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

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

= x (\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

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

簡素化

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

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