\begin{align*}
(1)\ & [u(x) \pm v(x)]' = u'(x) \pm v'(x) \\[10pt]
(2)\ & [u(x)v(x)]' = u'(x)v(x) + u(x)v'(x) \\[10pt]
(3)\ & \left[\frac{u(x)}{v(x)}\right]' = \frac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}
\end{align*}
证明
(1) 加减法则
\begin{align*}
[u(x) \pm v(x)]'
&= \lim_{\Delta x \to 0} \frac{[u(x + \Delta x) \pm v(x + \Delta x)] - [u(x) \pm v(x)]}{\Delta x} \\
&= \lim_{\Delta x \to 0} \frac{[u(x + \Delta x) - u(x)] \pm [v(x + \Delta x) - v(x)]}{\Delta x} \\
&= \lim_{\Delta x \to 0} \frac{u(x + \Delta x) - u(x)}{\Delta x} \pm \lim_{\Delta x \to 0} \frac{v(x + \Delta x) - v(x)}{\Delta x} \\
&= u'(x) \pm v'(x)
\end{align*}
(2) 乘法法则
\begin{align*}
[u(x)v(x)]'
&= \lim_{\Delta x \to 0} \frac{u(x + \Delta x)v(x + \Delta x) - u(x)v(x)}{\Delta x} \\
&= \lim_{\Delta x \to 0} \left[
\frac{u(x + \Delta x) - u(x)}{\Delta x} \cdot v(x + \Delta x)
+ u(x) \cdot \frac{v(x + \Delta x) - v(x)}{\Delta x}
\right] \\
&= u'(x)v(x) + u(x)v'(x)
\end{align*}
(3) 除法法则
\begin{align*}
\left[\frac{u(x)}{v(x)}\right]'
&= \lim_{\Delta x \to 0} \frac{\frac{u(x + \Delta x)}{v(x + \Delta x)} - \frac{u(x)}{v(x)}}{\Delta x} \\
&= \lim_{\Delta x \to 0} \frac{u(x + \Delta x)v(x) - u(x)v(x + \Delta x)}{\Delta x \cdot v(x)v(x + \Delta x)} \\
&= \lim_{\Delta x \to 0} \frac{
\frac{[u(x + \Delta x) - u(x)]v(x)}{\Delta x}
- \frac{u(x)[v(x + \Delta x) - v(x)]}{\Delta x}
}{v(x)v(x + \Delta x)} \\
&= \frac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}
\end{align*}
反函数的导数
\begin{align*}
[f^{-1}(x)]'
= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}
= \lim_{\Delta y \to 0} \frac{1}{\frac{\Delta x}{\Delta y}}
= \frac{1}{f'(y)}
\end{align*}
复合函数导数
\begin{align*}
y &= f(u) \\
u &= g(x) \\
y &= f[g(x)] \\[10pt]
\frac{dy}{dx} &= f'(u) \cdot g'(x)
= \frac{dy}{du} \cdot \frac{du}{dx}
\end{align*}
隐函数求导
例1
\begin{align*}
x + y^3 - 1 &= 0 \\
1 + 3y^2 \cdot \frac{dy}{dx} &= 0
\end{align*}
例2
\begin{align*}
e^y + xy - e &= 0 \\
e^y \frac{dy}{dx} + y + x\frac{dy}{dx} &= 0
\end{align*}
参数方程导数
运动学示例
\begin{align*}
\begin{cases}
x = v_1 t \\
y = v_2 t - \frac{1}{2}gt^2
\end{cases}
\end{align*}
一般形式
\begin{align*}
\begin{cases}
x = \varphi(t) \\
y = \psi(t)
\end{cases}
\quad \Rightarrow \quad
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\psi'(t)}{\varphi'(t)}
\end{align*}
