Related Content

Thank you!

On behalf of our dedicated team, we thank you for your continued support. It's fulfilling to see so many people using Voovers to find solutions to their problems. Thanks again and we look forward to continue helping you along your journey!

Nikkolas and Alex
Founders and Owners of Voovers

Difference Quotient

Lesson Contents

What is the Difference Quotient?

The difference quotient is a method for finding the average rate of change of a function over an interval. It calculates an approximated form of a derivative. The difference quotient is given as:

difference quotient

Where f(x) is the function and h is the step size. This calculates the average rate of change of the function f(x) over the interval [x, x + h]. We apply the difference quotient to our function, which creates a new function of the variables x and h.

How to use the Difference Quotient

Let’s step  through an example of using the difference quotient.
Find the difference quotient of the function f(x) = 3x2 + 4. Then, determine the average rate of change on the interval x = [2, 4] and x = [5, 11].

1.) First we plug the function into the difference quotient.
f(x + h)hf(x)h = [3(x + h)2 + 4]h[3x2 + 4]h
= [3(x + h)2 – 3x2]h

2.) [3(x + h)2 – 3x2]h is our new function that we can use for calculating the average rate of change for 3x2 + 4.

3.) Let’s calculate for the interval x = [2, 4] first.
[3(x + h)2 – 3x2]h[3(2 + 2)2 – 3(22)]2 = 18.

4.) Now, let’s do the same for the interval x = [5, 11].
[3(x + h)2 – 3x2]h[3(5 + 6)2 – 3(52)]6 = 48.

When to use the Difference Quotient

If we are given a function and must find the slope at a point, we can make an approximation by using the difference quotient. To approximate the slope, we pick our x limits on either side of the point. Imagine the point is right in the middle of the interval. The closer the interval x limits are to the point, the more accurately the difference quotient will approximate the slope at that point. In other words, a narrower interval = a more accurate approximation.

As shown earlier in the example, we also use the difference quotient to find the average rate of change over a range of x values for a function. Look out for questions that give a function and ask to find the average slope or average rate of change over an interval or range of x values. The difference quotient is especially useful when there are multiple points to perform this with because it saves time compared to using the slope formula.

Scroll to Top