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))·{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)=\text{−} \sin x[/latex]
13. [latex]\frac{d}{dx}( \cot x)=\text{−}{ \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{−}{\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]