# 微积分笔记（8）——导数计算与高阶导数

## 微积分笔记（8）——导数计算与高阶导数

Contents

### 导数

#### 定义

$$f(x) = \lim\limits_{\Delta x \to 0} \dfrac{f(x + \Delta x) – f(x)}{\Delta x} = \lim\limits_{\Delta x \to 0} \dfrac{\Delta f}{\Delta x}$$

#### Leibniz 记号

$$y = f(x),\Delta y = f(x + \Delta x) -f(x)$$

$$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \lim\limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x},f^{\prime}(x_0) = \dfrac{\mathrm{d} y}{\mathrm{d} x}\Big|_{x = x_0}$$

### 导数计算

#### 基本初等函数的导数

1. $f_1(x) \equiv C$：$f^{\prime}_1(x) = \lim\limits_{\Delta x \to 0} \dfrac{f(x + \Delta x) – f(x)}{\Delta x} = \lim\limits_{\Delta x \to 0} \dfrac{C – C}{\Delta x} = 0$

2. $f_2(x) = x^n(n = 1,2,\cdots)$：$f^{\prime}_2(x) = \lim\limits_{\Delta x \to 0} \dfrac{f(x + \Delta x) – f(x)}{\Delta x} = \lim\limits_{\Delta x \to 0} \dfrac{n x^{n-1} \Delta x +o(\Delta x)}{\Delta x} = n x^{n-1}$

注：当 $a \in \mathbb{R}$，$(x^a)^{\prime} = a x^{a-1}$，后面证，$x > 0$

3. $f_3(x) = \sin x$：$f^{\prime}_3(x) = \lim\limits_{\Delta x \to 0} \dfrac{\sin (x + \Delta x) – \sin x}{\Delta x} = \lim\limits_{\Delta x \to 0} \dfrac{\cos (x + \frac{\Delta x}{2}) \sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} = \cos x$

同理可得 $(\cos x)^{\prime} = -\sin x$

4. $f_4(x) = \ln x,x > 0$：$f^{\prime}_4(x) = \lim\limits_{\Delta x \to 0} \dfrac{1}{\Delta x} \ln (1 + \dfrac{\Delta x}{x}) = \lim\limits_{\Delta x \to 0} \ln [(1 + \dfrac{\Delta x}{x})^{\frac{x}{\Delta x} \cdot \frac{1}{x}}] = \dfrac{1}{x}$

#### 四则运算的导数

1. $(f \pm g)^{\prime}(x) = f^{\prime}(x) \pm g^{\prime}(x)$
2. $(fg)^{\prime}(x) = f^{\prime}(x)g(x) + f(x)g^{\prime}(x)$
3. $(\frac{f}{g})^{\prime}(x) = \dfrac{f^{\prime}(x)g(x) – f(x)g^{\prime}(x)}{g^2(x)},g(x) \not = 0$

#### 复合运算的导数（链式法则）

Leibniz 记号：记 $y = f(x),x = \varphi(t)$，则 $\dfrac{\mathrm{d} y}{\mathrm{d} t} = \dfrac{\mathrm{d} y}{\mathrm{d} x} \dfrac{\mathrm{d} x}{\mathrm{d} t}$，或者 $\dfrac{\mathrm{d} y}{\mathrm{d} t} = f^{\prime}(x) \varphi^{\prime}(t)$。

$\therefore \dfrac{\Delta y}{\Delta t} \to f^{\prime}(x) \varphi^{\prime}(t)$。

#### 反函数的求导

Leibniz 记号：$y = f(x),x = f^{-1}(y)$，$\dfrac{\mathrm{d}x}{\mathrm{d}y} = \dfrac{1}{\dfrac{\mathrm{d}y}{\mathrm{d}x}}$。

$$\dfrac{\mathrm{d}y}{\mathrm{d}x} \dfrac{\mathrm{d}x}{\mathrm{d}y} = 1$$

$y = x^a = e^{a \ln x} = e^u,u = a \ln x$

$$\left(u(x)^{v(x)}\right)^{\prime} = u(x)^{v(x)}\left(v^{\prime}(x) \ln u(x) + \dfrac{v(x)u^{\prime}(x)}{u(x)}\right)$$

#### 对数求导法

$$\dfrac{1}{y}y^{\prime} = (\ln|f(x)|)^{\prime}$$

$$y^{\prime} = f(x)(\ln|f(x)|)^{\prime}$$

### 高阶导数

#### 高阶导数

Leibniz 记号：$y = f(x)$，$\dfrac{\mathrm{d}}{\mathrm{d} x}$ 表示求导运算。

$$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \dfrac{\mathrm{d}}{\mathrm{d} x}\left(\dfrac{\mathrm{d}^{n-1} y}{\mathrm{d} x^{n-1}}\right)$$