Power means exponent, such as the 2 in x2
The Power Rule, one of the most commonly used derivative rules, 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 this little "dash" mark: ’
Using that mark we can write the Power Rule like this:
f’(xn) = nx(n−1)
Example: What is the derivative of x3 ?
f’(x3) = 3x3−1 = 3x2
"The derivative of" can also be shown by d dx
Example: What is d dx (1/x) ?
1/x is also x−1
Using the Power Rule with n = −1:
d dx xn = nxn−1
d dx 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|