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