# Basic Formulas of Derivatives

General Derivative Formulas:

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

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

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

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

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

10) $\frac{d}{{dx}}[\frac{{f(x)}}{{g(x)}}] = \frac{{g(x)\frac{d}{{dx}}f(x) - f(x)\frac{d}{{dx}}g(x)}}{{{{[g(x)]}^2}}}$ is called the 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) $\frac{d}{{dx}}Co\sec {h^{ - 1}}x = \frac{{ - 1}}{{x\sqrt {1 + {x^2}} }},{\text{ }}x > 0$