In continuation of our exploration into the power rule, we will need to learn the chain rule. That is, we will need to be able to the determine the derivative of a function if it is the composition of functions we know the derivative. After explaining how to do this and why the method works, we will then proceed to show that the power rule holds for rational exponents in addition to the integer exponents we have already shown. With a little extra work we will also look at irrational exponents, and, after all this time, we will finally have shown that the power rule will work for any real number exponent.
We will be looking at the situation where we have a composition of functions f(g(x)) and we want to the find the derivative. As was the case with our previous rules, I like to lead my students to the answer by first looking at an example. In order to be consistent, I will again use f(x)=3x2 and g(x)=x5 as I did for both the Product Rule and the Quotient Rule.
As I did in both of those cases, I will relabel these using letters that are easier to keep track of. In the case of the chain rule, we have an outer and an inner function, so I let the outer function be o(i) and the inner function be i(x). I do want to point out that I did not originally like these choices, as o looks too much like 0 and i should be an index or √(-1). Therefore, uses this in the context of function just feels wrong to me and I was worried it would further confuse my students. However, I have noticed that as I’ve taught this way, I was generally the only one that had issue with the choices. The students haven’t yet learned that these are poor choices, so using outside and inside functions really does seem to help them keep track of what is going on. Therefore, I persist with this choice.
Once I’ve explained my choices of function names to my students, and why the book doesn’t use the same notation, I will proceed to find,
Now that we have all of these on the board, I ask them how we may combine the functions and their derivatives in order to get out the correct answer. I then give them a short time to think about and proceed with. “Well, it hasn’t worked so far, but can we just multiply the derivatives?” No one expects this to work, but then we have,
At which point I ask, “Is this what we were looking for?” I normally get a not quite, but some people have already seen that the coefficient is correct, the only problem is that there is an i in the second answer we gave, but not in the derivative. It may take a little prodding, but they will notice that they can plug in what i is in terms and x and get,
Now, the ones that hadn’t caught on yet tend to look in disbelief since we did in fact get the correct answer by just multiplying the derivatives. In order to reinforce this, I point if that if we write this in Leibniz’s notation, we would have
Now, I tell them, “While there is more going on here, and we aren’t able to just multiplying two fractions and cancel things out, in a hand wavy kind of way, this is essentially what happens.” I know that’s not very precise, but seeing this written down and making the association with multiplication does indeed help them to memorize the chain rule, because it at least gives them an informal method of seeing what is going on.
At this point I normally move on to examples to help them understand the implementation of the chain rule. While I have gone through a fomral limit definition proof of the chain rule with the students in the past, I don’t really feel they get much more out of it than they do from the above explanation. Whether or not to do a proof really seems to be a balancing act in Calculus. While just giving the students equations reinforces the idea that math is just a toolbox full of equations they just need to memorize, being too rigorous with proofs leads many students to tune out, and they don’t really come back to paying attention after the proof. Therefore, in this case, I feel a hand wavy reasoning of the answer is a good middle ground for these students.
Once we have gone through some specific examples, I will then move on to the following example. Find
At this point, I will have done the square root using the limit definition; however, we will not have done anything with this using any rules. In particular, the above is not a product, quotient, or composition of any functions we have rules for. Therefore, I try to get them to think that will need to go through a limit definition of derivative to find this. They seem really relieved when I tell them that we don’t have to if we get a bit creative. So I tell them, remember that if n is even, then the domain is only non-negative numbers, but if the n is odd we have the domain is all real numbers. If we limit ourselves to the corresponding domains and ranges, then we notice that
The discussion regarding domain and range is important here, because I will be using this same technique in the next lecture to find the derivatives of inverses. Since I want to be able to recall this example, I need to them to see that while x2 and √x aren’t inverses since x2 is not invertible, they are inverses over the restricted domains and ranges. Once I point this out, I then point out that since both sides of the equation are equal, then they form the same function. Therefore, if we take the derivative of a function in different ways, we would still need to get the same answer.
Therefore, in order to find
we let o(i)=in and i(x)=n√x. We then get that
Now, since we know that
Combining these two together, we then get that,
Now, I remind them that the nth root can be written as x1/n. Using this, and exponent properties, we can then simplify. After doing so, I use the same trick I used last time of letting m=1/n, and we finally realize that, if m is of the form 1/n, then
We have therefore shown that the power rules also works for roots. However, what about rational exponents in general? Well, I am generally able to get them through the previous proof because I threaten that having to use the limit definition of derivative. Therefore, I don’t push my luck, but I point out that
therefore we can again use the chain rule to show that the power rule works for all rational exponents.
Now, in order to finish my thought, I tell remind them that I had defined irrational exponents for them as the limit of rational exponents, therefore, the power rule will work for each of the functions in the limit, so it will also work for the limiting function. Yes, this is rather vague, but I feel it is at the appropriate level for a calculus class. I do, however, let them know if they’d like to see all the details behind these last two areas, they are welcome to come by during office hours and I will be happy to go through the details with them. I have yet to have anyone take me up on the offer.
We have finally shown the power rule for all real number exponents. In this theorem we have the seemingly simple statement that, for all real numbers m,
However, as we have seen through 9 posts now, (see Power Rule in the Archive for all of these posts) that there is much more going on with this problem than initially thought. Because of this, if you take the time to go through all of these related topics with your students, you can provide a motivation for exploration a plethora of topics. On the other hand, if you just give your students the formula and tell you to trust you, you end up shutting down their curiosity and you miss out on a great opportunity.
The next time you will be presenting something new to your class, perhaps think about all of the background information that is needed to make the statement you are telling them. I know that is it often difficult to get to everything you need during a class, but, try to think about how you can work all of these topics into existing lectures so that the students can start to make the same connections that you do. I hope this series helps you to come up with your own idea of how to keep your students curious about why things are true.
I do want to point out that after my lecture on the chain rule, I go back to working on a Calculus Labs with the students, because I feel we’ve spent a lot of time on theory and they need a break with some applications. In particular, I look at the Skydiving and Derivatives lab. When using this lab, I tell them the story of when I went skydiving and ask if any of them have done so as well. Even talking about jumping out a plane seems to get everyone excited and they do well working together. I even get them to work with some parameters. If you’d like to use a copy of the lab in your class, you can get a preview of the lab on the Calculus Labs page.