(\partial)/(\partial y)(((ye^x))/(xe^y))

解

\frac{\partial}{\partial y} (\frac{( y e ^{x} )}{x e ^{y}})

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

解答ステップ

\frac{\partial}{\partial y} (\frac{( y e ^{x} )}{x e ^{y}})

x

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

= \frac{e ^{x}}{x} \cdot \frac{\frac{\partial}{\partial y} ( y ) e ^{y} - \frac{\partial}{\partial y} ( e ^{y} ) y}{( e ^{y} ) ^{2}}

\frac{\partial}{\partial y} (y) = 1

\frac{\partial}{\partial y} (e^{y}) = e^{y}

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

簡素化

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

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