Derivatives

 

General Formulas

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

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

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

4. ddx(xn)=nxn−1,for real numbersn$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1},\text{for real numbers}n$

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

6. ddx(f(x)−g(x))=f′(x)−g′(x)$\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)=f^{\prime }\left(x\right)-g^{\prime }\left(x\right)$

7. ddx(f(x)g(x))=g(x)f′(x)−f(x)g′(x)(g(x))2$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right)f^{\prime }\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)}{{\left(g\left(x\right)\right)}^2}$

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

Trigonometric Functions

9. ddx(sinx)=cosx$\frac{d}{dx}\left(\text{sin}x\right)=\text{cos}x$

10. ddx(tanx)=sec2x$\frac{d}{dx}\left(\text{tan}x\right)={\text{sec}}^{2}x$

11. ddx(secx)=secxtanx$\frac{d}{dx}\left(\text{sec}x\right)=\text{sec}x\text{tan}x$

12. ddx(cosx)=−sinx$\frac{d}{dx}\left(\text{cos}x\right)=\text{−}\text{sin}x$

13. ddx(cotx)=−csc2x$\frac{d}{dx}\left(\text{cot}x\right)=\text{−}{\text{csc}}^{2}x$

14. ddx(cscx)=−cscxcotx$\frac{d}{dx}\left(\text{csc}x\right)=\text{−csc}x\text{cot}x$

Inverse Trigonometric Functions

15. ddx(sin−1x)=11−x2√$\frac{d}{dx}\left({\text{sin}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{1-x^2}}$

16. ddx(tan−1x)=11+x2$\frac{d}{dx}\left({\text{tan}}^{\mathrm{-1}}x\right)=\frac{1}{1+x^2}$

17. ddx(sec−1x)=1|x|x2−1√$\frac{d}{dx}\left({\text{sec}}^{\mathrm{-1}}x\right)=\frac{1}{\left|x\right|\sqrt{x^2-1}}$

18. ddx(cos−1x)=−11−x2√$\frac{d}{dx}\left({\text{cos}}^{\mathrm{-1}}x\right)=-\frac{1}{\sqrt{1-x^2}}$

19. ddx(cot−1x)=−11+x2$\frac{d}{dx}\left({\text{cot}}^{\mathrm{-1}}x\right)=-\frac{1}{1+x^2}$

20. ddx(csc−1x)=−1|x|x2−1√$\frac{d}{dx}\left({\text{csc}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{x^2-1}}$

Exponential and Logarithmic Functions

21. ddx(ex)=ex$\frac{d}{dx}\left(e^x\right)=e^x$

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

23. ddx(bx)=bxlnb$\frac{d}{dx}\left(b^x\right)=b^x\text{ln}b$

24. ddx(logbx)=1xlnb$\frac{d}{dx}\left({\text{log}}_{b}x\right)=\frac{1}{x\text{ln}b}$

Hyperbolic Functions

25. ddx(sinhx)=coshx$\frac{d}{dx}\left(\text{sinh}x\right)=\text{cosh}x$

26. ddx(tanhx)=sech2x$\frac{d}{dx}\left(\text{tanh}x\right)={\text{sech}}^{2}x$

27. ddx(sechx)=−sechxtanhx$\frac{d}{dx}\left(\text{sech}x\right)=\text{−sech}x\text{tanh}x$

28. ddx(coshx)=sinhx$\frac{d}{dx}\left(\text{cosh}x\right)=\text{sinh}x$

29. ddx(cothx)=−csch2x$\frac{d}{dx}\left(\text{coth}x\right)=\text{−}{\text{csch}}^{2}x$

30. ddx(cschx)=−cschxcothx$\frac{d}{dx}\left(\text{csch}x\right)=\text{−csch}x\text{coth}x$

Inverse Hyperbolic Functions

31. ddx(sinh−1x)=1x2+1√$\frac{d}{dx}\left({\text{sinh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{x^2+1}}$

32. ddx(tanh−1x)=11−x2(|x|<1)$\frac{d}{dx}\left({\text{tanh}}^{\mathrm{-1}}x\right)=\frac{1}{1-x^2}\left(\left|x\right|<1\right)$

33. ddx(sech−1x)=−1x1−x2√(0<x<1)$\frac{d}{dx}\left({\text{sech}}^{\mathrm{-1}}x\right)=-\frac{1}{x\sqrt{1-x^2}}\left(0<x<1\right)$

34. ddx(cosh−1x)=1x2−1√(x>1)$\frac{d}{dx}\left({\text{cosh}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{x^2-1}}\left(x>1\right)$

35. ddx(coth−1x)=11−x2(|x|>1)$\frac{d}{dx}\left({\text{coth}}^{\mathrm{-1}}x\right)=\frac{1}{1-x^2}\left(\left|x\right|>1\right)$

36. ddx(csch−1x)=−1|x|1+x2√(x≠0)$\frac{d}{dx}\left({\text{csch}}^{\mathrm{-1}}x\right)=-\frac{1}{\left|x\right|\sqrt{1+x^2}}\left(x\ne 0\right)$