Basic Formulas of Derivatives

General Derivative Formulas:

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

2) \frac{d}{{dx}}{x^n} = n{x^{n - 1}} It is called Power Rule of Derivative.

3) \frac{d}{{dx}}x = 1

4) \frac{d}{{dx}}{[f(x)]^n} = n{[f(x)]^{n - 1}}\frac{d}{{dx}}f(x) Power Rule for Function.

5) \frac{d}{{dx}}\sqrt x= \frac{1}{{2\sqrt x }}

6) \frac{d}{{dx}}\sqrt {f(x)} = \frac{1}{{2\sqrt {f(x)}  }}\frac{d}{{dx}}f(x) = \frac{1}{{2\sqrt {f(x)} }}f'(x)

7) \frac{d}{{dx}}c  \cdot f(x) = c\frac{d}{{dx}}f(x) = c \cdot f'(x)

8) \frac{d}{{dx}}[f(x) \pm g(x)] = \frac{d}{{dx}}f(x)  \pm \frac{d}{{dx}}g(x) = f'(x) \pm g'(x)

9) \frac{d}{{dx}}[f(x) \cdot g(x)] =  f(x)\frac{d}{{dx}}g(x) + g(x)\frac{d}{{dx}}f(x) It is called Product Rule.

10) \frac{d}{{dx}}[\frac{{f(x)}}{{g(x)}}] =  \frac{{g(x)\frac{d}{{dx}}f(x) - f(x)\frac{d}{{dx}}g(x)}}{{{{[g(x)]}^2}}} It is called Quotient Rule.

 

Derivative of Logarithm Functions: 

11) \frac{d}{{dx}}\ln x = \frac{1}{x}

12) \frac{d}{{dx}}{\log _a}x = \frac{1}{{x\ln a}}

13) \frac{d}{{dx}}\ln f(x) =  \frac{1}{{f(x)}}\frac{d}{{dx}}f(x)

14) \frac{d}{{dx}}{\log _a}f(x) = \frac{1}{{f(x)\ln  a}}\frac{d}{{dx}}f(x)

 

Derivative of Exponential Functions:

15) \frac{d}{{dx}}{e^x} = {e^x}

16) \frac{d}{{dx}}{e^{f(x)}} = {e^{f(x)}}\frac{d}{{dx}}f(x)

17) \frac{d}{{dx}}{a^x} = {a^x}\ln a

18) \frac{d}{{dx}}{a^{f(x)}} = {a^{f(x)}}\ln  a\frac{d}{{dx}}f(x)

19) \frac{d}{{dx}}{x^x} = {x^x}(1 + \ln x)

 

Derivative of Trigonometric Functions:

20) \frac{d}{{dx}}Sinx = Cosx

21) \frac{d}{{dx}}Cosx = Sinx

22) \frac{d}{{dx}}Tanx = Se{c^2}x

23) \frac{d}{{dx}}Cotx = - Co{\sec ^2}x

24) \frac{d}{{dx}}Secx = Secx \cdot Tanx

25) \frac{d}{{dx}}Co\sec x = - Co\sec x \cdot Cotx

 

Derivative of Hyperbolic Functions:

26) \frac{d}{{dx}}Sinhx = Coshx

27) \frac{d}{{dx}}Coshx = Sinhx

28) \frac{d}{{dx}}Tanhx = Sec{h^2}x

29) \frac{d}{{dx}}Cothx =- Co\sec {h^2}x

30) \frac{d}{{dx}}Sechx =- Sechx \cdot Tanhx

31) \frac{d}{{dx}}Ce\sec hx =- Co\sec hx \cdot Cothx

 

Derivative of Inverse Trigonometric Functions:

32) \frac{d}{{dx}}Si{n^{ - 1}}x = \frac{1}{{\sqrt {1 -  {x^2}} }},{\text{ }} - 1 < x < 1

33) \frac{d}{{dx}}Co{s^{ - 1}}x = \frac{{ - 1}}{{\sqrt  {1 - {x^2}} }},{\text{ }} - 1 < x  < 1

34) \frac{d}{{dx}}Ta{n^{ - 1}}x = \frac{1}{{1 + {x^2}}}

35) \frac{d}{{dx}}Co{t^{ - 1}}x = \frac{{ - 1}}{{1 +  {x^2}}}

36) \frac{d}{{dx}}Se{c^{ - 1}}x = \frac{1}{{x\sqrt  {{x^2} - 1} }},{\text{ }}\left| x \right| > 1

37) \frac{d}{{dx}}Co{\sec ^{ - 1}}x = \frac{{ -  1}}{{x\sqrt {{x^2} - 1} }},{\text{  }}\left| x \right| > 1

 

Derivative of Inverse Hyperbolic Functions:

38) \frac{d}{{dx}}Sin{h^{ - 1}}x = \frac{1}{{\sqrt {1 +  {x^2}} }}

39) \frac{d}{{dx}}Cos{h^{ - 1}}x = \frac{1}{{\sqrt  {{x^2} - 1} }}

40) \frac{d}{{dx}}Tan{h^{ - 1}}x = \frac{1}{{1 -  {x^2}}},{\text{ }}\left| x \right| <  1

41) \frac{d}{{dx}}Cot{h^{ - 1}}x = \frac{1}{{{x^2} -  1}},{\text{ }}\left| x \right| > 1

42) \frac{d}{{dx}}Sec{h^{ - 1}}x = \frac{{ -  1}}{{x\sqrt {1 - {x^2}} }},{\text{ }}0  < x < 1

43) \frac{d}{{dx}}Co\sec {h^{ - 1}}x = \frac{{ -  1}}{{x\sqrt {1 + {x^2}} }},{\text{ }}x  > 0

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