基本函数导数及推导

基本函数导数

1. $c' = 0$

2. $(x^n)' = nx^{n-1}$

   • $(\sqrt x)' = \dfrac{1}{2\sqrt x}$

   • $(\dfrac{1}{x})' = -\dfrac{1}{x^2}$

3. $(\sin x)' = \cos x$

4. $(\cos x)' = -\sin x$

5. $(\ln x)' = \dfrac{1}{x}$

6. $(\log_a x)' = \dfrac{\log_a e}{x}$

7. $(a^x)' = a^x\ln a$

   • $(e^x)' = e^x$

8. $(\tan x)' = \sec ^2 x$

9. $(\cot x)' = -\csc^2 x$

10. $(\sec x)' = \sec x\tan x$

11. $(\csc x)' = -\csc x\cot x$

12. $(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}$

13. $(\arccos x)' = -\dfrac{1}{\sqrt{1 - x^2}}$

14. $(\arctan x)' = \dfrac{1}{1 + x^2}$

15. $(\text{arccot} x)' = -\dfrac{1}{1 + x^2}$

$(x^n)' = nx^{n-1}$

由导数的定义：$f'(x)=\lim_{\Delta x\to 0}\dfrac{f(x + \Delta x) - f(x)}{\Delta x}$

$$\begin{align} \left( x^n \right)' &=\lim_{\Delta x\rightarrow 0}\dfrac{\left( x+\Delta x \right) ^n-x^n}{\Delta x} \notag \\ &=\lim_{\Delta x\rightarrow 0}\dfrac{\sum_{m=1}^n{C}_{n}^{m}x^m\Delta x^{n-m}-x^n}{\Delta x} \notag \\ &=\lim_{\Delta x\rightarrow 0}\dfrac{C_{n}^{0}x^0\Delta x^n+C_{n}^{1}x^1\Delta x^{n-1}+……+C_{n}^{n-1}x^{n-1}\Delta x^1+C_{n}^{n}x^n\Delta x^0-x^n}{\Delta x} \notag \\ &=\lim_{\Delta x\rightarrow 0}\dfrac{C_{n}^{0}x^0\Delta x^n+C_{n}^{1}x^1\Delta x^{n-1}+……+C_{n}^{n-1}x^{n-1}\Delta x^1}{\Delta x} \notag \\ &=nx^{n-1} \end{align}$$

$(\sin x)' = \cos x$

$$\begin{align} (\sin x)' &=\lim_{\Delta x\to 0}\dfrac{\sin (x + \Delta x) - \sin x}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\dfrac{2\cos\dfrac{x + \Delta x + x}{2}\sin\dfrac{x + \Delta x - x}{2}}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\dfrac{2\cos (x + \dfrac{\Delta x}{2})\sin\dfrac{ \Delta x}{2}}{2\dfrac{\Delta x}{2}} \notag \\ &=\lim_{\Delta x\to 0}\cos (x + \dfrac{\Delta x}{2}) \notag \\ &=\cos x \end{align}$$

$(\cos x)' = -\sin x$

$$\begin{align} (\cos x)' &=\lim_{\Delta x\to 0}\frac{\cos (x + \Delta x) - \cos x}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{-2\sin\frac{x + \Delta x + x}{2}\sin\frac{x + \Delta x - x}{2}}{\Delta x} \notag \\ &= \lim_{\Delta x\to 0}\frac{-2\sin (x + \frac{\Delta x}{2})\sin\frac{ \Delta x}{2}}{2\frac{\Delta x}{2}} \notag \\ &=\lim_{\Delta x\to 0}-\sin (x + \frac{\Delta x}{2}) \notag \\ &=-\sin x \end{align}$$

$(\ln x)' = \dfrac{1}{x}$

$$\begin{align} (\ln x)' &=\lim_{\Delta x\to 0}\frac{\ln (x + \Delta x) - \ln x}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{\ln(\frac{x + \Delta x}{x})}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{\ln(1 + \frac{\Delta x}{x})}{\frac{\Delta x}{x}x} \notag \\ &=\frac{1}{x} \end{align}$$

