Calculus: Combining Computational Techniques

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Computational Functions
The Chain Rule
Combining Computational Techniques Page [1 of 3]
Well, here we are. We are now examining fancy methods for finding derivatives of more elaborate functions. We've seen a couple of them. We've seen the, the Product Rule, which allows us to take the derivatives of functions that are made up of the product of two functions. And we've now seen the other method, another method, the Quotient Rule, which allows us to take derivatives of objects that are composed of the quotient of two functions. And these are elaborate formulas that we, we get down, but allow us to find derivatives of really, really complicated things.
Well, now, what I'd like to do is tell you about, in my opinion, what I think, is probably one of the fanciest methods of taking derivatives. And this is called the Chain Rule; this is the Chain Rule. But I thought before actually explaining the Chain Rule in, sort of, mathematical language, I thought we should take a moment here to do a little bit of cooking with Professor Burger.
So, let's take a moment here to sort of inspire the Chain Rule to first put on our aprons and, and go into the kitchen. Okay, so what do I want us to think about? I want us to think about the kitchen and food in terms of a calculus motif. Let's think about it as it really is. Taking the derivative is really getting to the very fruit, the very heart of the matter, getting to exactly what you want. Let me illustrate that with, say, a banana. So, if you want to eat a banana, there it is given to you. Imagine that the inside of the banana, the very thing that you want, is the derivative, and how do you get there? Well, you have to peel off, and that requires, you know, either a Product Rule or a Quotient Rule, or maybe just taking the derivative, or using a factor that's a sum or difference of two things and then, you're there. Voila, umm, the beauty and tastiness of the derivative. Great, very sweet, very sweet.
Now, there are other things, other ways of looking at it. There are other techniques we've seen like, like I said, you could use the Quotient Rule for something. Well, again, it's the same idea. You have to peel off, and once you peel off in terms of doing that work required, and you can see this here, once you peel it away, what you've got is the fruit of the derivative. And there it is. You can do a little bit of computation, you use the Quotient Rule, maybe you have the sum of two functions, and then, there you are, and you can eat the orange, eat the derivative. It's great, again, a tasty snack, good for the body and the mind.
But now what I'd like us to think about is something new, something different. What happens after you peel away something, you're left with something that's still not quite the fruit that you seek. Well, then you have to peel away again, and peel away again. Let me illustrate that with an onion. This is a beautiful Bermuda onion. Let me cut off the edges here. Look at that, it's a beautiful onion, very nice, cut off the edges here. Of course, you want to rip off the skin if you can, peel off the skin. Now, how many Calculus lectures have you ever been to where they actually do this right there, you know? How many books have you seen this in? Well, not a lot, probably because it's not a very good idea, but, anyway, we're going to do it. We're peeling off the onion here, by the way, this is, uh, harder to do than you may have thought, peel off that onion. Here, let me see if I can use the, the knife here to help me peel off that layer, there we go, okay, now I'm making progress.
Now, what's the difference between an onion and the other things we've looked at? The other things we've looked at -- once you peel it away, once you peel that away, you hit the fruit. Great, not a problem. Now, with the onion, however, now look at that, look how great, look at the beautiful color, can you see all that? That's just beautiful. You can take this, by the way, and put it in your compost heap. I'm trying to be very cooking show like. If this is a real cooking show, by the way, I'm going to pull out a pie right about now, but okay. Anyway, now suppose you want to take the derivative of this. Suppose you wanted to peel this away. When you peel off this layer then what do you notice underneath? You notice underneath another layer that looks basically just like it.
Now, here's the principle that I want you to think about here, and the principle is that one way to untangle this particular vegetable would be to do the following. In your mind, think about this as the orange and say, "Okay, I know that there's a layer underneath this one, but for now I'm going to just pretend that underneath this is the very thing I seek and I peel it away." So I just peel it away, pretending that the thing that's left over inside there is what I want. And then once I peel that away, you see, I say, "Oh, well, by the way, that's not quite what I wanted." So, now I'm going to peel this away and treat the inside thing as the thing that I want. Do you see how I'm peeling these things away? In fact, the very word peeling these things away really captures the spirit of what I want you to think about. This notion of peeling, and then once you peel, you get down, in fact, this one I noticed, by the way, actually bifurcates, ooh, I wasn't expecting that. And then you can peel some more, and some more, and some more. So, this, wow, by the way, this is a real onion, it's not a prop onion, and, uh, it's a little moving here. But you get, I hope you get the idea, wow, okay. So, let me (laugh), these things are really strong. I don't mean to get so emotional with you, but I want you to keep that idea alive. Let me just move this over here, and let me try to pack up my, uh, kitchen accouterment (laugh). See how, see how I can be in terms of - and my hands, by the way, are very smelly right now, but luckily, there's no "smell surround." Remember that in the movies they'd actually have like "smell surround," you could actually smell things. Maybe that was before your time. Anyway, okay, I'll, I'll clean up here in a second.
