(\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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

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}