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Calculus: The Derivative


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:15
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 121 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: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Intro to Derivatives (10 lessons, $17.82)
Calculus: Understanding the Derivative (4 lessons, $6.93)

This lesson is a review of how we arrive at the derivative. We start with secants and tangents and then move to average and instantaneous rates before we conclude with the definition of the derivative. We know that tangent lines are a graphical representation of instantaneous rates of change. To find the slope of the tangent line, we take the limit as the change in the independent variable (delta X) approaches zero. The derivative is the function that gives you the instantaneous rate and slope of the tangent line at a particular point. The derivative gets its name because it is derived from another function. The derivative of a function at x is equal to the limit as delta x approaches zero of [f(x+ delta x) - f(x)]/delta x (provided the limit exists).

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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An Introduction to Derivatives
Understanding the Derivative
The Derivative Page [1 of 2]

Okay, you have made it! You have made it to the summit. And now it's time to just nail down all the stuff that we've done. What I'd like to do is actually show you how to find instantaneous rates of change of any old function, not just the function that tells how far I’ve travel on a bicycle. It turns out that there's no more work to be done. We did all the work with that previous example. All we have to do is just write down exactly what we did. So basically, I'm just going to go through and do the exact same thing that I did but instead of with a particular function, I'm going to do a general function.

So first, let me draw you the picture that we had. The picture that we had was the following, so of course, I have my axes, and now instead of using time, t and p, I'm just going to call it f of x. I'm just going to call it a function and if x happens to be time, great, and if it's not time, you know what, great! So now I've got some whacko function here, I'm going to draw a graph of it for you here. It's some whacko function, maybe it looks like this, and that's the whacko function f of x, so that's f of x. And my mission in life, right now, is to figure out at a particular point, let me call that point, let's say, call this point, x not, just x maybe. What is the instantaneous rate of change of that function at that point? So, if you march up to here, there's the point, you can march this way, and you can actually know the value there, that would be the f of x, plug in x.

How do I find the instantaneous rate? Well, let me just recap all the ideas that we've developed. Again, nothing new, no new ideas, same old idea. If I want to find sort of an approximation to the instantaneous rate, what do I do? I pick a point nearby, I go up to there and find the average rate between those two points, and the average rate, I remind you, would just be the slope of that line. The slope of the secant line, average rate equals the slope of the secant line. Again, nothing new here, but now with a more exotic, more general function.

All right, well now, what do I want to do? I want to turn that average rate into an instantaneous rate. So what do I do? Well, I want to take that point, which in some sense, in the eyes of instantaneous rates are actually light years away from this point, and I want to push that in. And as I push that in, what happens to this line? Well, the slope starts to change, watch that as I move that point in here, watch what happens. The line sort of goes up a little bit, you see it? And then starts to come down a little bit, and it heads toward what? It heads toward the line that just grazes the curve, at that point, just grazes the curve, don't believe me? Take a look for yourself – just grazes the curve. Well I guess it doesn't quite graze the curve, but all right, there we go; now it just grazes the curve. It's the tangent line, it's the tangent line, and remember that the instantaneous rate is in fact the slope of the tangent line. So that's what we're after.

How do we get that? We take the average rate and then take the limit as that point drifts into x. So I'm just going to do the exact same thing that we did before. The exact same thing that we did before let me remind you how that goes. I take the average rate by taking x and I displace it slightly, watch me displace it right now. There I go, I displaced it by a little bit. What should I call that little bit? I'll call that delta x. I'll call that a small change in x. So I took x and I added a little displacement. Watch, I'll do it again. Okay, that displacement is delta x. Now if I come up to here, I can mark that point and I can actually tell you the value of the function there. Now I admit it's not going to be a very pretty looking thing, but what is it? It's just, what f is, evaluated at that point. So I plug in that big “garbagy” thing into f and out spits that value. So this is actually, I'll write it on the side here, because it's so big, f of x plus delta x. That's the value there.

Okay, and what's the slope of that line? Well, the slope of that line is easy to see, it's just the change in y over the change in x, and that will give me the average rate between those two points. So that's going to be the change in y or the change in x, so it's going to be f of x plus delta x minus f of x, that is just the change in the y, notice it's just this value minus that value. It looks long, it looks impressive, it may even look intimidating, but all it is, is looking at this difference here, it slopes.

Now to divide it, to make it slope, to finish off the slope, I have to divide it by that, and what's that difference? Well, it's just this minus x, just delta x; it's just a change in x. So that gives me, that gives me in fact, the average rate. How do I make that into instantaneous rate? Well, I let that delta x get smaller and smaller and smaller so that this point, right here, drifts closer and closer, into here. And then what happens, as that thing drifts, I now approach this beautiful tangent line which is the thing I seek.

