derivative of (x(1+{z}(x))/(1-{z)(x)})

解

\frac{d}{d x} (\frac{x ( 1 + z ( x ) )}{1 - z ( x )})

\frac{2 x \frac{d}{d x} ( z ( x ) ) - z ( x ) ^{2} + 1}{( 1 - z ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ( 1 + z ( x ) )}{1 - z ( 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 ( 1 + z ( x ) ) ) ( 1 - z ( x ) ) - \frac{d}{d x} ( 1 - z ( x ) ) x ( 1 + z ( x ) )}{( 1 - z ( x ) ) ^{2}}

\frac{d}{d x} (x (1 + z (x))) = 1 + z (x) + x \frac{d}{d x} (z (x))

\frac{d}{d x} (1 - z (x)) = - \frac{d}{d x} (z (x))

= \frac{( 1 + z ( x ) + x \frac{d}{d x} ( z ( x ) ) ) ( 1 - z ( x ) ) - ( - \frac{d}{d x} ( z ( x ) ) ) x ( 1 + z ( x ) )}{( 1 - z ( x ) ) ^{2}}

簡素化

\frac{( 1 + z ( x ) + x \frac{d}{d x} ( z ( x ) ) ) ( 1 - z ( x ) ) - ( - \frac{d}{d x} ( z ( x ) ) ) x ( 1 + z ( x ) )}{( 1 - z ( x ) ) ^{2}} : \frac{2 x \frac{d}{d x} ( z ( x ) ) - z ( x ) ^{2} + 1}{( 1 - z ( x ) ) ^{2}}

= \frac{2 x \frac{d}{d x} ( z ( x ) ) - z ( x ) ^{2} + 1}{( 1 - z ( x ) ) ^{2}}