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About this Lesson
 Type: Video Tutorial
 Length: 23:17
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 250 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: Special Functions (10 lessons, $15.84)
Calculus: Exponential Functions (2 lessons, $4.95)
The power rule for differentiation does not apply to exponential functions (e.g. the derivative of 2^x does NOT equal x*2^(x1)). In this lesson, we will return to the limit definition of the derivative to discover how to differentiate exponential functions like 2^x. Professor Burger will graph the exponential function and calculate the slope of the tangent line. In the end, we will arrive at the fact that the derivative of an exponential function is the product of the function and the natural log of its base. You will also see how to arrive at e = 2.71828..., which is the only constant for which its derivative, when raised to any power is equal to the same thing. Thus, the derivative of e^x is equal to e^x and the e^(ln x) = e. The derivative of N^x is equal to N^x*ln N. The constant 'e' turns out to be very important in math, and Dr. Burger will explain some of its uses as a critical constant.
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 http://www.thinkwell.com/student/product/calculus. 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 UTAustin 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, padic 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
 Thinkwell
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11/13/2008
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Special Functions
Exponential Functions
Derivatives of Exponential Functions Page [1 of 1]
Now what I want to do is think about how we can find the derivative of an exponential function. So, let’s take a look at a function; say for example, f of x equals—oh, I don’t know—how about two to the x. Now, what’s the derivative of that function? Well, here, I think, is a super guess. I want to show you a super guess. And I like to do my superguesses in green—in a green font—because I think this is a guess that makes an awful lot of sense to me. So, you learned this already.
What you do is you bring down the exponent, and then you have two to the exponent minus one. I think that’s a super guess. But that is a guess that is incorrect, because when we establish this rule, the promise that we made and the understanding that we had was that, in fact, the variable—the unknown—was this piece here. And that number was actually a real number. And here, the roles of variable and constant have swapped position. If this were x squared, then this procedure would probably work, but this is two to the x. That’s the variable. The problem is this method is wrong. And so I want you to really take a look at this and realize that even though it sounds great, it’s wrong.
And I had mentioned this, and I actually spent a little time on this, because I see folks constantly taking derivatives of this by bringing down the x times two to the x minus one. Please don’t do that. Always remember that if I have x to a power—like x to the seventeen—the derivative of that is seventeen x to the sixteen. But if I have seventeen raised to the x power, that’s much different. It’s a completely different function, and the derivative technique that you used for the previous one – example will not work for that.
So, please never ever do this kind of thing. Okay. In fact—this one—I don’t even want to have this around. So, how could we figure out what the derivative of this is? Well, let me begin by just sketching a graph of this again. We took a look at these graphs before in the little review. Let me just remind you how these look, roughly speaking. This would be—I’m not going to put in any points—label any points here. But just to give you this general flavor of this. That might look like the two to the x curve. And to figure out the derivative, what we could do is think about the fact that the derivative is—represents anyway, the slope of tangent lines. So, let’s take a look at some tangent lines.
So if, for example, I pick a point here, the tangent line looks like it should be right around there. You’ll notice that is positively sloped and reasonably steep. As I move further out, what happens to steepness? The steepness, actually, starts to increase quite dramatically. What happens when I move this way? The steepness begins to decrease. It becomes less steep. Even if I go into the negative terrain, those tangent lines are even “smallerly” steep, right? Almost zero, but not quite zero.
And you’ll notice that they’re always positive. So, one thing we know for certain is that the derivative of this function is always going to be a positive function, because never, anywhere, is the slope negative. Never is this thing pitching down. It’s always pitching up—always pitching up. So, that’s sort of interesting.
But now, what exactly is it? What exactly is it? Well, it isn’t that obvious. So, what I’d like us to do is just to try to see if we can figure out what it should be. And the only way we know how to do this kind of thing—and I hope this is okay with you folks—is I’d like to actually return to the very definition of this. So, I want to—oh, and see, all the camera people are saying they don’t even want to see this. But I’ll do it for you. You don’t have to do this, and you won’t be responsible for this.
What I’d like for you to do is just to watch me go through this process, and I want you to sort of listen to what I have to say. Don’t worry about memorizing this technique or anything, but I want the actual answer that we’re going to inevitably find to make some sense to you. So just watch me—sit back and watch—and I’ll take a quick stab at this, and we’ll see what we can do.