So how do I get that tangent line? What I have to do is let that delta x approach zero. And so what I need to do here is write the limit as delta x approaches zero, and that quantity represents the instantaneous rate of change of this function at that point, or it also represents the slope of that red tangent line.

This mathematical object that we just discovered is the answer to that first question of Calculus. How do you find instantaneous rates of change? This is how you find it; this is how you find it. And in fact, mathematicians have a name for this; I'd like to tell you about that name right now. In fact, that formula is so beautiful, I wanted to show it to you in a beautiful looking font, and in fact, we call that long thing, we have a name for that, we call it f prime of x. You'll notice that little prime there, that's just the name for this complicated thing. It's just a name for this thing. So if I want to talk about the whole thing, instead of saying to you, the limit as delta x approaches zero of f of x plus delta x, minus f of x, all over delta x, I could just say f prime of x because that's what it means. And in fact, there's a name for this object; we call this the derivative, so in fact that is the definition of the derivative. Maybe some of you, maybe in fact you, have actually heard of the word derivative somehow, and maybe some people I know for sure, hear this word and they immediately panic or they go into sort of fits of terror. But all it really is, if you think about it, is just a slope, it's the change in y's divided by the change in x's. But it’s the slope of something really neat; it’s the slope of a tangent line. And so that's why we have to have that limit there. So when I, when you think of derivative, what I hope that you'll think about someday, is that represents an instantaneous rate of change, it represents a slope, a slope of a tangent. If you look at the very, very heart of the idea, it's not that hard. Oh, I admit that the limit is hard and I admit that this thing looks really threatening, and I admit that some of these problems, actually to solve, require a lot of work. But the basic idea, the overall idea which is really important for you to grab on to is not that hard, it's just slope, with all these new ideas thrown in to make the derivative.

So now let me recap what we've seen so far. The first thing we've seen, is that the derivative of f of x at a point, let's say x not a particular point, represents the slope of the tangent line at that point. So if you want to find the slope of the tangent line of a curve, at a point, what you can do is find the derivative, f prime of x.

Another way to think about the derivative is to think about the derivative as follows. The derivative of f of x at a point, let’s say x not, represents the instantaneous rate of change of the function f of x at x not. You'll notice these two things look almost the exact same, and that's because, in fact, they really are equivalent. And when I think of the derivative personally, I think of both of these things, basically simultaneously. If I need a slope of a tangent line, then I think of the slope of the tangent line. If I need an instantaneous rate of change, I think of it as instantaneous rate of change. The point is these two things both capture the very essence, the very spirit of the derivative, of this mathematical object. And we saw an example of this, actually, in the previous discussion, where we actually computed the derivative at the particular point I was interested in for my particular journey. But now you can see, we can compute these things for any journey at any point.

Up next, what I want to do is begin to embark upon a long journey, through a lot of examples. I hope that you're building sort of an intuition for this idea, but now we need to sort of get to the practical end of things and actually begin to actually compute these things, find them, work through them.

How do we do that? Well, we have to set up a problem and then actually just do a limit. So with all that limit stuff, so all that warm-up that we've been doing – do you remember in fact, way back when we were talking about limits and I said to you, "What you got to do is you've got to practice those limits and you've got to do every single problem in your book. There's like sixty of them, you've got to do them all." And you thought I was joking, and you sort of rolled your eyes, and thought I was a complete whacko? Well now you can see the value of those, if you actually did them, cause I knew that the way to find the derivative, by definition, is to actually compute a limit. So we actually will be doing a lot of limits, and all those limits, I'll tell you right now, will be of indeterminate form. A lot of fun, so reflect on this stuff, think about it and get ready for a whole bunch of examples.

And as we leave, as we leave this really momentous moment, I think it's time to really think of ourselves as journeying, and being on the top. It's as though we're at the Mount Rushmore of Calculus, and that's why our background is appropriate. I really feel like we are there now, and in fact, if I stand the right way, I wonder if I could, it almost looks like I'm on Mount Rushmore. That would be sort of cool, I think that's a good idea. Hey, why don't we start a write-in campaign right now, why don't you write to your congressperson and say that you think that Professor Edward Burger should be replacing, I guess it would be Thomas Jefferson, on Mount Rushmore. I like that idea, in fact so let's close with two really important ideas. The first one is the idea of the derivative and the value of it. And the second idea is that I believe I should be on Mount Rushmore. Okay, I'll see you in a bit and we'll take a look at some examples. Bye.

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