derivative of (e^y/(x+y))

解

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

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

解答ステップ

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

\frac{d}{d x} (e^{y}) = e^{y} \frac{d y}{d x}

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

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