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:

$$\frac{f(x+h)-f(x)}{h}$$

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)}⁄_h - ^f(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.