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

解

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

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

解答ステップ

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

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^{x y}

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

\frac{\partial}{\partial x} (x) = \frac{1}{2 x}

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

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

\frac{1}{2 x} e^{x y} = \frac{e ^{x y}}{2 x}

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