derivative of 2x*e^{x^2}

解答

\frac{d}{d x} (2 x \cdot e^{x^{2}})

2 (e^{x^{2}} + 2 e^{x^{2}} x^{2})

求解步骤

\frac{d}{d x} (2 x e^{x^{2}})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d x} (x e^{x^{2}})

使用微分乘法定则:

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

f = x, g = e^{x^{2}}

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

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

\frac{d}{d x} (e^{x^{2}}) = e^{x^{2}} \cdot 2 x

= 2 (1 \cdot e^{x^{2}} + e^{x^{2}} \cdot 2 xx)

化简

2 (1 \cdot e^{x^{2}} + e^{x^{2}} \cdot 2 xx) : 2 (e^{x^{2}} + 2 e^{x^{2}} x^{2})

= 2 (e^{x^{2}} + 2 e^{x^{2}} x^{2})

x \to 25 lim (x - 5)

\frac{d}{d x} (\frac{x}{3 + e ^{x}})

\int 2 + x - x^{2} d x

\int \frac{e ^{4 x}}{4 - e ^{4 x}} d x

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