1. \quad \dfrac{d}{dx}\left(c\right)=0
2. \quad \dfrac{d}{dx}\left(f(x)+g(x)\right)=f′(x)+g′(x)
3. \quad \dfrac{d}{dx}\left(f(x)g(x)\right)=f′(x)g(x)+f(x)g′(x)
4. \quad \dfrac{d}{dx}\left(x^n\right)=nx^{n−1},\quad \text{for real numbers }n
5. \quad \dfrac{d}{dx}\left(cf(x)\right)=cf′(x)
6. \quad \dfrac{d}{dx}\left(f(x)−g(x)\right)=f′(x)−g′(x)
7. \quad \dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f′(x)−f(x)g′(x)}{(g(x))^2}
8. \quad \dfrac{d}{dx}\left[f(g(x))\right]=f′(g(x))·g′(x)
Trigonometric Functions
9. \quad \dfrac{d}{dx}\left(\sin x\right)=\cos x
10. \quad \dfrac{d}{dx}\left(\tan x\right)=\sec^2x
11. \quad \dfrac{d}{dx}\left(\sec x\right)=\sec x\tan x
12. \quad \dfrac{d}{dx}\left(\cos x\right)=−\sin x
13. \quad \dfrac{d}{dx}\left(\cot x\right)=−\csc^2x
14. \quad \dfrac{d}{dx}\left(\csc x\right)=−\csc x\cot x
Inverse Trigonometric Functions
15. \quad \dfrac{d}{dx}\left(\sin^{-1}x\right)=\dfrac{1}{\sqrt{1−x^2}}
16. \quad \dfrac{d}{dx}\left(\tan^{-1}x\right)=\dfrac{1}{1+x^2}
17. \quad \dfrac{d}{dx}\left(\sec^{-1}x\right)=\dfrac{1}{|x|\sqrt{x^2−1}}
18. \quad \dfrac{d}{dx}\left(\cos^{-1}x\right)=\dfrac{-1}{\sqrt{1−x^2}}
19. \quad \dfrac{d}{dx}\left(\cot^{-1}x\right)=\dfrac{-1}{1+x^2}
20. \quad \dfrac{d}{dx}\left(\csc^{-1}x\right)=\dfrac{-1}{|x|\sqrt{x^2−1}}
Exponential and Logarithmic Functions
21. \quad \dfrac{d}{dx}\left(e^x\right)=e^x
22. \quad \dfrac{d}{dx}\left(\ln|x|\right)=\dfrac{1}{x}
23. \quad \dfrac{d}{dx}\left(b^x\right)=b^x\ln b
24. \quad \dfrac{d}{dx}\left(\log_bx\right)=\dfrac{1}{x\ln b}
Hyperbolic Functions
25. \quad \dfrac{d}{dx}\left(\sinh x\right)=\cosh x
26. \quad \dfrac{d}{dx}\left(\tanh x\right)=\text{sech}^2 \,x
27. \quad \dfrac{d}{dx}\left(\text{sech} x\right)=−\text{sech} \,x\tanh x
28. \quad \dfrac{d}{dx}\left(\cosh x\right)=\sinh x
29. \quad \dfrac{d}{dx}\left(\coth x\right)=−\text{csch}^2 \,x
30. \quad \dfrac{d}{dx}\left(\text{csch} \,x\right)=−\text{csch} x\coth x
Inverse Hyperbolic Functions
31. \quad \dfrac{d}{dx}\left(\sinh^{-1}x\right)=\dfrac{1}{\sqrt{x^2+1}}
32. \quad \dfrac{d}{dx}\left(\tanh^{-1}x\right)=\dfrac{1}{1-x^2}\quad (|x|<1)
33. \quad \dfrac{d}{dx}\left(\text{sech}^{-1}\,x\right)=\dfrac{-1}{x\sqrt{1-x^2}}\quad (0<x<1)
34. \quad \dfrac{d}{dx}\left(\cosh^{-1}x\right)=\dfrac{1}{\sqrt{x^2-1}}\quad (x>1)
35. \quad \dfrac{d}{dx}\left(\coth^{-1}x\right)=\dfrac{1}{1-x^2}\quad (|x|>1)
36. \quad \dfrac{d}{dx}\left(\text{csch}^{−1}\,x\right)=\dfrac{-1}{|x|\sqrt{1+x^2}}\quad (x≠0)
