(\partial)/(\partial x)(5xye^{(x+y)})

解

\frac{\partial}{\partial x} (5 x y e^{(x + y)})

5 y (e^{x + y} + e^{x + y} x)

解答ステップ

\frac{\partial}{\partial x} (5 x y e^{(x + y)})

y

を定数として扱う

定数を除去:

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

= 5 y \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}

= 5 y (\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}) = e^{x + y}

= 5 y (1 \cdot e^{x + y} + e^{x + y} x)

簡素化

= 5 y (e^{x + y} + e^{x + y} x)