)....(-n) *(x-1)^(-n-1)

所以
y的n阶导数
=2*(-1)(-2)(-3)....(-n) *(x-2)^(-n-1) - (-1)(-2)(-3)....(-n) *(x-1)^(-n-1)
=(-1)(-2)(-3)....(-n) *[2(x-2)^(-n-1) -(x-1)^(-n-1)]

追问

不好意思指出一下，第二行是否为=2x/(x-2)-x(x-1)

追答

当然不是的呢，
x/(x^2-3x+2)
=2/(x-2) -1/(x-1)
而
2x/(x-2)- x/(x-1)
=x^2 /(x^2-3x+2)

Answer 2

y=x/(x^2-3x+2)
=x/(x-1)(x-2)=2/(x-2)-1/(x-1)
由于1/(x-2)的n阶导数=(-1)^n*n!/(x-2)^(n+1)
1/(x-1)的n阶导数=(-1)^n*n!/(x-1)^(n+1)
所以：y的n阶导数
=2(-1)^n*n!/(x-2)^(n+1)-(-1)^n*n!/(x-1)^(n+1)
=(-1)^n*n![2/(x-2)^(n+1)-1/(x-1)^(n+1)]