1 导数的乘法法则
$(uv)'=u'v+v'u$
乘法法则推导Part 1 $$\begin{align} \Delta(uv) & = u(x+\Delta x)v(x+\Delta x)-u(x)v(x) \ \\ & = [u(x+\Delta x)-u(x)]v(x+\Delta x)+u(x)[v(x+\Delta x)-v(x)] \ \\ & = \Delta u\times v(x+\Delta x)+u(x)\Delta v \end{align}$$
乘法法则推导Part 2 $$\begin{align} \lim_{\Delta x \to 0}\frac{\Delta(uv)}{\Delta x} & = \lim_{\Delta x \to 0}\frac{\Delta u}{\Delta x} v(x+\Delta x)+\frac{u(x)}{\Delta x}\Delta v \end{align}$$ 即$\frac{d(uv)}{dx}=\frac{d(u)}{dx}v+\frac{d(v)}{dx}u$
2 导数的除法法则
在$v\neq 0$的前提下：$(\frac{u}{v})'=\frac{u'v-uv'}{v^2}$
除法法则推导Part 1 $$\begin{align} \Delta(\frac{u}{v}) & = \frac{u(x+\Delta x)}{v(x+\Delta x)}-\frac{u(x)}{v(x)} \ \\ & = \frac{u(x+\Delta x)v(x)-v(x+\Delta x)u(x)}{v(x+\Delta x)v(x)} \end{align}$$
除法法则推导Part 2 $$\begin{align} \lim_{\Delta x \to 0}(\frac{\Delta\frac{u}{v}}{\Delta x}) & = \frac{u(x+\Delta x)v(x)-v(x+\Delta x)u(x)}{v(x+\Delta x)v(x)} \end{align}$$ 即$\frac{d}{dx}(\frac{u}{v})== \frac{\frac{du}{dx}v(x)-\frac{dv}{dx}u(x)}{v^2}$
3 链式法则 chain rule
$$\frac{dy}{dt}=\frac{dy}{dx}\times \frac{dx}{dt}$$
- 链式法则使用的关键在于换元
- 通过换元法将一个复杂函数导数拆解成多个简单函数导数的链式表示
4 高阶导数
用符号$D$表示求导运算$\frac{d}{dx}$
示例：对函数$x^n$进行n阶求导 $$D^nx^n=\frac{d}{dx}x^n=n!$$
5 参考
