运算法则

加减乘除

$$[u(x) \pm v(x)]' = u'(x) \pm v(x)$$

$$[u(x) v(x)]' = u'(x)v(x) + u(x)v'(x)$$

$$[\dfrac{u(x)}{v(x)}]' = \dfrac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}$$

提示

提示: 乘法和除法中都有 $u'(x)v(x)$ 和 $u(x)v'(x)$

复合(链式法则)

$$\begin{align} [f(g(x))]' &= f'(g(x)) \cdot g'(x) \\ &= f'(x)|_{x=g(x)} \cdot g'(x) \end{align}$$

$$\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$$

亦见链式法则#一阶导

幂指函数(对数求导法)

乘除法 $\to$ 加减法 (简化计算) 多个乘除法/幂指函数 $\to$ 对数求导法

$$y = f(x)^{g(x)} \\ \implies \ln y = g(x) \ln f(x) \\ 同时求导, \implies \dfrac{y'}{y} = g'(x) \ln f(x) + g(x) \dfrac{f'(x)}{f(x)} \\ \implies y' = [g'(x) \ln f(x) + g(x) \dfrac{f'(x)}{f(x)}] \cdot f(x)^{g(x)}$$

反函数

$$(f^{-1})'(x) = \dfrac{1}{f'(y)}$$

$$\dfrac{dy}{dx} = \dfrac{1}{\dfrac{dx}{dy}}$$

例如: $y = \arcsin x$ 是 $x = \sin y$ 的反函数, $\dfrac{dx}{dy} = \dfrac{d(\sin y)}{dy} = \cos y$, $\dfrac{dy}{dx} = \dfrac{1}{\cos y} = \dfrac{1}{\sqrt{1 - \sin^2 y}} = \dfrac{1}{\sqrt{1-x^2}}$ 注意最后要尽量写成以 $x$ 为自变量的形式

参数方程

$$\begin{cases} x &= f(t), \\ y &= g(t) \end{cases} (t \in I)$$

$$\dfrac{dy}{dx} = \dfrac{f'(t)}{g'(t)} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$$

极坐标方程

$$\begin{cases} x &= r(\theta) \cos \theta, \\ y &= r(\theta) \sin \theta \end{cases} (\alpha \leq \theta \leq \beta)$$

$$\dfrac{dy}{dx} = \dfrac{y'(\theta)}{x'(\theta)} = \dfrac{r'(\theta) \sin \theta + r(\theta) \cos \theta}{r'(\theta) \cos \theta - r(\theta) \sin \theta}$$

注意

$\dfrac{dr}{d\theta}$ 非切线斜率