1 00:00:00,012 --> 00:00:04,196 [music] We've seen df/dx and we've seen d/dx. 2 00:00:04,196 --> 00:00:07,781 So now, we're going to think about just dx. 3 00:00:07,781 --> 00:00:11,976 This sort of object is called a differential. 4 00:00:11,976 --> 00:00:18,732 [inaudible] about dx not d/dx, right? We've thought about that little bit 5 00:00:18,732 --> 00:00:23,866 already, right? As an operator, this the thing that does 6 00:00:23,866 --> 00:00:28,656 the differentiation. This object is what we're going to study 7 00:00:28,656 --> 00:00:31,228 right now. It's called a differential. 8 00:00:31,228 --> 00:00:35,986 But what does that even mean? Geometrically, these differentials, so dy 9 00:00:35,986 --> 00:00:41,145 and dx, represented some change in the linear approximation that I get using the 10 00:00:41,145 --> 00:00:44,732 tangent line. So here, I've got some function that I've 11 00:00:44,732 --> 00:00:48,255 graphed, y equals f of x. And here's a point, x, f of x. 12 00:00:48,255 --> 00:00:51,311 And I've got a tangent line with a curve there. 13 00:00:51,311 --> 00:00:55,906 And here's some change in x, which I'm calling dx, and here's some change in y, 14 00:00:55,906 --> 00:00:59,170 which I'm calling dy. And if you think about, what is the 15 00:00:59,170 --> 00:01:02,324 derivative? Well, I mean the derivative really is dy 16 00:01:02,324 --> 00:01:07,390 over dx, right, in some sense, you know. If you take the derivative and multiply it 17 00:01:07,390 --> 00:01:12,558 by how much the input changed by, I mean, you don't actually get how much the output 18 00:01:12,558 --> 00:01:17,726 changed by but you get an approximation to how much the output changed by and as for 19 00:01:17,726 --> 00:01:23,068 this dy here is being used as, this dy is the change in the linear approximation. 20 00:01:23,068 --> 00:01:27,764 Okay, here's a statement in words. In words, dy is a change in the 21 00:01:27,764 --> 00:01:32,604 linearization, or the linear approximation, or the tangent line 22 00:01:32,604 --> 00:01:38,354 approximation, to this function that you're thinking of as y, depending on x. 23 00:01:38,354 --> 00:01:42,781 That's what dy is. I should emphasize that your dy's had 24 00:01:42,781 --> 00:01:49,229 better include a dx in there somewhere. At different points in your life, you'll 25 00:01:49,229 --> 00:01:54,116 develop hm, different ideas about what dx and dy really mean. 26 00:01:54,116 --> 00:01:58,696 So, in a Calculus class, in this Calculus class, if you really want to, you can 27 00:01:58,696 --> 00:02:03,064 think of dx and dy as infinitesimal quantities, very small quantities. 28 00:02:03,064 --> 00:02:06,969 It's difficult to make this sort of thinking entirely rigourous. 29 00:02:06,969 --> 00:02:11,667 But that's certainly an okay way to think if you're trying to get some intuition for 30 00:02:11,667 --> 00:02:15,987 what these things are representing. Now, later on in your life, when you take 31 00:02:15,987 --> 00:02:20,537 some course, say, a differential geometry course, dx will then be given some better 32 00:02:20,537 --> 00:02:24,004 foundation. Dx will be revealed to be a differential 33 00:02:24,004 --> 00:02:26,822 form or a covector. I mean, you'll have some actual 34 00:02:26,822 --> 00:02:30,917 interpretation of dx but for the time being, you know, if you really want to, 35 00:02:30,917 --> 00:02:34,211 you can regard these things as infinitesimal quantities. 36 00:02:34,211 --> 00:02:39,233 But honestly, I wouldn't worry too much about how to actually think about these 37 00:02:39,233 --> 00:02:42,977 things and focus more on how to compute with these objects. 38 00:02:42,977 --> 00:02:48,811 The basic trick is that if y is f of x, then dy is the derivative of f dx, and you 39 00:02:48,811 --> 00:02:53,799 don't even need necessarily to differentiate f in a lot cases. 40 00:02:53,799 --> 00:03:00,589 The differential satisfies many of the same rules or analogous rules as just 41 00:03:00,589 --> 00:03:07,456 differentiation satisfies. So, for example, d of u and v, or u plus v 42 00:03:07,456 --> 00:03:13,982 I should say, is du plus dv. D of u times v, is a product rule, is du 43 00:03:13,982 --> 00:03:19,416 times v plus u dv, right? And there's a quotient rule. 44 00:03:19,416 --> 00:03:26,184 I mean, this differential satisfied all the same or let's just say analogous rules 45 00:03:26,184 --> 00:03:29,888 to derivatives. Let's take a look at an example. 46 00:03:29,888 --> 00:03:35,943 For example maybe y is something here like x squared and, in that case, what's dy? 47 00:03:35,943 --> 00:03:42,197 Well, dy is the derivative 2x times dx and yes, this does relate some change in x and 48 00:03:42,197 --> 00:03:48,097 change in y in terms of linearizations. But you don't even need necessarily to 49 00:03:48,097 --> 00:03:51,803 write it this way. You could if you wanted to write dx 50 00:03:51,803 --> 00:03:57,823 squared and then think that there is power rule of a differential just like the usual 51 00:03:57,823 --> 00:04:01,143 power rule, which then says that this is 2x dx. 52 00:04:01,143 --> 00:04:05,906 So, in general, d of x to the n would be nx to the n minus 1 dx. 53 00:04:05,906 --> 00:04:12,360 We can cook up a more complicated example. For example, let's calculate d of x sine 54 00:04:12,360 --> 00:04:15,221 x. Well, this is d of a product and I can 55 00:04:15,221 --> 00:04:20,291 compute that by using the product rule for differentials, right? 56 00:04:20,291 --> 00:04:24,875 What does that tell me? Well, it will be d of the first thing 57 00:04:24,875 --> 00:04:30,617 times the second thing plus the first thing times d of the second thing. 58 00:04:30,617 --> 00:04:35,740 Now, instead of writing dx times sin x, I could write that as sin x dx. 59 00:04:35,740 --> 00:04:39,734 And what's d of sin of x? Well, this is d of a function. 60 00:04:39,734 --> 00:04:42,939 And remember, how do I take d of a function? 61 00:04:42,939 --> 00:04:49,879 Well, it's the derivative times dx. So, this is x times the derivative, which 62 00:04:49,879 --> 00:04:55,145 is cosine x times dx. Now, if I like, I could factor out the dx 63 00:04:55,145 --> 00:05:01,130 and I could write this as sin x plus x cos x dx, now, because I haven't done anything 64 00:05:01,130 --> 00:05:05,390 new here, right? I mean, I could have just differentiate x 65 00:05:05,390 --> 00:05:11,375 sin x but the point is I'm writing it down without ever, you know, taking derivative. 66 00:05:11,375 --> 00:05:16,653 I'm sort of using the rule for how to compute differentials, right? 67 00:05:16,653 --> 00:05:23,589 Like the differential product rule and the fact that the differential of a function 68 00:05:23,589 --> 00:05:28,489 is the derivative dx. I'm being a little bit vague with how to 69 00:05:28,489 --> 00:05:32,206 deal with these differentials at this point. 70 00:05:32,206 --> 00:05:38,399 But we're going to be seeing more of these differentials in the coming weeks.