Calculus I: The Quotient Rule

Have you adequately studied the previous lesson about the Product Rule? It’s an important one, so make sure you do!

What’s the quotient rule? What does that even mean? Is it difficult to use? All your questions will be answered! Remember the product rule, and what it was used for? The quotient rule is very similar, but has a different formula. The quotient rule can be used when you have a function of x, divided by another function of x. Basically, that means a fraction. Here are a couple of examples:

See? If you have a fraction with x’s in both the numerator and denominator, this is the time for the quotient rule. Keep an eye out for these types.

What is the quotient rule?

(Like I showed you above, this just means if you have x’s on the top and bottom)

The quotient rule says:

“y prime equals the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.” This is a mouthful. You really should say this out loud over and over again, until you fully understand it. You should do your best to memorize that sentence, and the formula.

Yes, you should do your best to memorize that long sentence. I know that it is kind of boring and hard to remember, but it’s very important. However, this site wouldn’t be called CalculusTricks.com if I didn’t have a trick for you. I took calculus I in high school, and then I took it again in college as a refresher. The second time I took it, my teacher gave us an awesome little trick that helped us out a lot. Please bear with me, as it’s not going to seem like that big of a deal at first, but it is extemely helpful!

Say the following: “Low Dee High minus High Dee Low, all over Low squared.”

You have to say it with some rhythm. The “Dee” means derivative of. Like we say “dee-why-dee-exx” for . “Low” and “High” refer to the denominator and the numerator respectively. This is a much shorter sentence to remember, and feels much easier to say. This is something you would say to yourself, in your head, while you are taking a test. If you repeat this over and over, you won’t mess up the formula when you need to use it! This is the one you NEED to memorize, if you want to score well on your tests.

So how do we use it?

Let’s start with a very simple example.

All that I did here was follow the formula from above. You have to recognize what part is the denominator, and which is the numerator (this is very self explanatory). Then, say the trick out loud a couple of times as you take the derivative. “Low Dee High minus High Dee Low, all over Low squared.” I didn’t leave a single step out when I was simplifying, so that you could see everything I did very clearly. Do you see how everything I did matched the forumla exactly? I’ll show you a couple more examples, in case you are still confused.

Something a little bit harder, perhaps? Remember, apply the forumla exactly as we did above.

Simplify the most that you can, but don’t worry if it still looks a bit messy. Some of them just come out that way. I used the quotient rule the same way that I did in the first example. This time, the fraction was just a bit more difficult. We can handle this, right?

Okay, I think it might be time to step it up a notch. For my final example, I’m going to show you something you need to be on the lookout for. What if you need to use the product rule inside of the quotient rule?!? Did you even know that that could be possible? Other text books will try to trick you with a problem like this, but I’ll be straight forward with you and warn you about these. In the example below, I’ll do the most difficult problem you can get with the quotient rule. Ready?

You need the product rule for the numerator!!! This is going to be really messy. All I ask is to please try to follow me. This is as hard as it gets. Remember and apply the formula!


For this problem, I did it on the fly (I did the product rule directly inside the quotient rule as I was going through the problem. I did not stop and do the product rule off on the side. You can do that if you’d like to, though). This is exactly how I would have solved this problem, if it were assigned to me. If you wanted to, you could have broken it down into smaller parts to make it easier for yourself. You could say f(x) equals what? g(x) equals what? And the same thing for f ‘(x) and g ‘(x). Then, you can insert those answers directly into the formula. I did a similar process on the last example of my Product Rule post. You should try to factor as best as you can, in order to simplify the problem. Occasionally, you will get an answer where something in the numerator can cancel out something in the denominator. That is rare, but most text books have a few problems set up specifically to do that.

I hope this lesson has been helpful to you. Try some problems, use my trick, and practice some more. If you then have any specific questions, be sure to email me at admin@calculustricks.com. I check my throughout the day, and I’ll respond at my earliest convenience.

Good luck!