Table of Derivatives

General Formulas

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. $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1},\text{for real numbers}n$

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

Trigonometric Functions

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

Inverse Trigonometric Functions

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

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

Exponential and Logarithmic Functions

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

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

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

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

Hyperbolic Functions

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

Inverse Hyperbolic Functions

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

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

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

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

You can also download for free at http://cnx.org/contents/9a1df55a-b167-4736-b5ad-15d996704270@5.1

Attribution:

• For questions regarding this license, please contact partners@openstaxcollege.org.
• If you use this textbook as a bibliographic reference, then you should cite it as follows: OpenStax College, Calculus. OpenStax CNX. http://cnx.org/contents/9a1df55a-b167-4736-b5ad-15d996704270@5.1.
• If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: "Download for free at http://cnx.org/contents/9a1df55a-b167-4736-b5ad-15d996704270@5.1."
• If you redistribute part of this textbook, then you must retain in every digital format page view (including but not limited to EPUB, PDF, and HTML) and on every physical printed page the following attribution: "Download for free at http://cnx.org/contents/9a1df55a-b167-4736-b5ad-15d996704270@5.1."