(\partial)/(\partial y)(1/y e^{xy})

解

\frac{\partial}{\partial y} (\frac{1}{y} e^{x y})

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

解答ステップ

\frac{\partial}{\partial y} (\frac{1}{y} e^{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 = \frac{1}{y}, g = e^{x y}

= \frac{\partial}{\partial y} (\frac{1}{y}) e^{x y} + \frac{\partial}{\partial y} (e^{x y}) \frac{1}{y}

\frac{\partial}{\partial y} (\frac{1}{y}) = - \frac{1}{y ^{2}}

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

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

簡素化

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

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