(\partial)/(\partial x)(8xe^{sqrt(xy)})

解

\frac{\partial}{\partial x} (8 x e^{x y})

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

解答ステップ

\frac{\partial}{\partial x} (8 x e^{x y})

y

を定数として扱う

定数を除去:

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

= 8 \frac{\partial}{\partial x} (x e^{x y})

積の法則を適用:

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

f = x, g = e^{x y}

= 8 (\frac{\partial}{\partial x} (x) e^{x y} + \frac{\partial}{\partial x} (e^{x y}) x)

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

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

= 8 (1 \cdot e^{x y} + \frac{y e ^{y x}}{2 x} x)

簡素化

8 (1 \cdot e^{x y} + \frac{y e ^{y x}}{2 x} x) : 8 (e^{x y} + \frac{e ^{x y} x y}{2})

= 8 (e^{x y} + \frac{e ^{x y} x y}{2})