derivative of (e^x(x-2)/(x^3))

解

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

\frac{e ^{x} ( x ^{2} - 4 x + 6 )}{x ^{4}}

解答ステップ

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

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

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

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

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

簡素化

\frac{( e ^{x} x - e ^{x} ) x ^{3} - 3 x ^{2} e ^{x} ( x - 2 )}{( x ^{3} ) ^{2}} : \frac{e ^{x} ( x ^{2} - 4 x + 6 )}{x ^{4}}

= \frac{e ^{x} ( x ^{2} - 4 x + 6 )}{x ^{4}}