derivative (x-sqrt(x))(x+sqrt(x))

解答

导数

(x - x) (x + x)

2 x - 1

求解步骤

\frac{d}{d x} ((x - x) (x + x))

使用微分乘法定则:

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

f = x - x, g = x + x

= \frac{d}{d x} (x - x) (x + x) + \frac{d}{d x} (x + x) (x - x)

\frac{d}{d x} (x - x) = 1 - \frac{1}{2 x}

\frac{d}{d x} (x + x) = 1 + \frac{1}{2 x}

= (1 - \frac{1}{2 x}) (x + x) + (1 + \frac{1}{2 x}) (x - x)

化简

(1 - \frac{1}{2 x}) (x + x) + (1 + \frac{1}{2 x}) (x - x) : 2 x - 1

= 2 x - 1

x \to 0 lim ((1 - \frac{3}{x})^{x})

\int \frac{1}{r ^{2}} d r

\int_{- 1}^{1} (x - x^{3}) d x

\frac{\partial}{\partial x} (t e^{\frac{- x ^{2}}{2 t}})

x \to 0 lim (e^{8 π})