# 关于自然指数的知识

## 指数函数的导数 #

$2^x$ 的导数是什么？

$$\lim_{dx \to 0} \frac{2^{x + dx} - 2^x}{dx} = \frac{2^{dx} - 1}{dx} \cdot 2^x$$

def derivative_const(dx):
return (2**dx - 1)/dx

derivative_const(0.001), derivative_const(0.0001), derivative_const(0.0000001)

(0.6933874625807412, 0.6931712037649973, 0.6931472040783149)


$$\lim_{dx \to 0} \frac{8^{x + dx} - 8^x}{dx} = \frac{8^{dx} - 1}{dx} \cdot 8^x$$


def derivative_const(dx):
return (8**dx - 1)/dx

derivative_const(0.001), derivative_const(0.0001), derivative_const(0.0000001)

(2.0816050796328422, 2.0796577605231015, 2.079441758784384)


$$\lim_{dx \to 0} \frac{e^{dx} - 1}{dx} = 1$$

$$2^x = e^{ln(2)\cdot x}$$

$2^x = e^{ln(2)\cdot x}$ 的导数为

$$ln(2) \cdot e^{ln(2)\cdot x} = ln(2)\cdot 2^x$$

## 重新看数字 e 的定义 #

$$\lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n = e \tag{1}$$

$$\lim_{{n \to \infty}} \left(1 + n\right)^\frac{1}{n} = e \tag{2}$$

$$\lim_{{n \to \infty}} \left(1 + n\right)^\frac{x}{n} = e^x$$

$\frac{x}{n} = t$

$$\lim_{{n \to \infty}} \left(1 + \frac{x}{t}\right)^t = e^x$$

$t$$n$ 表示可能更加易于记忆：

$$\lim_{{n \to \infty}} \left(1 + \frac{x}{n}\right)^n = e^x \tag{3}$$

#统计