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}{2\sqrt{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} a\frac{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}{x\sqrt{x^{2}-1}},\left | x \right |> 1

30.  \frac{d}{dx}\textup{sec}^{-1} x = \frac{1}{x\sqrt{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