Calculus

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# Difference Quotient

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## 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:

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) = 3x^{2} + 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 3x^{2} + 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.