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

解

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

- \frac{e ^{\frac{y}{x}} y}{x ^{2}}

解答ステップ

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

x

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= \frac{e ^{\frac{y}{x}}}{x} - \frac{e ^{\frac{y}{x}} y + x e ^{\frac{y}{x}}}{x ^{2}}

簡素化

\frac{e ^{\frac{y}{x}}}{x} - \frac{e ^{\frac{y}{x}} y + x e ^{\frac{y}{x}}}{x ^{2}} : - \frac{e ^{\frac{y}{x}} y}{x ^{2}}

= - \frac{e ^{\frac{y}{x}} y}{x ^{2}}