导数推导

发表: 5/12/2025 更新: 5/22/2026 字数: 0 字 时长: 0 分钟

导数定义:$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$

1. $f(x)=\sin x$
   $\begin{array}{rl}{f}^{\prime }(x)& =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\sin (x+\mathrm{\Delta }x)-\sin x}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\sin x\cos \mathrm{\Delta }x+\cos x\sin \mathrm{\Delta }x-\sin x}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\cos x\sin \mathrm{\Delta }x}{\mathrm{\Delta }x}\\ & =\cos x\end{array}$

$\mathrm{作}\mathrm{圆}\mathrm{心}\mathrm{在}\mathrm{原}\mathrm{点}\mathrm{的}\mathrm{单}\mathrm{位}\mathrm{圆}，\mathrm{取}\mathrm{圆}\mathrm{心}\mathrm{角}x\in (0,\frac{\pi }{2}),A\mathrm{是}\mathrm{圆}\mathrm{上}\mathrm{一}\mathrm{点},B(1,0)$
$\mathrm{过}B\mathrm{点}\mathrm{作}x\mathrm{轴}\mathrm{的}\mathrm{垂}\mathrm{线}\mathrm{与}\mathrm{射}\mathrm{线}OA\mathrm{交}\mathrm{于}C\mathrm{点},\mathrm{作}\mathrm{外}\mathrm{切}Rt\mathrm{\Delta }OBC$
$\mathrm{如}\mathrm{图}\mathrm{所}\mathrm{示},\mathrm{显}\mathrm{然}\mathrm{圆}\mathrm{内}\mathrm{三}\mathrm{角}\mathrm{形}<\mathrm{扇}\mathrm{形}<\mathrm{外}\mathrm{切}\mathrm{三}\mathrm{角}\mathrm{形}:S_{\mathrm{\Delta }OAB}<S_{⌔OAB}<S_{\mathrm{\Delta }OBC}$
$S_{\mathrm{\Delta }OAB}=\frac{1}{2}AO\cdot BO\cdot \sin x=\frac{1}{2}\sin x$
$S_{⌔OAB}=\frac{1}{2}x$
$S_{\mathrm{\Delta }OBC}=\frac{1}{2}OB\cdot \tan x=\frac{1}{2}\tan x$
$\begin{array}{rcl}\Rightarrow \frac{1}{2}\sin x<& \frac{1}{2}x& <\frac{1}{2}\\ \sin x<& x& <\tan x\\ 1<& \frac{x}{\sin x}& <\frac{1}{\cos x},(x\in (0,\frac{\pi }{2}),\sin x>0)\\ \cos x<& \frac{\sin x}{x}& <1\end{array}$
$\underset{x\to 0^{+}}{lim}\cos x=\cos 0=1$
$\mathrm{由}\mathrm{夹}\mathrm{逼}\mathrm{准}\mathrm{则}\mathrm{得}\mathrm{到}:\underset{x\to 0^{+}}{lim}\frac{\sin x}{x}=1$
$\mathrm{令}t=-x,\mathrm{则}x\to 0^{-}⟺t\to 0^{+}$
$\underset{x\to 0^{-}}{lim}\frac{\sin x}{x}=\underset{t\to 0^{+}}{lim}\frac{\sin -t}{-t}=\underset{t\to 0^{+}}{lim}\frac{\sin t}{t}=1$
$\therefore \underset{x\to 0}{lim}\frac{\sin x}{x}=1,\mathrm{或}\mathrm{描}\mathrm{述}\mathrm{为}x\to 0\mathrm{时},\sin x\sim x$

图一 $\begin{array}{r}\underset{x\to 0}{lim}\frac{\sin x}{x}\end{array}$ 极限示意图

2. $f(x)=\cos x$
   $\begin{array}{rl}{f}^{\prime }(x)& =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\cos (x+\mathrm{\Delta }x)-\cos x}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\cos x\cos \mathrm{\Delta }x-\sin x\sin \mathrm{\Delta }x-\cos x}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{-\sin x\sin \mathrm{\Delta }x}{\mathrm{\Delta }x}\\ & =-\sin x\end{array}$

3. 其他三角函数
   ① $\tan x=\frac{\sin x}{\cos x}$
   $\begin{array}{rl}(\tan x{)}^{\prime }& =\frac{(\sin x{)}^{\prime }\cos x-\sin x(\cos x{)}^{\prime }}{{\cos }^{2}x}\\ & =\frac{\cos x\cdot \cos x-\sin x(-\sin x)}{{\cos }^{2}x}\\ & =\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}\\ & ={\sec }^{2}x\end{array}$
   ② $\cot x=\frac{\cos x}{\sin x}=\frac{1}{\tan x}$
   $\begin{array}{rl}(\cot x{)}^{\prime }& =-\frac{1}{{\tan }^{2}x}{\sec }^{2}x\\ & =-\frac{\cancel{{\cos }^{2}x}}{{\sin }^{2}x}\cdot \frac{1}{\cancel{{\cos }^{2}x}}\\ & =-{\csc }^{2}x\end{array}$
   ③ $\sec x=\frac{1}{\cos x}$
   $\begin{array}{rl}(\sec x{)}^{\prime }& =\cancel{-}\frac{1}{{\cos }^{2}x}(\cancel{-}\sin x)\\ & =\frac{1}{\cos x}\cdot \frac{\sin x}{\cos x}\\ & =\sec x\tan x\end{array}$
   ④ $\csc x=\frac{1}{\sin x}$
   $\begin{array}{rl}(\csc x{)}^{\prime }& =-\frac{1}{{\sin }^{2}x}\cos x\\ & =-\frac{1}{\sin x}\cdot \frac{\cos x}{\sin x}\\ & =-\csc x\cot x\end{array}$

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整理所有三角函数的导数:

$$\begin{array}{rl}(\sin x{)}^{\prime }& =\cos x\\ (\cos x{)}^{\prime }& =-\sin x\\ (\tan x{)}^{\prime }& ={\sec }^{2}x\\ (\cot x{)}^{\prime }& =-{\csc }^{2}x\\ (\sec x{)}^{\prime }& =\sec x\tan x\\ (\csc x{)}^{\prime }& =-\csc x\cot x\end{array}$$