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) $$\frac{d}{{dx}}Co\sec {h^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {1 + {x^2}} }},{\text{ }}x > 0$$
Khushi Shrestha
July 26 @ 4:57 pm
can u tell me about the d^2y/dx^2
lol
August 24 @ 12:32 pm
its a second order derivative.
in simple, the derivative of the derivative. i.e. dy/dx of y= x^3+29 is 3x^2 then d^2y/dx^2 will be 6x.
Vhia Berania
August 17 @ 11:20 am
How to answer: y²= b²/(2x+b) at (0,b)
The b² is over the 2x+b
Suraj Yadav
October 3 @ 8:06 am
Firstly u have take the derivative of given equation w.r.t x
Then find value of [dy/dx=••••••] only which contains some x terms and y terms.
Substitute x and y with given point’s coordinates i.e here ‘0’ as x and ‘b’ as y
Omkar
November 25 @ 9:13 am
Ans of e^2x
Ramsha
July 11 @ 10:09 am
2e^2x