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

解答

\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)

求解步骤

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

将

y

视为常数

使用微分加减法定则:

(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))

化简

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)

a re a x^{2}, \frac{2}{( x ^{2} + 1 )}

\frac{\partial}{\partial x} (cos (x + 7 y))

\frac{1}{x} \frac{d y}{d x} + 2 y - e^{- x^{2}} = 0

x \to 0 lim (\frac{\frac{6}{x + 6} - 1}{x})

x \to 10 lim (- 15)