Now, so the method I want to now describe to you, this last method in some sense, so we're going to talk about it for a while, is the Chain Rule. And, actually, the ridiculous onion cutting example, I think, really captures the spirit of what I want you to think about in terms of the Chain Rule.
Let me try to inspire the Chain Rule even more so that I just did with a particular example. So, let's take a look at the following question. Find the derivative of this function. F of x equals three x squared plus one squared. And your question is to find the derivative. Well, you can do this. In fact, you know what? Why don't you do this right now? Find the derivative of this using any method that you want -- that you know that you know is correct. Try it right now. See if you can get an answer. Okay.
All right. Well, how'd you go about doing this problem? There are actually a variety of different ways. Let me point out a few of them and let's pick one and then, and then try it. One way would be to take this quantity and square it out. Take this and multiply it by itself and FOIL it out, and then take the derivative of each term. That works great.
Another way would be to think about this as a product. This multiplied by itself and then use the Product Rule. Either way will work and give you the same answer. I'll actually use the Product Rule, just to sort of remind you of the Product Rule and get you in the habit of using it. Again, you might have done it the other way, which would have been great, by the way. So, to use the Product Rule, I'll think about this in terms of thinking purposes, I'll write this out as f of x equals three x squared plus one, multiplied by three x squared plus one. It's the same thing as this, but now I can really see the product here. The f and the g are now the same, but that's okay.
So, the derivative would be the following. F prime of x equals -- and I do the chant. The first, three x squared plus one, multiplied by the derivative of the second, and the derivative of this, two times three is six x to the two minus one is one, plus, and the derivative of a constant is zero, so I'm not going to write that in, and then, plus the second, which is three x squared plus one, times the derivative of the first. Well, that's the same as we just did before, so it's six, again, that's six x plus zero. And, there, we get the answer. Actually, you could simplify this a little teeny bit if you noticed that, in fact, I have this number and I'm adding it to itself. These are the same. So, if I have something and I add it to itself, I have two of them. So, in fact, I could report that the derivative equals two times three x squared plus one times six x. And I did that using the Product Rule. Okay, great.
Well, actually, it's not so great because I have a slight confession to make. And the confession is that when I wrote this problem, I wrote it down wrong. There was a typo in the problem. The original function wasn't supposed to be three x squared plus one quantity squared. The problem was to find the derivative of the function f of x, which equals three x squared plus one quantity to the two hundredth power. Sorry about that, just a little mistake. It was supposed to be two hundredth power, two hundred. Well, that changes things a little bit. In fact, I could ask you to go off and try to find this one now on your own, but I won't. Let's think about this together.
Well, no matter which method you used to find the derivative of the first question I asked, I think that method is going to fall a little bit flat on this actual question. Because notice that if you use the method of just squaring that out and taking the derivative to multiply all this out two hundred times is going to keep you busy, probably for the rest of your college career. So, not a great strategy. Well, what about my very sophisticated Product Rule strategy? Not a great strategy either, because that's a lot of products there. We're going to have to do the Product Rule an awful lot of times. Not extremely practical. So, it turns out that the actual purple function that I was looking at, the purple and green function is actually difficult to take the derivative of.
So, what do we do? What we do is we try to regroup, look at what we can do, and see if there's some sort of fundamental pattern we can detect that can be generalized to do more challenging problems. Now, what we could do, and what we were able to do just moments ago was to find the derivative of this function. Let me cover this one piece up here.
