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

解

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

y

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

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

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

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

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

簡素化

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