derivative (((4/x)^2))/(1-4/x)

解

導関数

\frac{( ( \frac{4}{x} ) ^{2} )}{1 - \frac{4}{x}}

\frac{- 32 x + 64}{x ^{2} ( x - 4 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{( \frac{4}{x} ) ^{2}}{1 - \frac{4}{x}})

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{4}{x} ) ^{2} ) ( 1 - \frac{4}{x} ) - \frac{d}{d x} ( 1 - \frac{4}{x} ) ( \frac{4}{x} ) ^{2}}{( 1 - \frac{4}{x} ) ^{2}}

\frac{d}{d x} ((\frac{4}{x})^{2}) = - \frac{32}{x ^{3}}

\frac{d}{d x} (1 - \frac{4}{x}) = \frac{4}{x ^{2}}

= \frac{( - \frac{32}{x ^{3}} ) ( 1 - \frac{4}{x} ) - \frac{4}{x ^{2}} ( \frac{4}{x} ) ^{2}}{( 1 - \frac{4}{x} ) ^{2}}

簡素化

\frac{( - \frac{32}{x ^{3}} ) ( 1 - \frac{4}{x} ) - \frac{4}{x ^{2}} ( \frac{4}{x} ) ^{2}}{( 1 - \frac{4}{x} ) ^{2}} : \frac{- 32 x + 64}{x ^{2} ( x - 4 ) ^{2}}

= \frac{- 32 x + 64}{x ^{2} ( x - 4 ) ^{2}}