附录 B 导数表

一般公式

1. $\frac{d}{dx}\left(c\right)=0$

2. $\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)=f^{\prime }\left(x\right)+g^{\prime }\left(x\right)$

3. $\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)=f^{\prime }\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime }\left(x\right)$

4. $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1},\text{其中 n 为实数}$

5. $\frac{d}{dx}\left(cf\left(x\right)\right)=c{f}^{\prime }\left(x\right)$

6. $\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)=f^{\prime }\left(x\right)-g^{\prime }\left(x\right)$

7. $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right)f^{\prime }\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)}{{\left(g\left(x\right)\right)}^2}$

8. $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]=f^{\prime }\left(g\left(x\right)\right)·g^{\prime }\left(x\right)$

三角函数

9. $\frac{d}{dx}\left(\text{sin}x\right)=\text{cos}x$

10. $\frac{d}{dx}\left(\text{tan}x\right)={\text{sec}}^{2}x$

11. $\frac{d}{dx}\left(\text{sec}x\right)=\text{sec}x\text{tan}x$

12. $\frac{d}{dx}\left(\text{cos}x\right)=\text{−}\text{sin}x$

13. $\frac{d}{dx}\left(\text{cot}x\right)=\text{−}{\text{csc}}^{2}x$

14. $\frac{d}{dx}\left(\text{csc}x\right)=\text{−csc}x\text{cot}x$

反三角函数

15. $\frac{d}{dx}\left({\text{sin}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{1-x^2}}$

16. $\frac{d}{dx}\left({\text{tan}}^{\mathrm{-1}}x\right)=\frac{1}{1+x^2}$

17. $\frac{d}{dx}\left({\text{sec}}^{\mathrm{-1}}x\right)=\frac{1}{\left|x\right|\sqrt{x^2-1}}$

18. $\frac{d}{dx}\left({\text{cos}}^{\mathrm{-1}}x\right)=-\frac{1}{\sqrt{1-x^2}}$

19. $\frac{d}{dx}\left({\text{cot}}^{\mathrm{-1}}x\right)=-\frac{1}{1+x^2}$

20. $\frac{d}{dx}\left({\text{csc}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{x^2-1}}$

指数函数和对数函数

21. $\frac{d}{dx}\left(e^x\right)=e^x$

22. $\frac{d}{dx}\left(\text{ln}\left|x\right|\right)=\frac{1}{x}$

23. $\frac{d}{dx}\left(b^x\right)=b^x\text{ln}b$

24. $\frac{d}{dx}\left({\text{log}}_{b}x\right)=\frac{1}{x\text{ln}b}$

双曲函数

25. $\frac{d}{dx}\left(\text{sinh}x\right)=\text{cosh}x$

26. $\frac{d}{dx}\left(\text{tanh}x\right)={\text{sech}}^{2}x$

27. $\frac{d}{dx}\left(\text{sech}x\right)=\text{−sech}x\text{tanh}x$

28. $\frac{d}{dx}\left(\text{cosh}x\right)=\text{sinh}x$

29. $\frac{d}{dx}\left(\text{coth}x\right)=\text{−}{\text{csch}}^{2}x$

30. $\frac{d}{dx}\left(\text{csch}x\right)=\text{−csch}x\text{coth}x$

反双曲函数

31. $\frac{d}{dx}\left({\text{sinh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{x^2+1}}$

32. $\frac{d}{dx}\left({\text{tanh}}^{\mathrm{-1}}x\right)=\frac{1}{1-x^2}\left(\left|x\right|<1\right)$

33. $\frac{d}{dx}\left({\text{sech}}^{\mathrm{-1}}x\right)=-\frac{1}{x\sqrt{1-x^2}}\left(0<x<1\right)$

34. $\frac{d}{dx}\left({\text{cosh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{x^2-1}}\left(x>1\right)$

35. $\frac{d}{dx}\left({\text{coth}}^{\mathrm{-1}}x\right)=\frac{1}{1-x^2}\left(\left|x\right|>1\right)$

36. $\frac{d}{dx}\left({\text{csch}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{1+x^2}}\left(x\ne 0\right)$