 So now we're ready to introduce the derivative, so here it is. Actually, this goes back to one of the fundamental questions of calculus, which is the problem of the tangent line. So remember, the secant line is the line between two points on the graph of some function. And the closer that second point is to the first point, the better the secant line, which in this case actually is running between those two points on the graph, the better that the secant line approximates the tangent line, which just touches one point on the graph. Now, if we consider this problem of a tangent line, we have two important facts here, which is that one. We know where that tangent line touches the graph, because it has to go through some point on the graph. So the only question that we need to answer to write the equation for the tangent line is, what's the slope of the tangent line? And once we have the slope, and once we know the point on the tangent line, then we can write the equation for the tangent line. So let's see how we can find that. Now, the slope of the secant line is easy. Because I know the two points that the secant line runs through, a point on the graph, a point that's not too far away, then because I know the coordinates of the two points, I can find the slope of the secant line. That's just going to be my difference in my y-coordinates divided by my difference in the x-coordinates. And that's going to give me the slope of the secant line. And so, as that second point gets closer to the first point, then the slope of the secant line gets closer to the slope of the tangent line. So that's the question we're trying to address here, is what happens as those two points get close? Well, mathematically, that corresponds to finding a limit. So let's take a look at the limit as b approaches a, as this x-coordinate gets close to that x-coordinate, this point gets close to that point. And what happens to the slope of the secant line as b closes in on a? So there's the geometric idea, the slope of the tangent line. It's also tied into an algebraic equivalent, which is the derivative of my function at x equal to a. And we're going to write that in one of two ways. One way is we're going to remind ourselves that this derivative is somehow related to our original function. And one way of doing that is to use the prime. So this is f prime of a, and this is what's called prime notation. The other way is to remind us that the derivative really is a slope, and it looks like a change in a y-value, divided by a change in an x-value. And so our differential notation for the derivative looks like this, dy over dx. And this vertical bar here we're going to be using quite a bit in calculus. This means that what we're going to do is we're going to find the value of this thing when x is equal to a. And that's called differential notation. Now, notations exist and persist for a reason, and mostly that reason is that they remind us of something useful. The prime notation persists because it reminds us that the derivative is related to our original function. The differential notation exists and persists because it tells us something very useful about the units of our derivative. Now, while we could find this limit as b approaches a of our difference quotient, it's usually easier to find an equivalent limit in the following way. So instead of taking two points, one of them at x equal to a, and the other one at x equal to b, we're going to take two points at x equal to a, and x equal to a plus some amount. And again, my y value is my function value, so y is going to be my function evaluated at a plus h. And so my graph is going to look like this. So here's my first point, here's my second point, someplace over someplace else, and my derivative is still going to be the limit of the slope of the secant line. But the slope of the secant line is going to have a slightly different expression here. Difference in the y values f of a plus h minus f of a, difference in the x values a plus h minus a is h, and my secant line has a somewhat simpler expression. And so my derivative, again, is going to be the limit of the slopes of the secant lines, so that's just going to be the limit as h goes to zero. If h goes to zero, these two points close in on each other, and the slope of the secant line approaches the slope of the tangent line. For example, let's find the slope of the time tangent to the graph of y equals x squared at the point where x is equal to... Now again, one of the fundamental things you want to be able to do is to transition between the algebraic and the geometric. So here we're being asked the geometric question, find the slope of the line tangent to the graph. And slope is a geometric idea, but algebraically what that slope is is the derivative. So the slope of the line tangent to the graph, well, it's really the derivative of y equals x squared at x equal to 4. And that connection is important, because in some cases it's easier to think about the derivative as a slope. In other cases, it's easier to think about the slope as a derivative. So let's take a look at that. So let's start out by sketching a graph. Well, maybe you know what y equals x squared looks like. You should. It's actually one of the easy graphs. But maybe you do, maybe you don't. If you do, that's great. That'll really help you, because one of the questions will come to you later. How do I know I got the right answer? If you don't know what y equals x squared is, it doesn't matter if we can sketch a generic graph. In general, your graph is going to be used as a way of organizing your information. So let's sketch a generic graph. I have no idea what y equals x squared looks like. I'll draw a couple of squiggles on the paper, and there's my y equals graph. So I do need to take two points. One of my points is going to be at x equals 4, but that's half a point. I need the y coordinate. Well, y is x squared, and since x is 4, y is 4 squared, y is 16. So my point is at x equals 4, y equals 16. So that's some point on the graph. And it really doesn't matter where it is, although it's kind of nice if you're at least in the right quadrant. So it's in the first quadrant someplace. And I'll put that on the graph. And then I'm going to take a second point at x equal to 4 plus h, and y equal to x squared again. That's 4 plus h squared, and that's going to be something. Again, if I'm using a generic graph, I don't have to worry too much about where the exact locations of these points are, because they're primarily used to organize our information. We want the slope of the line tangent to the graph, so I want to look at the limit of the slopes of the secant lines. So I have the two points. The secant line is going to run between them, so I'll draw the secant line. And again, the idea to keep in mind is that the secant line is going to approximate the tangent line as the second point gets closer to the first. Well, I need to find the slope of the secant line. Well, I have two points on the line, here and here, and I know how to find the slope between two points. That's just the difference in the y values over the difference in the x values. Rise over run, so I can find the slope of the secant line, difference in the y values over the difference in the x values, and after all the dust settles, I do a lot of algebra, and there's a nice formula for the slope of the secant line. And the slope of the tangent line will be the limit as h gets close to 0. So I'm going to take the limit as h gets close to 0, and that's going to be 8, and there's my slope of the tangent line.