In Power Rule, we saw that the power rule for derivatives worked for all natural numbers. Our goal is now to expand on this to show that the power rule also works for negative integers as well. In order to arrive at this, we will need the quotient rule. We will therefore look at the product rule today, so that we can do the quotient rule and expand upon the power rule next time.
Depending on my schedule for calculus I may be able to cover both the product and quotient rule in one lecture. In particular, if I have a class meeting of 75 minutes, I will plan the schedule so that I can cover both of these topics during this lecture. On the other hand, if I only have a 50-minute lecture, I will have to break these up into two lectures. I much prefer to teach these in the same day because it allows me to draw parallels between them, which in turn helps give students a better chance of understanding both topics. While I only present the product rule today, I will try to make these connect in the next post.
Finding a guess for the product rule
In order to introduce the product rule to students, I will always begin with an example that we can find without it. My normal example is, let f(x)=3x2 and g(x)=x5. I will then ask them to find f'(x), g'(x), f'(x)*g'(x) and (f(x)*g(x))’. We will get the following for these
The first thing I note to my students is that the derivative of f times g is not the derivative of f times the derivative of g. I try to make this as clear as possible, because inevitably there is someone that tries to distribute the derivative across a product. In fact, I point out, that there are differences between these two in both the power of x and the coefficient in front of the term.
Now that we know that we cannot distribute the derivative across a product, I ask them if there was some other way they see to combine the terms to get out the correct answer. Most of the time they stare at me blankly, so I let them do so for a short time, hoping that they are at least trying. After this awkward silence, I try to lead them to the answer by saying, “Can we multiply any of the terms to get out an x6 so that the power is at least correct?”
By allowing them the time to think after the first question, there are usually a few that have seen that if we multiply one of the original functions by the others derivative we will get out an x6. Note that, if you choose different functions you want to be careful of the case where one of the terms squared will give you what you want. While f(x) cubed or f'(x) to the sixth would work, these are less commonly noticed than a square. If they give an incorrect answer, you can work with it and try another example to see if it still works.
After I am able to get my students to make the connection (or I’ve waited an adequate time that I feel I should give it to them), I point out that we have gotten closer because we have the correct power, but not the correct coefficient. Through some prodding, I then get them to try the other mixture of function and derivative. Once this is done, they usually notice quickly that the two need to be added together.
Hence we arrive at the conclusion that
At this point, I would ask them if they believe this will work in general. Some will answer yes, some may answer no, most won’t answer. However, I will follow up with, “in order to check to see if its correct, we can try another example. If it works in this case, we would hope that it works in general. However, we won’t know this works in general unless we can provide a proof. Therefore, what would a proof look like?” Yes, most students tend to grumble at the idea that I am not going to just give them the answer and let them be on their way, but it is important that they notice that, while we have an educated guess, we do not know the answer is correct. Therefore, we must study the problem more.
Proof of product rule
If the students were unable to come up with the formula on their own, or if they seem unsure that it will work in general, I will sometimes do another quick example. After I have convinced most of them that the equation is correct, I will remind them that we must use the limit definition of derivative to prove that our answer is correct. Therefore, I write out
Then I sit and stare. I turn to them and say, “Can we simplify this?” They look back and usually say nothing, so I ask them, “What if it was your job to find an answer to this, that is you had to figure it out or you would get fired, what would you do?” This seems to get them a bit more into it as they start to consider such a situation. At this point, I state that, “As a mathematician, it is my job to determine things like this. Therefore, if I was posed with such a problem, and I didn’t know how to proceed, I would sit and stare at the problem and think until something came to my mind. I would try this, note that my guess was wrong and repeat the process until I finally got something to work after months of trying.” At this point, I’ve gotten more of them excited because they are left with a picture either of me working diligently on mathematics or of me getting fired.
Now I go on, “After getting it incorrect over and over, when you finally manage to get the correct answer out, the joy of doing so is just overwhelming, and that is, in essence, what mathematics research is. Getting things wrong over and over, until you finally get it right and the satisfaction at the end is worth all of the frustration.” I really think it helps them to see that they aren’t the only ones that struggle with math. The struggle is part of the process. If they don’t get the homework questions right away, that is normal. They need to keep trying because that is what it takes. I really try to use this lecture to remind them that its alright to make mistakes, you just need to keep working.
Now that I’ve finished trying to motivate to work harder on the homework and not get so easily frustrated, I move on to let them know that yes there is a way to ‘complify’ the problem. I took this word from one of my professor’s in grad school, Dr. Wim Ruitenberg. While not an actual word, it helps the students see that eventually we will be able to simplify the problem, if we first make it more complicated. To do this, I add 0. Therefore, we get,
I really try to make sure that they understand why I can do each step, even if they don’t understand why I’m doing each of these things. I tell them, I’m doing this because it works, but I just need you to see that I can do it.
At the end, however, I stand back and let them see what we’ve arrived at. Since h is going to 0, and f and g are differentiable, and therefore continuous, we have g(x)f'(x)+f(x)g'(x). Then I stand back, as let them know how amazing it was that this worked out exactly as we thought it would. That the formulation of the proof was indeed beautiful. While they may not see it that way, they do tend to feed off my enthusiasm and I feel this lecture really helps to get the students to see math in a new way. That is, it’s not just numbers and equations, but it’s a creative application of rules to get out something exciting.
In mathematics, I try hard to make my students memorize as little as possible. Instead, I hope that they can understand the rules well enough that they can determine the answer without having to refer back to a formula. With the product rule, however, there is just too much going on to allow them the time to re-derive the formula every time they need it. Therefore, this is one place I tell them to memorize things. I do have some things I have found really help in this case. In particular, instead of leaving the product rule as
I rewrite this as
That is, the derivative of a first function multiplied by a second is second d first plus first d second. In fact, I even sing the product rule to them. Note that I will use low notes for second and high notes for first. This allows for the association of the functions to musical tones. The students can then memorize the product rule by memorizing the melody. Furthermore, I make them sing this back to me, and I continue to sing the product rule throughout the semester every time we need it. I find this has helped many students memorize this rule much faster than if I leave it up to them to keep looking it over while doing their homework.
I should point out here that not all books like using F and S for first and second, and it is rare to see the product rule written in this order. I do this because I believe it makes both this rule and the quotient rule easier for the student to memorize. We will indeed see next time that this alignment will correspond with the quotient rule in such a way that it will facilitate memorizing of the quotient rule for the students.
If you teach calculus, I hope this gave you some ideas on how to present the product. Feel free to pick and choose from the ideas as you like. If you don’t teach calculus, I hope that the outline of presenting an example, making students find a solution, then providing a proof and following up with memorization techniques and examples will help you to plan out a lecture that helps your students learn whatever you are teaching them. While you may not want to sing in class, I will state that it drastically helps to increase the attention level.
Please let me know if you have any questions, if you used any of these techniques in class, or if you have suggestion on ways I could improve this lecture. I always enjoy the opportunity to talk about or improve my teaching process. If you liked the post let me know by liking it below. We’ll continue this discussion by looking at the quotient rule next time.