In Product Rule, I showed how I would present the product rule in one of my calculus classes. Today, we will continue this discussion as we look into the quotient rule. I will be referring to the previous post, because I normally give the lecture of the product and quotient rule on the same day. So, if you haven’t read Product Rule yet, I would suggest taking the time to go through that before reading this post.
Finding a guess for the quotient rule
At this point in the lecture I would have just finished my presentation of the product rule, and I would have followed this up with a few examples. Since I will have shown the derivative of 1/x and ex using the limit definition, I will use as a last example of the product rule: find
I usually don’t simplify after using the product rule to take the derivative. I feel that they are learning enough at the time that adding in the extra difficulty of recalling simplification rules distracts them from the topic at hand. In this case, however, I let them know that I want to use the answer for another question, so it is worth the time to simplify. Now, I ask, what if we had just considered ex as the top function and x as the bottom function. Then how would we have arrived at this answer using the original functions and the derivatives thereof?
The reason why I like this example is that the distinctions between ex and x are very clear and you can see what portions are coming from each question. However, there is an added degree of difficulty determining exactly what’s going on because the derivative of ex is the original function ex. Despite this, they can normally come up with
I am using the parenthesis on the prime to denote that they are unsure which one to use. If you follow through the product rule outline, you can get them to arrive at the correct answer. However, I usually leave this an open-ended outline for them to use on the next example.
At this point, I will return to the equations I originally used for the product rule except I will switch f(x) and g(x) to get g(x)=3x2 and f(x)=x5. We then get that
Using these, I would get them to try the options we came up with before to find that,
I would have used the f(x) and g(x) to begin with to get the rule, but, as I did with the product rule, I will change these letters to things that are easier to memorize. That is, T for top and B for bottom functions.
I now take it a step further and make that, if we compare the quotient to the product rule, that we can associate T with F and B with S since we would normally consider the top before the bottom. If we substitute these into the product rule, we would get BT’+TB’. Here the pattern for the quotient rule is the same, but we have to subtract the second part instead of add, and when we are finished we have to divide by the bottom squared. This is the reason I use SF’+FS’ as the product rule, that is, it makes a connection between the patterns for the students so that it is easier for them to memorize which portions needs to be subtracted.
As an additional note, this association between top and first and bottom and second is recalled from the singing of the product rule. That is, we did a low note, a high note, a high note, a low note. Associating T with high and B with low, we get that the lyrical association ties in very well in this context. Yes, I also sing the quotient rule for them. The pattern of notes is then low d high – high d low over low squared.
Here I want to note I am not a very good singer. I did actually take voice lessons while in undergraduate school in order to help with my guitar playing, but I never mastered the voice. I point this out because you can sing for your students, even if you aren’t a great singer, and your students can sing back to you, even if they aren’t confident in their singing. As such, I do make them sing back to me as a class so that I can further make the association in their minds between melody and rules. As I stated in Product Rule, I really do see the difference in the ability of the students to memorize the material using this trick. Therefore, if you still don’t want to sing, feel free to use my recording.
Here I will note that I don’t actually provide the proof of the quotient rule to my students in class. I had already gone through the proof of the product rule with them, and it is extremely difficult to keep their attention through two proofs in the same day. Therefore, I let them know that we came up with the correct formula, and if they’d like to see the proof, they are welcome to come by during office hours and I will walk through it with them.
Now that we have the quotient rule, we will now be able to look at functions with negative exponents. That is, I go through the following with them; find
for any natural number n. I do make sure to point out that we can’t use the power rule, because we only showed the power rule worked for positive exponents. However, we don’t have to go back to the limit definition of derivative if we rewrite this and use a quotient rule. Therefore, we find that
I now take a second to let them look at that. Then I say, let’s let m=-n. Then if we substitute this above what do we get? Well we get that the above gives a proof that the power rule not only works for 0 and positive integers, but it also works for negative integers. They are often surprised that this constitutes a proof, but they are encouraged by the fact. Furthermore, realizing that they will have fewer cases where they have to use the limit definition of derivative is also exciting.
Even though we were able to show that the power rule for derivatives was correct for natural numbers prior to learning the product and quotient rules, we see that these rules allow us to expand upon the product rule and show it works for negative exponents in general. We now have that relatively simple derivative rule has motivated counting, the binomial theorem, the definition of limits, the definition of derivative, the product rule and the quotient rule. That is, because we didn’t just accept that something was true because we were told it was true, we were able to learn and understand a great deal about other topics as well. Note, however, that we still aren’t done. That is, we don’t know what happens if we have rational (or irrational) exponents. Therefore, our journey to understand the power rule isn’t done. We still have more to learn, therefore, we will move on the chain rule next time.
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