(\partial)/(\partial x)((x+y)(2x+y)^2)

解

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

12 x^{2} + 16 x y + 5 y^{2}

解答ステップ

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

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 + y, g = (2 x + y)^{2}

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

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

\frac{\partial}{\partial x} ((2 x + y)^{2}) = 4 (2 x + y)

= 1 \cdot (2 x + y)^{2} + 4 (2 x + y) (x + y)

簡素化

1 \cdot (2 x + y)^{2} + 4 (2 x + y) (x + y) : 12 x^{2} + 16 x y + 5 y^{2}

= 12 x^{2} + 16 x y + 5 y^{2}