Power Rule

POWER RULE

The power functions are the functions of the form

f(x) = x^c
where c is a fixed nonzero real number, called the exponent. You can consult the Field Guide for basic facts related to power functions. Our purpose here is to present one of the most frequently used differentiation rules of all. We have already used it many times.

Power Rule

x^c

cx^c-1

The power rule can be applied for any nonzero exponent c. Here are some specific applications.

Applications of the Power Rule

c=2

(x²)

c=3

(x³)

3x²

c=796

(x⁷⁹⁶)

796x⁷⁹⁵

(x)

x^-1

c=1/2

(x^1/2)

x^-1/2

c=-7/8

(x^-7/8)

-7

x^-15/8

The list is endless. It is worth thinking about why the power rule is true in general. There are several steps in verifying the power rule.

Our final task in this stage is to consider the use of our differentiation techniques as they apply to two special classes of functions: the algebraic and the trigonometric functions.

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