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Calculus: Undoing the Chain Rule


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:30
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 92 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Basics of Integration (14 lessons, $23.76)
Calculus: Integration by Substitution (2 lessons, $3.96)

In this video, you will learn to do integration by substitution, which is a fancy way of saying that we'll learn to solve integration problems by undoing the chain rule, which is a fundamental technique we should know from differentiation. Since integration (antidifferentiation) and differentiation are inverse operations, we see many of the same patterns in integrals that we do with derivatives. Professor Burger will show you how to recognize integral problems that you may be able to solve by untangling the chain rule.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Recent Reviews

Professor Burger, Calculus
~ Benjamin9

If you don't get it the first time, or the second or the third or the fourth - fear not. Professor Burger is always interesting and fun to listen to. And eventually you'll get it!

Professor Burger, Calculus
~ Benjamin9

If you don't get it the first time, or the second or the third or the fourth - fear not. Professor Burger is always interesting and fun to listen to. And eventually you'll get it!

The Basics of Integration
Undoing the Chain Rule Page [1 of 1]
Okay, so we just saw a number of examples where we use the notion of anti-derivative to look at velocity of acceleration, positions, the relations to each other, and in particular, with the respect to vehicle motion where there the acceleration is always constant. I thought before we move forward to our next adventure, I’d actually allow you to, sort of, try a little warm up here, go through a warm up, before you get going. So in fact, I have three warm up challenges for you, I want to give them to you right now, I want you to try them. The first warm up challenge is define dy/dx if y= (5x3 + 2x) 4. Notice this is not an integral problem, this is not an anti-differentiation problem; this is a derivative problem. It’s an old kind of problem, but I want you to warm up with it. The next small question I want you to try is find dy/dx if y = the natural log of the quantity sin of the quantity 3x2, okay, that’s the next warm up, and the last one, this is a challenge problem now. Here’s the challenge problem. Find this integral, so this is the integral of look at the thing, it’s really long, 4 times the quantity (15x2 + 2(5x3 + 2x)3. Now remember, let me just remind you what the integral or anti-differentiation means. If I write down the integral, don’t start doing these problems yet, by the way, just watch me. The integral of f(x)dx, what does that integral or anti-derivative means? It means, this is a function, I’ll call it f(x), where if you take the derivative of f(x), you get the original function back. And also, you may remember, I hope, that we have to add a constant here because the derivative constant is zero. And so basically, the anti-derivative of f(x) is F, and capital F is defined to be anti-derivative in the sense that if you take the derivative of it, you get back the original function. This one will seem really hard, don’t do it until you do the first two. Try these problems right now, warm up to them, we’re going to come back and talk about all of them just in a second. So, go for a walk, stretch out, and I’ll see you in a bit.
Okay, well how did you make out on these? Let me see if I can do them. It’s been awhile since I’ve taken derivatives here because we’ve been talking about these integrals, so let’s give it a shot together and see if we can make some progress.
So, the first one we take a look at is to find dy/dx and what I notice is, is that this is a chain rule problem. To where I’ve got you see, blop, all the way up to the fourth. So to take the derivative of that, I’ve got to use the chain rule so, dy/dx equals, so I see blopped to the fourth. The derivative of that is 4 blop, cubed. Now remember, the only thing I write in here is the blop which is 5x3 + 2x, I put here 5x3 + 2x. You have to multiply that, I peeled off, I just peeled of the four, you have to multiply that by the derivative of the blop, which in this case is 15x2 + 2 because that’s the derivative of this. Okay, great.
The second question, is to find the derivative of that function there, natural log of sin of 3x2, again that is a nestedness here. I’ll have to use the chain rule, so dy/dx equals... Well, I see natural log of blop. The derivative of the natural log of blop is one over the blop, so I have one over the blop, and what’s the blop? Well, the blop is sin of 3x2. Sin of 3x2, and then I peeled off, I just peeled off the natural log, now I take the derivative of sin, so, what’s the derivative of the sin and stuff, well, that’s another blop, there that would be the sin of blop. So multiply this by the derivative of sin blop which is cos of blop and what’s the blop? Let’s see, 3x2, I just peeled off now the sin, and I finally take the derivative of 3x2 and multiply all this by 6x. So, there’s a lot of chain rule usage there, and you were actually able to figure out this derivative.
Okay, now the last question. The last question was this really, really complicated integral, and now I think the thing I want to tell you about this very complicated integral, the thing you will always have to remember when you see things that look really complicated like this, is to think Elur Niach. Think Elur Niach? What does that mean? Well, Elur Niach is what? It that right now, why don’t we take a little pop quiz, I’m going to give you four options right there what you think Elur Niach means? Who that person is? Give it a shot right now just for fun. Elur Niach is the chain rule backwards.
So, in fact, a problem like this, we might be inspired to notice the following: the very first question I asked you is to take the derivative of that function, and what do you notice about that derivative? That derivative is identical to the same that I’m supposed to integrate here. This is the derivative of this function and remember what integral means. Integral means find the function such that if you take the derivative of it, you get what’s in here. Well, this is the answer of this. So, in fact, without doing any work, since I gave you a good warm up problem, now you appreciate the warm up. This problem was just a warm up and for a dead rat to pull you off the fence of what I really wanted to get at here, but here we see that on this problem, the answer to this must be this plus the constant and why? Because the derivative of it is the function of here. So the answer to this question we are now able to note, this equals exactly what we discovered, namely (5x3 + 2x)4 + C. And if you don’t believe me, just take the derivative of this and see if you get that. And of course, that was the first warm up problem, so we already did that.
Well, great, so what’s the point of all this? Well, the point of all this is that now, if we’re careful, we may be able to find the integral of even more complicated functions than we’ve looked at so far, and the idea is to realize that maybe some problems are just a chain rule problem in reverse. A chain rule problem in reverse. In fact, if you look at this problem, you can begin to see that a little bit and let me show you why you might be able to notice that. Remember how the chain rule goes, I’ve got a blop, so if I take the derivative of the outside with the blop in there, and then I multiply it by the derivative of the blop. Now, look down here and notice something? If I take a look at just this piece right here, it seems like sort of an inside of something. Notice that the derivative of that thing, which is what? 15x2 = 2, that derivative is actually sitting right here. You see that? So, in fact, here I see a very complicated function, but it has the interesting feature that there’s some little inside and the derivative of that inside is sitting out multiplying everything else. So, this actually looks like some derivative that was taking using the chain rule. I’ve got a set of blop part here and a derivative of the blop. Well, when we see things like this, we can use a technique to actually untangle or reverse the chain rule and that technique is called substitution, and the idea here is to take this whole blop and just call it that. Call it blop, and then reduce this problem hopefully to an easier problem that we can figure out.
So, what I’m going to do up next is work through sort of the details of how to actually go about going from this to this without ever starting with a warm up problem which doesn’t actually give us the answer. So, that’s an inspiration, how do we take integrals of very complicated things where we notice that an inside is there and it’s derivative is sitting right in front. We’ll see how to deal with these things, and see how to deal with the blop up next. I’ll meet you there, bye.

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