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AP Calculus AB & BC: Help and Review17 chapters | 158 lessons

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Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will explain how to find the difference quotient of functions with fractions. We will look at the steps involved in this process, and we will apply these steps to an example to help solidify our understanding.

The **difference quotient** for a function *f*(*x*) is given by the formula

[*f*(*x* + *h*) - *f*(*x*)] / *h*

This difference quotient calculates the slope of the secant line through the points (*x*, *f*(*x*)) and (*x* + *h*, *f*(*x* + *h*)), and it is heavily used when dealing with derivatives in calculus.

Finding the difference quotient for a function *f*(*x*) is simply a matter of finding *f*(*x*+*h*) and plugging both *f*(*x*) and *f*(*x*+*h*) into the difference quotient formula, then simplifying.

This all may be old news to you; however, what you may not be familiar with is how to find the difference quotient of a function that has fractions. Thankfully, other than the simplification part, the process of finding the difference quotient for functions with fractions really isn't too much different than it is for functions without fractions.

In fact, the first step in finding the difference quotient for a function with fractions is exactly the same as for functions without fractions. We find *f*(*x* + *h*) and plug both *f*(*x*) and *f*(*x* + *h*) into the formula for the difference quotient. Once we do this, things will look a little different and seem a little trickier.

You see, your difference quotient, which is a fraction itself, will have fractions within it, so you have fractions within fractions. Well, that sounds confusing! Don't worry though! There is a nice trick for eliminating these fractions within the difference quotient, making simplification much easier. All we have to do is find the **common denominator** of the fractions within the difference quotient, and then multiply both the numerator and denominator by that common denominator.

This is okay to do, because it is the same as multiplying by 1. Therefore, we're not changing any values or anything, and what's great is that doing this will eliminate the fractions within the difference quotient, so we can simplify as we normally would.

This might sound like a bit much. I don't know about you, but understanding is always easier for me when I've got a nicely stepped out process and I can see that process used on an actual example, so let's summarize these steps into a nice organized list, and then we'll look at an example to help solidify our understanding of how to do each of the steps.

To find the difference quotient for a function, *f*(*x*), that contains fractions, we use the following steps:

- Find
*f*(*x*+*h*) and plug both*f*(*x*) and*f*(*x*+*h*) into the difference quotient, [*f*(*x*+*h*) -*f*(*x*)] /*h*. - Find the common denominator of all the fractions within the difference quotient and multiply the numerator and denominator of the difference quotient by that common denominator. Quick note: when you find the common denominator, keep it in factored form, and do not multiply it out. This will make simplification easier.
- Simplify the result of step 2.

Okay, that's nicely stepped out, but it will become even more clear when we actually look at applying it to an example, so let's go ahead and do just that.

Consider the function *f*(*x*) = 5 / (*x* - 4). This is a function that is a fraction, so let's use this function to illustrate the steps of finding the difference quotient of a function with fractions.

The first step is no different than finding the difference quotient for any function. We find *f*(*x* + *h*) by plugging in *x* + *h* anywhere we see *x* in the function.

We get that *f*(*x*+*h*) = 5 / (*x* + *h* - 4). Now we simply plug *f*(*x*) and *f*(*x* + *h*) into the difference quotient formula.

So far, so good!

Now comes the fun part! We want to eliminate those fractions within the difference quotient, so we find the common denominator of those fractions. The denominators of the fractions within the quotient are *x* + *h* - 4 and *x* - 4. The easiest way to find a common denominator is to just multiply the denominators together to get

- (
*x*+*h*- 4)(*x*- 4)

As our steps indicate, we should leave this in factored form. Multiplying it out will actually make things more confusing when we multiply the numerator and denominator by this expression. Speaking of which, that's the next step in this process, so let's do just that!

Ah-ha! No more fractions, just as we expected!

Now, we just need to simplify this expression.

We're done!

That wasn't so bad, and we see that finding the difference quotient of functions with fractions is really just a matter of the extra step of finding the common denominator of the fractions within and multiplying the numerator and denominator by that common denominator to eliminate fractions. Other than that, it's business as usual!

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AP Calculus AB & BC: Help and Review17 chapters | 158 lessons

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