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

解

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

\frac{2 - 4 x + x ^{2}}{e ^{x}}

解答ステップ

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

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

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

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

簡素化

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

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