Formulas of Derivatives

General Derivative Properties:

1.  frac{d}{dx}(c) = 0 where c is any constant.

2.  frac{d}{dx}x^{n}= nx^{n-1}  This is called power rule of Derivative.

3.  frac{d}{dx}x=1.

4.  frac{d}{dx}[f(x)]^{^{n}}=[f(x)]^{^{n-1}}frac{d}{dx}f(x). It is Power rule of function

5.  frac{d}{dx}sqrt{x}=frac{1}{2sqrt{x}}.

6.  frac{d}{dx}c.f(x)=c.frac{d}{dx}f(x).

7.   frac{d}{dx}[f(x)pm g(x)]=frac{d}{dx}f(x)pm frac{d}{dx}g(x).

8. frac{d}{dx}[f(x).g(x)]=f(x)frac{d}{dx}g(x)+ g(x)frac{d}{dx}f(x).

or frac{d}{dx}(u.v)=u.{v}'+v.{u}'.      It is called Product Rule.

9.  frac{d}{dx}left [ frac{f(x)}{g(x)} right ]=frac{g(x)frac{d}{dx}f(x)- f(x)frac{d}{dx}g(x)}{left [ g(x) right ]{^2}}.

or frac{d}{dx}left [ frac{f(x)}{g(x)} right ]= frac{v.{u}'-u{v}'}{v^{2}}. It is called Quotient Rule.

Derivative of Logarithm Functions: 

10.  frac{d}{dx}textup{ln}x = frac{1}{x}

11.  frac{d}{dx}textup{log}_{x}a= frac{1}{x textup{ln} a}

12.  frac{d}{dx}textup{log}_{a}u= frac{1}{u .textup{ln}a }frac{du}{dx}

13.  frac{d}{dx}textup{ln}u = frac{1}{u}frac{du}{dx}

Derivative of Exponential Functions:

14.  frac{d}{dx}e^{x}= e^{x}

15.  frac{d}{dx}e^{f(x)}= e^{f(x)}frac{d}{dx}f(x)

16.  frac{d}{dx}a^{(x)}=a^{x}textup{ln} a

17.  frac{d}{dx}a^{(u)}=a^{u}textup{ln} afrac{du}{dx}

18. frac{d}{dx}x^{(x)}=x^{x}(1+textup{ln} x)

Derivative of Trigonometric Functions:

19. frac{d}{dx}textup{sin} x = textup{cos} x

20. frac{d}{dx}textup{cos} x = -textup{sin} x

21.  frac{d}{dx}textup{tan} x = textup{sec}^{2}x

22.  frac{d}{dx}textup{cosec} x = -textup{csc}x. textup{cot}x

23. frac{d}{dx}textup{sec} x = textup{sec}x. textup{tan}x

24. frac{d}{dx}textup{cot} x = -textup{csc}^{2}x.

Derivative of Inverse Trigonometric Functions:

25.  frac{d}{dx}textup{sin}^{-1} x = frac{1}{sqrt{1-x^{2}}}, -1< x< 1

26. frac{d}{dx}textup{cos}^{-1} x = -frac{1}{sqrt{1-x^{2}}}, -1< x< 1

27. frac{d}{dx}textup{tan}^{-1} x = frac{1}{1+x^{2}}

28. frac{d}{dx}textup{cot}^{-1} x = frac{-1}{1+x^{2}}

29.  frac{d}{dx}textup{csc}^{-1} x = frac{-1}{xsqrt{x^{2}-1}},left | x right |> 1

30.  frac{d}{dx}textup{sec}^{-1} x = frac{1}{xsqrt{x^{2}-1}},left | x right |> 1

Derivative of Hyperbolic Functions:

31. frac{d}{dx}textup{sin} h x= textup{cos} h x

32.  frac{d}{dx}textup{cos} h x= textup{sin} h x

33.  frac{d}{dx}textup{tan} h x= textup{sec}^{2} h x

34. frac{d}{dx}textup{csc} h x= -textup{csc}hx.textup{cot} hx

35.  frac{d}{dx}textup{sec} h x= -textup{sec}hx.textup{tan} hx

36.  frac{d}{dx}textup{cot} h x= -textup{csc}^{2}hx