derivative of (x-y/(x+y))

解

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

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

解答ステップ

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

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

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

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

拡張

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

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