derivative of (e^xx-e^x/(x^2))

解

\frac{d}{d x} (\frac{e ^{x} x - e ^{x}}{x ^{2}})

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

解答ステップ

\frac{d}{d x} (\frac{e ^{x} x - e ^{x}}{x ^{2}})

商の法則を適用:

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

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

\frac{d}{d x} (e^{x} x - e^{x}) = e^{x} x

\frac{d}{d x} (x^{2}) = 2 x

= \frac{e ^{x} x x ^{2} - 2 x ( e ^{x} x - e ^{x} )}{( x ^{2} ) ^{2}}

簡素化

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

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