derivative of ((1/x+1/(x^2))/(x-1))

解

\frac{d}{d x} (\frac{( \frac{1}{x} + \frac{1}{x ^{2}} )}{x - 1})

\frac{- 2 x ^{2} - 2 x + 2}{x ^{3} ( x - 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{( \frac{1}{x} + \frac{1}{x ^{2}} )}{x - 1})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (\frac{1}{x} + \frac{1}{x ^{2}}) = - \frac{1}{x ^{2}} - \frac{2}{x ^{3}}

\frac{d}{d x} (x - 1) = 1

= \frac{( - \frac{1}{x ^{2}} - \frac{2}{x ^{3}} ) ( x - 1 ) - 1 \cdot ( \frac{1}{x} + \frac{1}{x ^{2}} )}{( x - 1 ) ^{2}}

簡素化

\frac{( - \frac{1}{x ^{2}} - \frac{2}{x ^{3}} ) ( x - 1 ) - 1 \cdot ( \frac{1}{x} + \frac{1}{x ^{2}} )}{( x - 1 ) ^{2}} : \frac{- 2 x ^{2} - 2 x + 2}{x ^{3} ( x - 1 ) ^{2}}

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