1 00:00:00,330 --> 00:00:05,670 Let us begin this section with a brief discussion of the mathematical definition of a derivative. 2 00:00:07,080 --> 00:00:12,930 I think you probably know what a derivative is, but in case you don't remember exactly, this is how 3 00:00:12,930 --> 00:00:15,240 we define a derivative mathematically. 4 00:00:16,200 --> 00:00:22,890 So we start from a function f of X, for example, this blue function here with it, which is a third 5 00:00:22,890 --> 00:00:24,180 order polynomial. 6 00:00:24,690 --> 00:00:28,500 And then we are asking about the derivative at a certain position. 7 00:00:28,500 --> 00:00:35,280 For example, here at X equal minus three point five, which is this red dot here that corresponds to 8 00:00:35,280 --> 00:00:36,180 the blue curve. 9 00:00:37,320 --> 00:00:43,690 And then what we can do is we can go along the positive x direction and to find another point. 10 00:00:44,040 --> 00:00:51,510 For example, here at X equal to minus one where we have a difference in X direction of two point five. 11 00:00:52,380 --> 00:00:58,890 So then we have a difference in the Y coordinates of these two points and a difference in the X coordinate 12 00:00:58,890 --> 00:00:59,820 and the two points. 13 00:01:00,180 --> 00:01:03,690 And this defines the slope of this orange curve here. 14 00:01:04,080 --> 00:01:10,140 And this is in a very bad approximation, the derivative of the function at the Red Point. 15 00:01:11,130 --> 00:01:18,810 And this derivative, this approximation of the derivative gets much better when we decrease the distance 16 00:01:18,810 --> 00:01:19,920 in the X direction. 17 00:01:20,010 --> 00:01:26,490 So this means when we try to move the Orange Point towards the Red Point, and this is what I have shown 18 00:01:26,490 --> 00:01:26,820 here. 19 00:01:26,830 --> 00:01:32,700 So imagine that like in the video, you take this Orange Point and you drag it along the curve and then 20 00:01:32,700 --> 00:01:39,390 you will find that this orange straight line here is linear function, but in the end, look like this. 21 00:01:40,020 --> 00:01:47,760 And so the slope of this function is really the derivative of F of X at the position of X equal to minus 22 00:01:47,760 --> 00:01:48,690 three point five. 23 00:01:49,590 --> 00:01:50,910 And then mathematical terms. 24 00:01:51,360 --> 00:01:58,830 We can write this down as the limit of some number h, which is the distance in the X direction, and 25 00:01:58,830 --> 00:02:01,230 the limit goes for h to zero. 26 00:02:01,230 --> 00:02:06,570 And then we have this term here, which is basically the difference in the Y coordinates and the difference 27 00:02:06,570 --> 00:02:07,620 in the X coordinates. 28 00:02:07,890 --> 00:02:10,259 So this is exactly the slope of this function. 29 00:02:11,850 --> 00:02:19,770 So there exist also equivalent definitions for this expression in case our function is continuous, 30 00:02:19,770 --> 00:02:22,260 which it will basically always be in physics. 31 00:02:22,830 --> 00:02:26,390 So we could also take a backward derivative. 32 00:02:26,400 --> 00:02:30,480 So this means we have two points in the origin point is this time on the left side? 33 00:02:31,110 --> 00:02:38,610 Or we could also do a center or a central point or a central definition for the derivative where we 34 00:02:38,610 --> 00:02:44,540 have two points, one along the positive and one along the negative direction, and we push both of 35 00:02:44,550 --> 00:02:46,410 these points towards the Red Point. 36 00:02:46,590 --> 00:02:52,800 And in all cases, we will end up with this red line, which has a slope and the slope is equal to the 37 00:02:52,800 --> 00:02:54,480 derivative at this point. 38 00:02:56,070 --> 00:03:01,990 So I think so far you already have learned what the derivative is, and I hope you remember how this 39 00:03:01,990 --> 00:03:02,610 is defined. 40 00:03:02,940 --> 00:03:04,710 But in case not, it's really easy. 41 00:03:04,710 --> 00:03:06,870 It's just as limit h to zero. 42 00:03:06,870 --> 00:03:14,670 And then the difference in y divided by the difference in X and in the next lecture, we're going to 43 00:03:14,670 --> 00:03:19,710 talk about the numerical implementations of this first or the derivative.