. Welcome to my presentation on derivatives I think you will always find that mathematics It became more fun than before, that is, before several previous lessons Okay, so let’s start with derivatives I know that at first glance it seems complicated Well, in general, if I have a straight line – let me see if I can draw a straight line precisely – if I have a straight line This is the coordinate system, and it is not straight This is a straight line . But when I have a straight line like this, I ask you to You find the tendency – I think you already know how to do this – It’s the change in y ÷ the change in x If you want to find the slope – in fact, I mean the slope is Same, because it’s a straight line, the slope is the same Along the whole line, but if I want to find the slope on any Point on this line, what I’m going to do is pick Point x – Let’s say I choose this point We’ll pick a different color – I’ll take this point, I’ll pick This point – it’s arbitrary, I can pick any Points, and I’ll know what the change in y is – this Is the change in y, the delta y, that’s another way Let’s express the change in y – this is the change in x Delta x We found that the slope is defined as The change in y ÷ the change in x . And another way to say this is a delta – it’s that triangle – Delta y ÷ delta x very clear Now what happens, though, if we’re not engaged With a straight line? Let me see if I have enough space to draw this That is, to draw another coordinate system It’s still messy, but I think you’ll get the idea . Now let’s assume, that instead of a regular font like this, that is It follows the criterion y = mx + b Let’s assume I have the curve y = x ^ 2 Let me paint it in a different color So y = x ^ 2 looks like this It’s a curve, and maybe that’s familiar to you until now And what I’m going to ask of you is, what is it Inclination of this curve? And think about that What does it mean to take the slope of the curve now? Well, for this line, the slope is the same as along This whole line But if you look at this curve The tendency doesn’t change, right? It’s flat here, and it’s getting more sharp Until it becomes very sharp And if you really get away, it gets quite severe So you might say, well, how do we find it Slope of a curve whose slope continues to change? Well, there is no definite slope of the curve as a whole For the line, the slope is for the whole line, because The tendency never changes But what we can do is find what it is Tilt at a certain point And the tilt at a certain point will be the same The slope of the seam For example – let me choose green – lean on this point Here it will be the same as the slope of this line is not it? Because this line is tangent to it It’s touching that curve, and at that particular point We’ll get – this blue curve, y = x ^ 2, will have The same slope as this green line But if we come back to the point here, though, that is Bad graph, slope will be Something like that The inclination of the tangent The slope will be negative, and here the slope is positive But if we take a point here, the slope will Be more positive So how do we find it? How do we find what is the slope at any point Along the curve y = x ^ 2? That’s where the derivative is used, and now For the first time you will see why it is considered the end Useful concept So let me redraw the curve Okay, so I’m going to draw the axes, that’s the y-axis – I’ll put it on First quadrant – and this – I really have to find Better tool for drawing – this is my x coordinate, so let me Draw the curve in yellow . So y = x ^ 2 looks like this I really focus on drawing it on Least well Okay Let’s suppose I want to find the slope at this point . Let’s call this point a At this point, x = a Of course this is f (a) f (a) So what we can do is try to find The slope of the cutting line A line that lies between – we take another point, and say it is fairly close From this point on the graph, let’s assume it’s here, and so We could have found the slope of this line, it would be sort of An approximate slope of the curve Right at this point So let me draw that line lets see So The cutout will look like this Let us assume that this point is a + h, where This distance is h, this is a + h, we We’re moving away by h from a, hence this point here Is f (a + h) . There is something wrong with the pen . So this would be an approximation of what Is the slope at the point And the closest that h reaches, that is, the point closest to it At this point, it is the best approximation Down to the point where we could get Slope, where h = 0, this is actually the slope Intraday tilt, at that point on the curve But how do we find what is the slope when h = 0? . So now, we’re assuming the slope is between these two The two points, the change will be in y, so what is Change in y? It is this, that is, that point here is The x coordinate is– the pen is still bad– The x-coordinate is a + h, and the y-coordinate is f (a + h) f (a + h) And this point here, the coordinate is a and f (a) If we only use the typical slope formula, as before We’ll say the change in y / the change in x Well, what is the change in y? It’s f (a + h) – this y-coordinate – this y-coordinate– – f (a / change in x) Well that change in x is this x-coordinate, so a + h, – This x-coordinate, i.e. – a And of course, this a and this a are omitted So it’s f (a + h) – f (a) / h This is the slope of this intersection If we want to obtain the slope of the tangent line, then We have to find what happens as the value of h gets smaller more and more And I think you know where to go We really want that, if we want to find a slope This tangent line, so we have to find the end of this The value as h approaches zero So, as h gets closer to zero, so does this cross line It will approach more than the slope of the tangent line And then we’re going to know the exact slope on The instantaneous point along the curve In fact, this appears to be a definition The derivative The derivative is nothing more than a mile Curve at a specified point This is very useful, because it is for the first time Everything we’ve talked about on this point is Inclination of the streak But now we can take any continuous curve, or Continuous curves, and we see the slope of that curve On a specific point So now that I’ve given you the definition of a derivative And I hope it might have given you intuition, in The next demo I will use this identification To apply it to some conjugations, such as x ^ 2 and others, and Give you some other questions I’ll see you in the next presentation .