基本函数导数及推导

基本函数导数

1. c^′ = 0

2. (xⁿ)^′ = 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^xln a

   • (e^x)^′ = e^x

8. (tan x)^′ = sec²x

9. (cot x)^′ = −csc²x

10. (sec x)^′ = sec xtan x

11. (csc x)^′ = −csc xcot 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. $(\text{arccot} x)' = -\dfrac{1}{1 + x^2}$

(xⁿ)^′ = 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^xln 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)^′ = sec²x

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

(cot x)^′ = −csc²x

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

(sec x)^′ = sec xtan 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 xcot 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 = arccotx，则x = cot y

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