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

解

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

\frac{y - x}{e ^{\frac{x}{y}} y}

解答ステップ

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

y

を定数として扱う

商の法則を適用:

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

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

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

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

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

簡素化

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

= \frac{y - x}{e ^{\frac{x}{y}} y}