So, we’re going to use this old definition of the derivative, and we’re going to apply it with this kind of function. So, let’s see. In fact, what I’d really like to do—if I had my way—is to consider the more general example where I don’t know what the number two actually equals. So, let me say f of x equals some number—could be two, could be three—to the x power. That’s a number now. It’s not a variable. So, it’s fixed. And remember that number has to bigger than zero, otherwise we can’t talk about this function.
Okay. Well, I’m going to use the definition of a derivative to try to attempt to figure out what that equals—the derivative. So, let’s see. The derivative f prime of x equals the limit as delta x approaches zero. And now, you can see the difference between this function and then other functions we’ve looked at. What is f of x plus delta x? Well, wherever I see an x here, I have replace it by x plus delta x. You’ll notice the x is in the exponent. So, in fact, this is going to be n raised to the power x plus delta x. That’s exactly this function evaluated at x plus delta x. I took the x, which appears here, and I replaced it by the big blop x plus delta x, and there it is.
Now, I’ve got to subtract out the function. So, minus n to the x, and I divide all that by delta x. So, that’s the limit I have to figure out. If we try to take the limit—let delta x go to zero right now—that term drops out, and I’m left with n to the x minus n to the x. Well, that’s zero divided by zero. I have, once again, an indeterminate form.
What can I do here? Well, I could remember the law of exponents, which says if I have n to a sum, I could replace that by n to the x multiplied by n to the delta x. I saw that in the previous little warmup review, that, in fact, this can be rewritten as n to the x times n to the delta x minus n to the x all over delta x. Because you just add the exponents to get that.
Now, I’ve noticed that I have a common factor of n to the x. So, I can factor that out. That’s going to be the limit as delta x approaches zero of n to the x. And if I factor that out, I’m left with n to the delta x minus one. Remember, you’re just supposed to be watching me and trying to follow. So, I’m doing all the work here. So, I just factor that common factor out, I got that minus one. If I distribute, I get that n to the x, which is right here.
Okay. Well now, here’s something interesting that I want you to notice. This term here has no delta x’s in it at all. It’s delta x free. So, when I take the limit, this is not going to change at all. There’s no delta x’s. Only delta x’s go to zero. So, in fact, what I could really do is actually pull that out in front of the limit and just say it’s that number times the limit. Because this is not going to change the limit answer at all. So, in fact, I could write this as n to the x multiplied by that limit—limit as delta x approaches zero of n to the delta x minus one all over delta x. Because remember, there’s no delta x there. So, I’m just going to take whatever the limit actually is and end up multiplying it by that number anyway, so why not write that number out front and then have this thing be here.
Now, what is that limit? Well, it turns out that limit is actually a little tricky to figure out. That limit is actually a little bit hard. But it’s something. It’s some number. I don’t know what the number is, but I’ll just call it something. I’ll call this number—oh, let’s see—. So, this is n to the x, and I’ll call that limit just l. And it depends upon n—whatever n is. And l’s a mystery. So, it’s a mysterious function. So, what is this equal too? I don’t know. But, it’s something. It’s some limit, and if we knew what the answer was, we’d actually know what it is.
So, that’s the derivative, in some sense. Of course there’s a big mystery piece, right now. I admit that this is a big mystery. But you have to admit that, in fact, that something interesting has happened. What do we see about the derivative? The original function was n to the x. And what do I see about its derivative. Its derivative is also n to the x time some mystery number.
So, the derivative of this function is actually equal to the original function itself multiplied by some number. And the question now is what’s the number. Well, let’s see what that number is. How can we figure out exactly what that number is? Well, it turns out that that number—well, let’s think about that for a second. How can we figure out that number?
What if you evaluated the derivative at zero? If I evaluated the derivative at zero, what would I see? Well, wherever I see an x, I would plug in zero. And where do I see an x here? Just there. There are no x’s in here. That’s just a number. So, put in a zero there. I would see n to the zero power and n to the zero is just one. So, what I would see is that f prime at zero would equal that mysterious number, because the n to the zero would just drop out.
So, what is f prime of zero? That represents the slope of the tangent line when x equals zero. So, in particular, if I go back to this picture for just a second, what I see is that l function—that mysterious l function—is just the slope of the tangent line where the curve crosses the yaxis. So, whatever that slope is, that slope is that mysterious number. That slope turns out to be this l. Because we’ve seen that f prime at zero equals just that number. So, that’s the slope at zero.
