Power Rule

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⁾

For x² we use the Power Rule with n=2:

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:

(xⁿ)' = nx⁽ⁿ⁻¹⁾

(x³)' = 3x³⁻¹ = 3x²

"The derivative of" can also be shown by ddx

1/x is also x⁻¹

Using the Power Rule with n = −1:

ddxxⁿ = nxⁿ⁻¹

ddxx⁻¹ = −1x⁻¹⁻¹ = −x⁻²

First, we write √x as x^½

Using the Power Rule with n = ½:

ddxx^½ = 12x^(½−1) = 12x^−½

How to Remember

Power rule diagram showing x cubed becoming 3 times x squared
"multiply by power
then reduce power by 1"

Here's the Power Rule with some sample values. See the pattern?

6800, 6801, 6802, 6803, 6804, 6805, 13384, 13385, 13386, 13387