Power Functions

A power function is any function of the form f(x) = x^a, where a is any real number. 

Sample Problem

The following are all power functions:

Sample Problem

The following are all power functions, written deceptively.The function

is a power function since it can be written as

f(x) = x^1/2

or

f(x) = x^.5.

The function

is also a power function, since this can be written as

g(x) = x^–6.

Sample Problem

The function 

f(x) = x^x

is not a power function, because the exponent is a variable instead of a constant.

We usually assume the exponent a isn't 0, because if a is 0 we find a power function. The function

f(x) = x⁰ = 1

is a constant function, and we already know how to deal with those.

Now that we've got an idea of what a power function is we can talk about their derivatives. Luckily, there's a handy rule we can use to find the derivative of any power function that we want.

Power Rule

Given a power function, f(x) = x^a, the power rule tells us that

f '(x) = ax^{a – 1}

To find the derivative, just take the power, put in front and then subtract 1 from the power.

Using this rule, we can quickly find the derivative of any power function. The derivative of x² is 2x, x^1.5 is 1.5x^0.5, and x^π is πx^{π – 1}.

No matter the power function, we can find its derivative.