derivative of (2xy/(x+y))

解

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

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

解答ステップ

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

定数を除去:

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

= 2 \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}}

= 2 \cdot \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) = y + x \frac{d y}{d x}

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

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

簡素化

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

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