<< Chapter < Page | Chapter >> Page > |
1. $\frac{d}{dx}\left(c\right)=0$
2. $\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)={f}^{\prime}\left(x\right)+{g}^{\prime}\left(x\right)$
3. $\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)={f}^{\prime}\left(x\right)g\left(x\right)+f\left(x\right){g}^{\prime}\left(x\right)$
4. $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1},\phantom{\rule{0.2em}{0ex}}\text{for real numbers}\phantom{\rule{0.2em}{0ex}}n$
5. $\frac{d}{dx}\left(cf\left(x\right)\right)=c{f}^{\prime}\left(x\right)$
6. $\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)={f}^{\prime}\left(x\right)-{g}^{\prime}\left(x\right)$
7. $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right){f}^{\prime}\left(x\right)-f\left(x\right){g}^{\prime}\left(x\right)}{{\left(g\left(x\right)\right)}^{2}}$
8. $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime}\left(g\left(x\right)\right)\xb7{g}^{\prime}\left(x\right)$
9. $\frac{d}{dx}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x$
10. $\frac{d}{dx}\left(\text{tan}x\right)={\text{sec}}^{2}x$
11. $\frac{d}{dx}\left(\text{sec}x\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}x$
12. $\frac{d}{dx}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212}\text{sin}\phantom{\rule{0.1em}{0ex}}x$
13. $\frac{d}{dx}\left(\text{cot}x\right)=\text{\u2212}{\text{csc}}^{2}x$
14. $\frac{d}{dx}\left(\text{csc}x\right)=\text{\u2212csc}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}x$
15. $\frac{d}{dx}\left({\text{sin}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$
16. $\frac{d}{dx}\left({\text{tan}}^{\mathrm{-1}}x\right)=\frac{1}{1+{x}^{2}}$
17. $\frac{d}{dx}\left({\text{sec}}^{\mathrm{-1}}x\right)=\frac{1}{\left|x\right|\sqrt{{x}^{2}-1}}$
18. $\frac{d}{dx}\left({\text{cos}}^{\mathrm{-1}}x\right)=-\frac{1}{\sqrt{1-{x}^{2}}}$
19. $\frac{d}{dx}\left({\text{cot}}^{\mathrm{-1}}x\right)=-\frac{1}{1+{x}^{2}}$
20. $\frac{d}{dx}\left({\text{csc}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{{x}^{2}-1}}$
21. $\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$
22. $\frac{d}{dx}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}\left|x\right|\right)=\frac{1}{x}$
23. $\frac{d}{dx}\left({b}^{x}\right)={b}^{x}\text{ln}\phantom{\rule{0.1em}{0ex}}b$
24. $\frac{d}{dx}\left({\text{log}}_{b}x\right)=\frac{1}{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}b}$
25. $\frac{d}{dx}\left(\text{sinh}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x$
26. $\frac{d}{dx}\left(\text{tanh}\phantom{\rule{0.1em}{0ex}}x\right)={\text{sech}}^{2}\phantom{\rule{0.1em}{0ex}}x$
27. $\frac{d}{dx}\left(\text{sech}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212sech}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.1em}{0ex}}x$
28. $\frac{d}{dx}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}x\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}x$
29. $\frac{d}{dx}\left(\text{coth}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212}{\text{csch}}^{2}\phantom{\rule{0.1em}{0ex}}x$
30. $\frac{d}{dx}\left(\text{csch}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212csch}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.1em}{0ex}}x$
31. $\frac{d}{dx}\left({\text{sinh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{{x}^{2}+1}}$
32. $\frac{d}{dx}\left({\text{tanh}}^{\mathrm{-1}}x\right)=\frac{1}{1-{x}^{2}}\left(\left|x\right|<1\right)$
33. $\frac{d}{dx}\left({\text{sech}}^{\mathrm{-1}}x\right)=-\frac{1}{x\sqrt{1-{x}^{2}}}\phantom{\rule{1em}{0ex}}\left(0<x<1\right)$
34. $\frac{d}{dx}\left({\text{cosh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{{x}^{2}-1}}\phantom{\rule{1em}{0ex}}\left(x>1\right)$
35. $\frac{d}{dx}\left({\text{coth}}^{\mathrm{-1}}x\right)=\frac{1}{1-{x}^{2}}\phantom{\rule{1em}{0ex}}\left(\left|x\right|>1\right)$
36. $\frac{d}{dx}\left({\text{csch}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{1+{x}^{2}}}\phantom{\rule{0.2em}{0ex}}\left(x\ne 0\right)$
Notification Switch
Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?