Power Rule

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)
   =  2x1
   =  2x

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 ddx

Example: What is d/dx(1/x) ?

1/x is also x-1

Using the Power Rule with n = −1:

d/dxxn = nxn−1

d/dxx−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?

ff’(xn) = nx(n-1)f’
x1x(1-1) = x01
x22x(2-1) = 2x12x
x33x(3-1) = 3x23x2
x44x(4-1) = 4x34x3
etc...  
   
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
etc...