derivative of y/(x-y)

解

\frac{d}{d x} (\frac{y}{x - y})

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

解答ステップ

\frac{d}{d x} (\frac{y}{x - y})

商の法則を適用:

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

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

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

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

拡張

\frac{d y}{d x} (x - y) - (1 - \frac{d y}{d x}) y : x \frac{d y}{d x} - y

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