Power Rule

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)
    =   2x1
    =   2x

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

power rule x^3 -> 3x^2
"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