A power function is any function of the form f(x) = xa, 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) = x1/2
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) = xx
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) = xa, the power rule tells us that
f '(x) = axa – 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, x1.5 is 1.5x0.5, and xπ is πxπ – 1.
No matter the power function, we can find its derivative.
Example 1
There's a skill needed for integrals (we'll explain integrals later) that we'll consider now: thinking backwards. Instead of taking a function and figuring out its derivative, think about looking at a derivative and figuring out what sort of function it came from. Try this out when looking over solutions to derivatives. If f ' (x) = 5x4, what could the original function f(x) be? |
Exercise 1
Find the derivative of the function f(x) = x2, using the limit definition of the derivative.
Exercise 2
Find the derivative of the function f(x) = x3, using the limit definition of the derivative.
Exercise 3
Find the derivative of the function f(x) = x4, using the limit definition of the derivative.
Exercise 4
Now it's time for pattern-finding. We know the following functions and their derivatives:
f(x) = x2 f ' (x) = 2x
f(x) = x3 f ' (x) = 3x2
f(x) = x4 f ' (x) = 4x3
What's the pattern?
Exercise 5
Find the derivative of the power function f(x) = x10.
Exercise 6
Find the derivative of the power function f(x) = x85.
Exercise 7
Find the derivative of the power function k(x) = x3.5.
Exercise 8
Find the derivative of the power function k(x) = x–6.
Exercise 9
Find the derivative of the power function
.
,
.Exercise 10
Find the derivative of the power function h(x) = x(π + e).
Exercise 11
Find the derivative of the power function
.
Exercise 12
Find the derivative of the power function
.
is an ugly constant, but it is a constant nonetheless.
Exercise 13
What is the derivative of the power function k(x) = x0.
Exercise 14
Find the derivative of the power function
.
Exercise 15
For the derivative f ' (x) = 3x2, find a possible original function.
Exercise 16
For the derivative f ' (x) = 8x7, find a possible original function.
Exercise 17
For the derivative f'(x) = –3x–4, find a possible original function.
Exercise 18
For the derivative g'(x) = –9x–10, find a possible original function.
Exercise 19
For the derivative
,
find a possible original function, h(x).
