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

解

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

- 3 y^{2} - 2 x^{2} y + x

解答ステップ

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

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^{2} + y, g = x - y^{2}

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

\frac{\partial}{\partial y} (x^{2} + y) = 1

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

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

簡素化

1 \cdot (x - y^{2}) + (- 2 y) (x^{2} + y) : - 3 y^{2} - 2 x^{2} y + x

= - 3 y^{2} - 2 x^{2} y + x