General Formulas

1. [latex]\frac{d}{dx}(c)=0[/latex]

2. [latex]\frac{d}{dx}(f(x)+g(x))={f}^{\prime }(x)+{g}^{\prime }(x)[/latex]

3. [latex]\frac{d}{dx}(f(x)g(x))={f}^{\prime }(x)g(x)+f(x){g}^{\prime }(x)[/latex]

4. [latex]\frac{d}{dx}({x}^{n})=n{x}^{n-1},\text{for real numbers}n[/latex]

5. [latex]\frac{d}{dx}(cf(x))=c{f}^{\prime }(x)[/latex]

6. [latex]\frac{d}{dx}(f(x)-g(x))={f}^{\prime }(x)-{g}^{\prime }(x)[/latex]

7. [latex]\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x){f}^{\prime }(x)-f(x){g}^{\prime }(x)}{{(g(x))}^{2}}[/latex]

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

Trigonometric Functions

9. [latex]\frac{d}{dx}( \sin x)= \cos x[/latex]

10. [latex]\frac{d}{dx}( \tan x)={ \sec }^{2}x[/latex]

11. [latex]\frac{d}{dx}( \sec x)= \sec x \tan x[/latex]

12. [latex]\frac{d}{dx}( \cos x)=- \sin x[/latex]

13. [latex]\frac{d}{dx}( \cot x)=-{ \csc }^{2}x[/latex]

14. [latex]\frac{d}{dx}( \csc x)=\text{-csc}x \cot x[/latex]

Inverse Trigonometric Functions

15. [latex]\frac{d}{dx}({ \sin }^{-1}x)=\frac{1}{\sqrt{1-{x}^{2}}}[/latex]

16. [latex]\frac{d}{dx}({ \tan }^{-1}x)=\frac{1}{1+{x}^{2}}[/latex]

17. [latex]\frac{d}{dx}({ \sec }^{-1}x)=\frac{1}{|x|\sqrt{{x}^{2}-1}}[/latex]

18. [latex]\frac{d}{dx}({ \cos }^{-1}x)=-\frac{1}{\sqrt{1-{x}^{2}}}[/latex]

19. [latex]\frac{d}{dx}({ \cot }^{-1}x)=-\frac{1}{1+{x}^{2}}[/latex]

20. [latex]\frac{d}{dx}({ \csc }^{-1}x)=-\frac{1}{|x|\sqrt{{x}^{2}-1}}[/latex]

Exponential and Logarithmic Functions

21. [latex]\frac{d}{dx}({e}^{x})={e}^{x}[/latex]

22. [latex]\frac{d}{dx}(\text{ln}|x|)=\frac{1}{x}[/latex]

23. [latex]\frac{d}{dx}({b}^{x})={b}^{x}\text{ln}b[/latex]

24. [latex]\frac{d}{dx}({\text{log}}_{b}x)=\frac{1}{x\text{ln}b}[/latex]

Hyperbolic Functions

25. [latex]\frac{d}{dx}(\text{sinh}x)=\text{cosh}x[/latex]

26. [latex]\frac{d}{dx}(\text{tanh}x)={\text{sech}}^{2}x[/latex]

27. [latex]\frac{d}{dx}(\text{sech}x)=\text{-sech}x\text{tanh}x[/latex]

28. [latex]\frac{d}{dx}(\text{cosh}x)=\text{sinh}x[/latex]

29. [latex]\frac{d}{dx}(\text{coth}x)=-{\text{csch}}^{2}x[/latex]

30. [latex]\frac{d}{dx}(\text{csch}x)=\text{-csch}x\text{coth}x[/latex]

Inverse Hyperbolic Functions

31. [latex]\frac{d}{dx}({\text{sinh}}^{-1}x)=\frac{1}{\sqrt{{x}^{2}+1}}[/latex]

32. [latex]\frac{d}{dx}({\text{tanh}}^{-1}x)=\frac{1}{1-{x}^{2}}(|x| < 1)[/latex]

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

34. [latex]\frac{d}{dx}({\text{cosh}}^{-1}x)=\frac{1}{\sqrt{{x}^{2}-1}}\phantom{\rule{1em}{0ex}}(x < 1)[/latex]

35. [latex]\frac{d}{dx}({\text{coth}}^{-1}x)=\frac{1}{1-{x}^{2}}\phantom{\rule{1em}{0ex}}(|x| < 1)[/latex]

36. [latex]\frac{d}{dx}({\text{csch}}^{-1}x)=-\frac{1}{|x|\sqrt{1+{x}^{2}}}(x\ne 0)[/latex]

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Calculus Volume 1 by OSCRiceUniversity is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book