derivative of xe^{x-y}

解答

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

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

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

化简

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

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

\int \frac{( 8 + 5 x )}{1 + x ^{2}} d x

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

\int_{- 1}^{x^{3}} 2 + t^{2} d t

(x^{2} + 1) y^{^{'}} + 6 x (y - 1) = 0, y (0) = 9