Okay. Well now, what does that equal? Well, for two, for example, I don’t know. Why don’t we try to estimate that number and estimate, in fact, the limit by evaluating it on a computer? Now, I actually have a computer here, so let’s do a little computer experiment right now. What we’re trying to do is figure out that mysterious number. That’s the goal.
And so, I have a computer right here for us. Let’s see if I can bring this up for you. Wow, look at this, we actually have a whole computer here. It’s hightech here folks. Let me turn it on. I hope it goes on. We’re not that hightech. Ah, okay, there we go.
Okay, and now, let’s try some examples. Now, remember what I’m trying to do here. What am I trying to compute? Let me just write this out. What I’m trying to compute is the following. I’m trying to figure out what this l is. Let me do it for two. So, I want to figure out l of two, which I remind you is the limit as delta x approaches zero of what? Well, of two to the delta x minus one all over delta x. That’s the limit I want. That’s the limit I want to figure out, which we’ve seen—actually is represented by the slope of the tangent line when the curve crosses the yaxis.
Well, what I’m going to do here, since I don’t know how to compute that limit, I’m just going to plug in some numbers for delta x that are very, very small, like we did in the very beginning of our discussions a way, way long time ago. If you can’t do a problem, try to estimate it—try to figure out roughly what the limit is by taking a look at the small values for delta x.
So, I actually wrote a little program to do that. Let’s put in some values right now. So, for example, let’s put in—let’s evaluate this for delta x being, let’s say, point one, which is a small number. That’s close to zero. Point one is a small number. And what do we see this equal in that case? Oh, it’s an old computer, folks. It’s sort of winds, cranks, cranks, cranks—oop, there it is. It turns out that it’s point seven one seven seven.
Okay. Let’s try even a smaller one and see what that equals. So, let’s try a smaller one. Let’s try—that was point one we tried. How about trying point o one? Point six nine five five. Got a little smaller. Let’s try another one. How about point o o one. Six nine three! Seems to be sort of leveling off here around six point six nine. Let’s try one last one, maybe.
How about point o o o one. That’s—notice that number is very, very close to zero. So, it’s very—this should be very close to the limit. Again, what I’m doing is, I’m taking that number for delta x and I’m plugging in two to the point o o o one minus one divided by point o o 1 and see what that equals point six nine three one, and so on. So, we can se this thing seems to really be heading to around, roughly point six nine three stuff. So roughly, it’s around point six nine. Okay. Great.
What does that mean, by the way? Let me just recap and show you exactly what that means. So many props here. I… What that means is that if I come back to here, the slope of this line right there—in fact, let me draw that in—the slope of this line—this slope is approximately point six nine three. That’s what we just saw.
Okay. Let’s try another example. How about if we looked at a different curve? What if we looked at three to the x? Now, remember what three to the x would look like. Since the three is bigger, the curve would be sharper. It grows faster. So, it would look something like this. It would start way down below and then come up and grow up. That would be three to the x—the sharper curve like that.
The tangents—notice there—are actually more steep than the other one. And in particular here, you’ll see the tangent of the orange one is actually steeper at x equals zero. Let me draw that one in. I’ll draw that one in purple. If I draw that one in, what does it look like? It’s a little steeper and that has some slope. And what is the slope of that? Well, the slope of that, we know. The slope of that would just be l evaluated at three—rather than at two—to find that limit for three.
Now, we know it’s going to be bigger than this number, because we see it’s steeper than that. Let’s see exactly what that is. So, let’s try that really fast. Let’s try to find l of three. That’s that special number we have to multiply to get the derivative. It’s the limit as delta x goes to zero of three to the delta x minus one, all over delta x.
So, let’s try that again on our little computer experiment really fast to see what that looks like. Bring back the computer. Oh, it’s sleeping. I have to wake it up. C’mon. Wake up. Oh, there it goes. Okay. And let’s try this one now. So, let’s try this. So, we’re going to plug in—let’s start off with point o one. Should be bigger than point six nine, like we saw at the last one. And let’s see what this is.
