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},\phantom{\rule{0.2em}{0ex}}\text{for real numbers}\phantom{\rule{0.2em}{0ex}}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}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(x\right)

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

11. \frac{d}{dx}\left(\text{sec}\left(x\right)\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right)

12. \frac{d}{dx}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=-\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)

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

14. \frac{d}{dx}\left(\text{csc}\left(x\right)\right)=-\text{csc}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}\left(x\right)

Inverse Trigonometric Functions

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

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

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

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

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

20. \frac{d}{dx}\left({\text{csc}}^{-1}\left(x\right)\right)=-\frac{1}{|x|\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}\phantom{\rule{0.1em}{0ex}}|x|\right)=\frac{1}{x}

23. \frac{d}{dx}\left({b}^{x}\right)={b}^{x}\text{ln}\phantom{\rule{0.1em}{0ex}}(b)

24. \frac{d}{dx}\left({\text{log}}_{b}(x)\right)=\frac{1}{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}(b)}

Hyperbolic Functions

25. \frac{d}{dx}\left(\text{sinh}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}\left(x\right)

26. \frac{d}{dx}\left(\text{tanh}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)={\text{sech}}^{2}\phantom{\rule{0.1em}{0ex}}\left(x\right)

27. \frac{d}{dx}\left(\text{sech}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{-sech}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.1em}{0ex}}\left(x\right)

28. \frac{d}{dx}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}\left(x\right)

29. \frac{d}{dx}\left(\text{coth}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{-}{\text{csch}}^{2}\phantom{\rule{0.1em}{0ex}}\left(x\right)

30. \frac{d}{dx}\left(\text{csch}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{-csch}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.1em}{0ex}}\left(x\right)

Inverse Hyperbolic Functions

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

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

33. \frac{d}{dx}\left({\text{sech}}^{-1}\left(x\right)\right)=-\frac{1}{x\sqrt{1-{x}^{2}}}\phantom{\rule{1em}{0ex}}\left(0<x<1\right)

34. \frac{d}{dx}\left({\text{cosh}}^{-1}\left(x\right)\right)=\frac{1}{\sqrt{{x}^{2}-1}}\phantom{\rule{1em}{0ex}}\left(x>1\right)

35. \frac{d}{dx}\left({\text{coth}}^{-1}\left(x\right)\right)=\frac{1}{1-{x}^{2}}\phantom{\rule{1em}{0ex}}\left(|x|>1\right)

36. \frac{d}{dx}\left({\text{csch}}^{-1}\left(x\right)\right)=-\frac{1}{|x|\sqrt{1+{x}^{2}}}\phantom{\rule{0.2em}{0ex}}\left(x\ne 0\right)

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Calculus: Volume 2 (Second University of Manitoba Edition) by Gilbert Strang and Edward 'Jed' Herman, modified by Varvara Shepelska is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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