5.1: A- Table of Derivatives
- Page ID
- 25978
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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. \(\quad \dfrac{d}{dx}\left(\text{csch}^{−1}\,x\right)=\dfrac{-1}{|x|\sqrt{1+x^2}}\quad (x≠0)\)