derivative of x/(1-e^t(1-x))

解

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

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

解答ステップ

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

t

を定数として扱う

商の法則を適用:

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

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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