(\partial)/(\partial x)(x^2e^{y/x})

解

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

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

解答ステップ

\frac{\partial}{\partial x} (x^{2} 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^{2}, g = e^{\frac{y}{x}}

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

簡素化

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

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