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

解

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

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

解答ステップ

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

y

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

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

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

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

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

簡素化

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

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