(\partial)/(\partial y)(2xe^{x+y}+x^2e^{x+y}-y^3sin(xy))

解

\frac{\partial}{\partial y} (2 x e^{x + y} + x^{2} e^{x + y} - y^{3} sin (x y))

2 e^{x + y} x + e^{x + y} x^{2} - 3 y^{2} sin (x y) - x y^{3} cos (x y)

解答ステップ

\frac{\partial}{\partial y} (2 x e^{x + y} + x^{2} e^{x + y} - y^{3} sin (x y))

x

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial y} (2 x e^{x + y}) + \frac{\partial}{\partial y} (x^{2} e^{x + y}) - \frac{\partial}{\partial y} (y^{3} sin (x y))

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

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

\frac{\partial}{\partial y} (y^{3} sin (x y)) = 3 y^{2} sin (x y) + cos (x y) x y^{3}

= 2 x e^{x + y} + x^{2} e^{x + y} - (3 y^{2} sin (x y) + cos (x y) x y^{3})

簡素化

2 x e^{x + y} + x^{2} e^{x + y} - (3 y^{2} sin (x y) + cos (x y) x y^{3}) : 2 e^{x + y} x + e^{x + y} x^{2} - 3 y^{2} sin (x y) - x y^{3} cos (x y)

= 2 e^{x + y} x + e^{x + y} x^{2} - 3 y^{2} sin (x y) - x y^{3} cos (x y)