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\(x = \cot y\)

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