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Welcome to the presentation on derivatives.

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I think you're going to find that this is when math starts

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to become a lot more fun than it was just a few topics ago.

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Well let's get started with our derivatives.

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I know it sounds very complicated.

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Well, in general, if I have a straight line-- let me see if I

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can draw a straight line properly-- if I had a straight

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line-- that's my coordinate axes, which aren't straight--

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this is a straight line.

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But when I have a straight line like that, and I ask you to

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find the slope-- I think you already know how to do this--

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it's just the change in y divided by the change in x.

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If I wanted to find the slope-- really I mean the slope is the

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same, because it is a straight line, the slope is the same

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across the whole line, but if I want to find the slope at any

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point in this line, what I would do is I would pick a

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point x-- say I'd pick this point.

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We'd pick a different color-- I'd take this point, I'd pick

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this point-- it's pretty arbitrary, I could pick any two

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points, and I would figure out what the change in y is-- this

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is the change in y, delta y, that's just another way of

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saying change in y-- and this is the change in x.

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delta x.

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And we figured out that the slope is defined really as

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change in y divided by change in x.

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And another way of saying that is delta-- it's that triangle--

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delta y divided by delta x.

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Very straightforward.

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Now what happens, though, if we're not dealing

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with a straight line?

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Let me see if I have space to draw that.

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Another coordinate axes.

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Still pretty messy, but I think you'll get the point.

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Now let's say, instead of just a regular line like this, this

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follows the standard y equals mx plus b.

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Let's just say I had the curve y equals x squared.

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Let me draw it in a different color.

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So y equals x squared looks something like this.

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It's a curve, you're probably pretty familiar with it by now.

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And what I'm going to ask you is, what is the

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slope of this curve?

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And think about that.

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What does it mean to take the slope of a curve now?

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Well, in this line, the slope was the same throughout

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the whole line.

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But if you look at this curve, doesn't the

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slope change, right?

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Here it's almost flat, and it gets steeper steeper steeper

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steeper steeper until gets pretty steep.

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And if you go really far out, it gets extremely steep.

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So you're probably saying, well, how do you figure out

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the slope of a curve whose slope keeps changing?

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Well there is no slope for the entire curve.

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For a line, there is a slope for the entire line, because

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the slope never changes.

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But what we could try to do is figure out what the

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slope is at a given point.

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And the slope at a given point would be the same as the

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slope of a tangent line.

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For example-- let me pick a green-- the slope at this point

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right here would be the same as the slope of this line.

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Right?

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Because this line is tangent to it.

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So it just touches that curve, and at that exact point, they

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would have-- this blue curve, y equals x squared, would have

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the same slope as this green line.

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But if we go to a point back here, even though this is a

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really badly drawn graph, the slope would be

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something like this.

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The tangent slope.

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The slope would be a negative slope, and here it's a positive

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slope, but if we took a point here, the slope would

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be even more positive.

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So how are we going to figure this out?

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How are we going to figure out what the slope is at any point

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along the curve y equals x squared?

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That's where the derivative comes into use, and now for the

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first time you'll actually see why a limit is actually

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a useful concept.

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So let me try to redraw the curve.

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OK, I'll draw my axes, that's the y-axis-- I'll just do it in

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the first quadrant-- and this is-- I really have to find a

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better tool to do my-- this is x coordinate, and then let

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me draw my curve in yellow.

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So y equals x squared looks something like this.

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I'm really concentrating to draw this at

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least decently good.

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OK.

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So let's say we want to find the slope at this point.

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Let's call this point a.

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At this point, x equals a.

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And of course this is f of a.

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So what we could try to do is, we could try to find

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the slope of a secant line.

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A line between-- we take another point, say, somewhat

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close, to this point on the graph, let's say here, and if

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we could figure out the slope of this line, it would be a

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bit of an approximation of the slope of the curve

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exactly at this point.

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So let me draw that secant line.

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Something like that.

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Secant line looks something like that.

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And let's say that this point right here is a plus h, where

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this distance is just h, this is a plus h, we're just going

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h away from a, and then this point right here

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is f of a plus h.

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My pen is malfunctioning.

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So this would be an approximation for what the

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slope is at this point.

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And the closer that h gets, the closer this point gets to

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this point, the better our approximation is going to be,

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all the way to the point that if we could actually get the

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slope where h equals 0, that would actually be the slope,

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the instantaneous slope, at that point in the curve.

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But how can we figure out what the slope is when h equals 0?

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So right now, we're saying that the slope between these two

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points, it would be the change in y, so what's

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the change in y?

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It's this, so that this point right here is-- the x

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coordinate is-- my thing just keeps messing up-- the x

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coordinate is a plus h, and the y coordinate is f of a plus h.

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And this point right here, the coordinate is a and f of a.

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So if we just use the standard slope formula, like before, we

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would say change in y over change in x.

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Well, what's the change in y?

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It's f of a plus h-- this y coordinate minus this y

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coordinate-- minus f of a over the change in x.

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Well that change in x is this x coordinate, a plus h, minus

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this x coordinate, minus a.

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And of course this a and this a cancel out.

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So it's f of a plus h, minus f of a, all over h.

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This is just the slope of this secant line.

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And if we want to get the slope of the tangent line, we would

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just have to find what happens as h gets smaller and

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smaller and smaller.

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And I think you know where I'm going.

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Really, we just want to, if we want to find the slope of this

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tangent line, we just have to find the limit of this

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value as h approaches 0.

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And then, as h approaches 0, this secant line is going to

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get closer and closer to the slope of the tangent line.

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And then we'll know the exact slope at the instantaneous

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point along the curve.

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And actually, it turns out that this is the definition

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of the derivative.

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And the derivative is nothing more than the slope of a

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curve at an exact point.

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And this is super useful, because for the first time,

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everything we've talked about to this point is

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the slope of a line.

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But now we can take any continuous curve, or most

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continuous curves, and find the slope of that curve

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at an exact point.

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So now that I've given you the definition of what a derivative

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is, and maybe hopefully a little bit of intuition, in the

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next presentation I'm going to use this definition to actually

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apply it to some functions, like x squared and others, and

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give you some more problems.

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I'll see you in the next presentation

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