The Power Rule, one of the most commonly used rules in Calculus, says:
The derivative of xn is nx(n-1)
Example: What is the derivative of x2 ?
For x2 we use the Power Rule with n=2:
|The derivative of x2||=||2x(2-1)|
Answer: the derivative of x2 is 2x
"The derivative of" can be shown with the little mark ’
So we get this definition:
f’(xn) = nx(n-1)
Example: What is the derivative of x3 ?
f’(x3) = 3x3−1 = 3x2
"The derivative of" can also be written d dx
Example: What is (1/x) ?
1/x is also x-1
Using the Power Rule with n = −1:
xn = nxn−1
x−1 = −1x−1−1 = −x−2
How to Remember
"multiply by power
then reduce power by 1"
A Short Table
Here is the Power Rule with some sample values. See the pattern?
|f||f’(xn) = nx(n-1)||f’|
|x||1x(1-1) = x0||1|
|x2||2x(2-1) = 2x1||2x|
|x3||3x(3-1) = 3x2||3x2|
|x4||4x(4-1) = 4x3||4x3|
|And for negative exponents:|
|x-1||-1x(-1-1) = -x-2||-x-2|
|x-2||-2x(-2-1) = -2x-3||-2x-3|
|x-3||-3x(-3-1) = -3x-4||-3x-4|