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

解

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

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

解答ステップ

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

x

を定数として扱う

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = e^{\frac{x}{y}}

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

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

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

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

簡素化

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

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