derivative of xye^{xy}

解答

\frac{d}{d x} (x y e^{x y})

e^{x y} y + x (e^{x y} \frac{d y}{d x} + e^{x y} (y + x \frac{d y}{d x}) y)

求解步骤

\frac{d}{d x} (x y e^{x y})

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = y e^{x y}

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

\frac{d x}{d x} = 1

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

= 1 \cdot y e^{x y} + (\frac{d y}{d x} e^{x y} + e^{x y} (y + x \frac{d y}{d x}) y) x

化简

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

\frac{\partial}{\partial x} (5 y - x^{3} y^{2})

\frac{d}{d x} (5 x^{2} sin (x) - x)

\frac{d}{d x} (cos (7))

x y^{^{'}} + y = - 4 x y^{2}

x \to 3 - lim (\frac{4 - x}{x - 3})