(\partial)/(\partial y)((x+y)*e^{x+2y})

解

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

e^{x + 2 y} + 2 e^{x + 2 y} (x + y)

解答ステップ

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

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

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

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

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

簡素化

= e^{x + 2 y} + 2 e^{x + 2 y} (x + y)