Oh, look. This is much bigger. One point four o one. So, let’s try even a closer—a smaller delta x. Let’s try point zero zero zero one. One point o nine eight—got a little bit smaller. Let’s do one last one and see if can see roughly what the limit is. How about point o o o o one. Point one point o nine eight.
So, it seems like this is roughly equal to one point o nine eight stuff, which you notice is bigger than the point six nine that we saw earlier. So, this slope has a slope a little bit bigger than one. Okay, let me move this over to here now and bring back my picture. So, what we now see is that the slope of this line is approximately equal to one point o nine eight stuff.
Okay. So, where are we? Where we are is the following. We’ve now seen that the derivative is equal to the function itself multiplied by this mysterious number, which in this case would be about point six nine three. And if I put a three in here, it would be about one point o eight o nine eight.
Well, if at the slope of this line is point six nine, and the slope of this steeper line is point—uh—one point o nine, there must be a line somewhere in between there that has slope exactly equal to one, right? This is too small—sort of like Goldilocks and the three be—this is too small, this is too big, but somewhere in between is just right. Just right, meaning that the slope of that line would be one.
So, there must be some number I could put in here. It would be bigger than two, but it would be smaller than three, which has the property that this mysterious function would actually be equal to one. Let me draw that curve in for you, if I can. See? I’m sort of running out of colors here, but let’s see if I can try to draw it in for you. It’s going to be in between these two curves. Going to be between those two curves. It’s hard to see this. But there’s a special number, which has the property that the slope at that point will be exactly one. And that special number, I’m going to call e.
And what would e have to equal? Well, e would have to be a number that’s bigger than two but smaller than three. And, actually, you think it’s going to be closer to three or closer to two? Remember the deal is that the slope of the line will equal one.
Notice that the slope at three is one point o nine, which is very, very close to one. It’s just a hair over; whereas, this is point six nine. That’s very, very far away from one. So, it seems like a good guess is that this special number e will actually be smaller than three, but pretty close to three—closer to three than it is to two.
And, well, what is that number? Well, that special number e is equal to—let’s take a look. That special number e is equal to two point seven one eight dip, dip, dip, dip. And that is the e that you have heard about. E equals two point seven one eight two and so on.
Notice it’s a little bit smaller than three but bigger than two. So, it fits right in between those two numbers. It’s right in between those two numbers. And that’s the very special number that has the property that that l function there is equal to one. The slope of the tangent there is one.
So, what have we just discovered? We’ve just discovered the following really neat thing. If I define the function f of x to be equal to this special number e raised to the x power—that e is around two point seven one eight—then the derivative of this function is what? Is itself  It’s e to the x multiplied by that special number, which in this case is one.
So, this number e is a very, very special constant. It’s the constant so that the derivative of this function turns out to be exactly the same as the function itself. The derivative of e to the x is e to the x—absolutely amazing. An example of a function whose derivative is the same as the function. That’s the special number e as the base. And it turns out this e is all around us, whenever things are growing.
And, in fact, here’s the basic idea when e turns out to be relevant. Take a look at the following very, very simple idea. Let’s look at population, for example. Suppose you’ve got just a few living things—could be people, could be bacteria, could be anything. But if you just have a few living things, then how much reproduction would you expect? Well, you’d expect a little reproduction, and so, therefore, you would expect that you’d have, in a while, some more of these things.
But what if you start out with a lot of things? Well, then you expect to be having reproduction all over the place, right? That is to say that the amount of change in the population should be roughly proportional to how much population there is. Lot of population, lot of reproduction. Little population, little reproduction.
Well, it turns out that when you solve those things, as we will, the number e is the number that turns out to be the relevant one. So, in terms at looking how your money grows, how people or other living things grow, it’s all this very special number e, which we now see how that e—that two point seven one—finally comes out of existence—comes into existence. And the answer is it’s that special number whose derivative of that number to the x turns out to be that number to the x.
Okay. Well, there’s the exponential function. And now, we see it’s derivative is just itself. Very, very simple idea. Derivative of e to the x is e to the x.
Okay. What I want to take a look at now is sort of the untangling of the exponential function—the logarithm function. And this gives people a lot of headaches. So what we’re going to do up is we’re going to first do a little review of the log, and then we’ll take a look at how to figure out what the derivative of the log function is.
Okay. So, stay with us and don’t get too nervous. See you.
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