derivative of (0.25e^x/(1-0.75e^x))

解

\frac{d}{d x} (\frac{0.25 e ^{x}}{1 - 0.75 e ^{x}})

\frac{0.25 e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{0.25 e ^{x}}{1 - 0.75 e ^{x}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.25 \frac{d}{d x} (\frac{e ^{x}}{1 - 0.75 e ^{x}})

商の法則を適用:

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

= 0.25 \cdot \frac{\frac{d}{d x} ( e ^{x} ) ( 1 - 0.75 e ^{x} ) - \frac{d}{d x} ( 1 - 0.75 e ^{x} ) e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}}

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

\frac{d}{d x} (1 - 0.75 e^{x}) = - 0.75 e^{x}

= 0.25 \cdot \frac{e ^{x} ( 1 - 0.75 e ^{x} ) - ( - 0.75 e ^{x} ) e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}}

簡素化

0.25 \cdot \frac{e ^{x} ( 1 - 0.75 e ^{x} ) - ( - 0.75 e ^{x} ) e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}} : \frac{0.25 e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}}

= \frac{0.25 e ^{x}}{( 1 - 0.75 e ^{x} ) ^{2}}