If-y-1-x-1-x-show-that-d-n-y-dx-n-n-1-n-x-n-1-1-1-x-n-1-

Question Number 127477 by ZiYangLee last updated on 30/Dec/20

If y = \frac{1}{x (1 - x)},

show that \frac{d^{n} y}{{d x}^{n}} = n! [\frac{{(- 1)}^{n}}{x^{n + 1}} + \frac{1}{{(1 - x)}^{n + 1}}]

Answered by Dwaipayan Shikari last updated on 30/Dec/20

y = \frac{1}{x} + \frac{1}{1 - x}

y^{'} = - \frac{1}{x^{2}} + \frac{1}{{(1 - x)}^{2}}

y^{″} = \frac{2}{x^{3}} + \frac{2}{{(1 - x)}^{3}} = \frac{2!}{x^{3}} + \frac{2!}{{(1 - x)}^{3}}

y^{‴} = - \frac{6}{x^{4}} + \frac{6}{{(1 - x)}^{4}} = - \frac{3!}{x^{4}} + \frac{3!}{{(1 - x)}^{4}}

y^{n} = n! (\frac{{(- 1)}^{n}}{x^{n + 1}} + \frac{1}{{(1 - x)}^{n + 1}})