So, let's return to that problem, and let's look at the solution and let's determine if, just looking for a pattern now, there's any way to get to this final answer without ever using the fact that this exponent was large. I mean, I'm sorry, was small. This exponent was small. See, I used the fact that the exponent was small. It was so small I could actually write the terms out that many times and use the Product Rule. It's so small that you might have actually squared that out and then taken the derivative. Can we look at the answer and see a pattern by which we could have gotten to this answer without ever using the fact that that exponent two is a tiny number? If we could actually extract such an observation, then that might lead us to a more general formulation which would allow us to deal with a more threatening two hundred exponent.
Let's go back here and take a look. Well, the answer we got was two times the quantity three x squared plus one times six x. And there's something, I think, that's quite nice here. All I'm in the business here of, folks, is pattern recognition. And I see something that I recognize. I see this quantity right here. That's the same thing as the inside of this. So, this is the inside here, this is the inside, and the outside is all squared, and that inside is right here. That's sort of an interesting coincidence. And then what's this term? Well, actually notice that that is the derivative of just that piece of it. In fact, that's just the derivative of the inside. So, this is actually the derivative of the inside. Well, that's sort of an interesting coincidence, again. And, what about that two? Well, now here's something that I think is interesting and worth thinking about. What if we take that entire inside there and treat it as one thing, and you know what I call it? I call it the big old blop. So, suppose that you just have a big old blop right there, okay. Well, then I'm looking at that thing squared. And what is the derivative of big old blop squared? The answer is two times big old blop - two times big old blop. Okay?
Now, what do I do now? Well that's two times big old blop, and then this seems to be the derivative, the derivative of the blop, right? So the derivative of the blop is right here and this is the blop. So, what, what's going on here? It seems as though one way to think about getting this particular answer is the following. It's to say, well, this is something inside, and then that inside thing is squared. And to find the derivative of that what I can do is the following. I can just treat that entire inside thing as one big old blop. And if I do that, well, then what? Well, then if I take the derivative of the big old blop squared it's two times big old blop. And then I'm thinking multiply that by the derivative of the inside, by the derivative of that blop, which is just six x plus zero. It turns out that that is exactly like my onion discussion. I'm peeling off a layer. I see this outside layer there. Think of that outside layer as the onion, as that layer of onion right there. And I want to peel off that two. See, this is sort of a hard problem, because I have this inside and this thing way out here. So, I want to peel off that two. The way I peel off that two is to treat the entire inside as the fruit that I'm seeking. So, I treat that as a big old blop, and I say, okay, forget about the inside, call it a big old blop. What's the derivative of blop squared? I'm peeling off the onion and I see two blop, and then I say, okay, this layer is now gone so what am I left with? I'm left with the blop. So, what's the derivative of that? I take the derivative of that and I multiply it here.
That basic idea is the fancy method known as the Chain Rule. That's the Chain Rule. So, now let me see if we can examine that Chain Rule, and the more elaborate question, that was the one I was supposed to ask a while back. And let me see if we can do this one now. So, if we're going to take a look at f of x equals three x squared plus one to the two hundred power, I claim that we can take the derivative of this without using this product method. Because what I'm going to do is I'm going to treat this as the onion, and I'm going to peel off the layers. I noticed that this thing in here is the inside of something, and then I have an outside. So I treat this whole thing as one big blop, and I say what's the derivative of blop to the two hundred power. And the answer is, well, you bring down the two hundred, two hundred blop to the two hundred minus one power. And so the derivative is two hundred times the blop raised to the one ninety-nine power. And what was the blop, well, I'll write that in, three x squared plus one.
But, now I have to take into consideration that I peeled off that layer. I'm still left with something. I now have to take the derivative of the blop. So I multiply this by the derivative of the inside. And the derivative of that inside is, well, six x plus zero. And, so, in fact, this is the derivative of the function that, just moments ago, we didn't know how to take the derivative of because this exponent was too large. And the secret is not to treat that exponent as this thing multiplied by itself that many times, but, instead, to look at this in a different point of view. To look at this as an inside object and an outside object, hold the inside as one big thing, and say I've got blop to the two hundred power. The derivative of that is two hundred blop to the one ninety-nine, multiplied by the derivative of the inside, which is six x.
Okay, I'm going to let you try one of these problems right now on your own and then I'm going to come back and do a couple more of these examples so you get a sense of this. The Chain Rule takes a little while to get comfortable with, but once you get comfortable with it the, the, the limits are almost unending. So, stay with us and I'll do some more examples when I come back. Have fun and I'll see you in a bit.