$(\log_a x)' = \dfrac{\log_a e}{x}$

$$\begin{align} (\log_a x)' &=\lim_{\Delta x\to 0}\frac{\log_a (x + \Delta x) - \log_a x}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{\log_a(1 + \frac{\Delta x}{x})}{\Delta x} =\lim_{\Delta x\to 0}\frac{\log_a (1 + \frac{1}{\frac{x}{\Delta x}})^{\frac{x}{\Delta x}\frac{\Delta x}{x}}}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{\log_a e^{\frac{\Delta x}{x}}}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{\frac{\Delta x}{x}\log_a e}{\Delta x} \notag \\ &=\frac{\log_a e}{x} \end{align}$$

$(a^x)' = a^x\ln a$

$$\begin{align} (a^x)' &=\lim_{\Delta x\to 0}\frac{a^{x + \Delta x} - a^x}{\Delta x} \notag \\ &=\lim_{\Delta x\to 0}\frac{a^x(a^{\Delta x} - 1)}{\Delta x} \notag \\ &=a^x\lim_{\Delta x\to 0}\frac{\Delta x\ln a}{\Delta x} \notag \\ &=a^x \ln a \end{align}$$

$(\tan x)' = \sec ^2 x$

$$\begin{align} (\tan x)'&= (\frac{\sin x}{\cos x})' \notag \\ &=\frac{(\sin x)'\cos x - \sin x(\cos x)'}{\cos^2 x} \notag \\ &=\frac{\cos^2 x + \sin^2 x}{\cos^2 x} \notag \\ &=\sec^2 x \end{align}$$

$(\cot x)' = -\csc^2 x$

$$\begin{align} (\cot x)' &= (\frac{1}{\tan x})' \notag \\ &=(\frac{\cos x}{\sin x})' \notag \\ &=\frac{(\cos x)'\sin x - \cos x(\sin x)'}{\sin^2 x} \notag \\ &=-\csc^2 x \end{align}$$

$(\sec x)' = \sec x\tan x$

$$\begin{align} (\sec x)' = (\frac{1}{\cos x})' \notag \\ &=\frac{1'\cos x - 1(\cos x)'}{\cos^2 x} \notag \\ &=\frac{\sin x}{\cos^2 x} \notag \\ &=\sec x \tan x \end{align}$$

$(\csc x)' = -\csc x\cot x$

$$\begin{align} (\csc x)' &= (\frac{1}{\sin x})' \notag \\ &=\frac{1'\sin x - 1(\sin x)'}{\sin^2 x} \notag \\ &=-\frac{\cos x}{\sin^2 x} \notag \\ &=-\csc x\cot x \end{align}$$

$(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}$

设 $y = \arcsin x$，则$x = \sin y$

$$\begin{align} (\arcsin x)' &= (\frac{1}{\sin y})' \notag \\ &=\frac{1}{\cos y} \notag \\ &=\frac{1}{\sqrt{1 - \sin^2 y}} \notag \\ &=\frac{1}{\sqrt{1 - x^2}} \end{align}$$

$(\arccos x)' = -\dfrac{1}{\sqrt{1 - x^2}}$

设 $y = \arccos x$，则$x = \cos y$

$$\begin{align} (\arccos x)' &= (\frac{1}{\cos y})' \notag \\ &=-\frac{1}{\sin y} \notag \\ &=-\frac{1}{\sqrt{1 - \cos^2 y}} \notag \\ &=-\frac{1}{\sqrt{1 - x^2}} \end{align}$$

$(\arctan x)' = \dfrac{1}{1 + x^2}$

设 $y = \arctan x$，则$x = \tan y$

$$\begin{align} (\arctan x)' &= (\frac{1}{\tan y})' \notag \\ &=\frac{1}{\sec^2 y} \notag \\ &=\frac{1}{1 + \tan^2 y} \notag \\ &=\frac{1}{1 + x^2} \end{align}$$

$(\text{arccot} x)' = -\dfrac{1}{1 + x^2}$

设 $y = \text{arccot}x$，则$x = \cot y$

$$\begin{align} (\text{arccot} x)' &= (\frac{1}{\cot y})' \notag \\ &=-\frac{1}{\csc^2 y} \notag \\ &=-\frac{1}{1 + \cot^2 y} \notag \\ &=-\frac{1}{1 + x^2} \end{align}$$