Table of Derivatives

Gilbert Strang and Edward 'Jed' Herman, modified by Varvara Shepelska

Table of Derivatives

General Formulas

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

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

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

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

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

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

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

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

Trigonometric Functions

9. $\dfrac{d}{dx}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(x\right)$

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

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

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

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

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

Inverse Trigonometric Functions

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

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

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

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

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

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

Exponential and Logarithmic Functions

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

22. $\dfrac{d}{dx}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}|x|\right)=\dfrac{1}{x}$

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

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

Hyperbolic Functions

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

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

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

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

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

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

Inverse Hyperbolic Functions

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

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

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

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

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

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