Ta có
$y = \dfrac{1}{x-x^2} = \dfrac{1}{x} + \dfrac{1}{1-x} = x^{-1} + (1-x)^{-1}$
Vậy
$y' = (-1) x^{-2} + (-1)(1-x)^{-2} . (-1) = (-1)^1 x^{-2} + (-1)^2 (1-x)^{-2}$
$y'' = (-1) (-2) x^{-3} + (-1)^2(-2) (1-x)^{-3} (-1) = (-1)(-2) x^{-3} + (-1)^3(-2) (1-x)^{-3}$
..
$y^{(n)} = (-1)(-2)...(-n)x^{-(n+1)} + (-1)^n (-1)(-2)...(-n) (1-x)^{-(n+1)}$
$= (-1)^n.n!.x^{-n-1} + (-1)^{2n}.n! (1-x)^{-n-1}$
$= n! [(-1)^n . x^{-n-1} + (1-x)^{n-1}]$
Vậy
$y^{100} = 100! x^{-101} + 100! (1-x)^{-101} = 100! \left( \dfrac{1}{x^{101}} + \dfrac{1}{(1-x)^{101}} \right)$
