# General Formulas

1. $\frac{d}{dx}(c)=0$

2. $\frac{d}{dx}(f(x)+g(x))={f}^{\prime }(x)+{g}^{\prime }(x)$

3. $\frac{d}{dx}(f(x)g(x))={f}^{\prime }(x)g(x)+f(x){g}^{\prime }(x)$

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

5. $\frac{d}{dx}(cf(x))=c{f}^{\prime }(x)$

6. $\frac{d}{dx}(f(x)-g(x))={f}^{\prime }(x)-{g}^{\prime }(x)$

7. $\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x){f}^{\prime }(x)-f(x){g}^{\prime }(x)}{{(g(x))}^{2}}$

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

# Trigonometric Functions

9. $\frac{d}{dx}( \sin x)= \cos x$

10. $\frac{d}{dx}( \tan x)={ \sec }^{2}x$

11. $\frac{d}{dx}( \sec x)= \sec x \tan x$

12. $\frac{d}{dx}( \cos x)=- \sin x$

13. $\frac{d}{dx}( \cot x)=-{ \csc }^{2}x$

14. $\frac{d}{dx}( \csc x)=\text{-csc}x \cot x$

# Inverse Trigonometric Functions

15. $\frac{d}{dx}({ \sin }^{-1}x)=\frac{1}{\sqrt{1-{x}^{2}}}$

16. $\frac{d}{dx}({ \tan }^{-1}x)=\frac{1}{1+{x}^{2}}$

17. $\frac{d}{dx}({ \sec }^{-1}x)=\frac{1}{|x|\sqrt{{x}^{2}-1}}$

18. $\frac{d}{dx}({ \cos }^{-1}x)=-\frac{1}{\sqrt{1-{x}^{2}}}$

19. $\frac{d}{dx}({ \cot }^{-1}x)=-\frac{1}{1+{x}^{2}}$

20. $\frac{d}{dx}({ \csc }^{-1}x)=-\frac{1}{|x|\sqrt{{x}^{2}-1}}$

# Exponential and Logarithmic Functions

21. $\frac{d}{dx}({e}^{x})={e}^{x}$

22. $\frac{d}{dx}(\text{ln}|x|)=\frac{1}{x}$

23. $\frac{d}{dx}({b}^{x})={b}^{x}\text{ln}b$

24. $\frac{d}{dx}({\text{log}}_{b}x)=\frac{1}{x\text{ln}b}$

# Hyperbolic Functions

25. $\frac{d}{dx}(\text{sinh}x)=\text{cosh}x$

26. $\frac{d}{dx}(\text{tanh}x)={\text{sech}}^{2}x$

27. $\frac{d}{dx}(\text{sech}x)=\text{-sech}x\text{tanh}x$

28. $\frac{d}{dx}(\text{cosh}x)=\text{sinh}x$

29. $\frac{d}{dx}(\text{coth}x)=-{\text{csch}}^{2}x$

30. $\frac{d}{dx}(\text{csch}x)=\text{-csch}x\text{coth}x$

# Inverse Hyperbolic Functions

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

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

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

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

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

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