1 00:00:02,522 --> 00:00:03,960 We know how to take derivatives of 2 00:00:03,960 --> 00:00:07,440 functions. If I apply the derivative operator to 3 00:00:07,440 --> 00:00:12,635 X^2, I get 2X. 4 00:00:12,635 --> 00:00:15,630 Now if I also apply the derivative operator to 5 00:00:15,630 --> 00:00:20,036 X^2+1, I also get 2X. 6 00:00:20,036 --> 00:00:23,943 If I apply the derivative operator to X^2 7 00:00:23,943 --> 00:00:27,235 X^2+pi, I also get 2X. The derivative of X^2 8 00:00:27,235 --> 00:00:31,714 is 2X, the derivative with respect to X of pi of a constant 9 00:00:31,714 --> 00:00:34,249 is just 0. The derivative with respect to X of 1 10 00:00:34,249 --> 00:00:36,281 is just a constant, is just 0. 11 00:00:36,281 --> 00:00:39,082 So once again, this is just going to be equal to 2X. 12 00:00:39,082 --> 00:00:43,187 In general, the derivative with respect to x of X^2 13 00:00:43,187 --> 00:00:47,309 plus any constant, any constant, is going to be 14 00:00:47,309 --> 00:00:50,569 equal to 2X. The derivative of X^2 with respect to X is 2X 15 00:00:50,569 --> 00:00:54,838 Derivative of a constant with respect to X, a constant does not change 16 00:00:54,838 --> 00:00:57,277 with respect to X, so it's just equal 17 00:00:57,277 --> 00:01:00,901 to 0. So you have, you take the derivative, you apply the derivative operator 18 00:01:00,901 --> 00:01:04,391 to any of these expressions, and you get 2X. 19 00:01:04,391 --> 00:01:08,974 Now let's go the other way around. Let's think about the antiderivative. 20 00:01:08,974 --> 00:01:13,254 Anti-derivative. And one way of 21 00:01:13,254 --> 00:01:14,782 thinking about it, it's the, it's the, 22 00:01:14,782 --> 00:01:18,294 we're doing the opposite of the derivative operator. 23 00:01:18,294 --> 00:01:20,174 The derivative operator, you get an expression, 24 00:01:20,174 --> 00:01:21,785 and you find find this derivative. 25 00:01:21,785 --> 00:01:24,861 Now what we want to do, is, given some expression, 26 00:01:24,861 --> 00:01:25,810 we want to find 27 00:01:25,810 --> 00:01:28,260 what it could be the derivative of. 28 00:01:28,260 --> 00:01:30,045 What it could be the derivative of. 29 00:01:30,045 --> 00:01:32,357 So if someone were to tell you, 30 00:01:32,357 --> 00:01:33,980 were to give you 2X. 31 00:01:33,980 --> 00:01:36,213 If someone were to say, 2X 32 00:01:36,213 --> 00:01:41,132 So if someone were to ask you: what is 33 00:01:41,132 --> 00:01:44,901 what is 2X 34 00:01:44,901 --> 00:01:49,432 the derivative - derivative - 35 00:01:49,432 --> 00:01:53,072 of? They're essentially asking you for 36 00:01:53,072 --> 00:01:55,464 the antiderivative! 37 00:01:55,464 --> 00:01:57,302 And so you could say: well, 2X is 38 00:01:57,302 --> 00:01:59,518 the derivative of X^2. 39 00:01:59,518 --> 00:02:01,624 2X is the derivative of X^2. 40 00:02:01,624 --> 00:02:05,020 But you could say: 2X is the derivative of X^2 plus 1. 41 00:02:05,020 --> 00:02:08,014 You could also say that 2X is the derivative of X^2 plus pi. 42 00:02:08,014 --> 00:02:09,612 I think you get the general idea. 43 00:02:09,612 --> 00:02:11,934 So if you wanted to write it in the most general sense. 44 00:02:11,934 --> 00:02:12,991 you would write: 45 00:02:12,991 --> 00:02:14,433 that 2X is the derivative of X^2 46 00:02:14,433 --> 00:02:18,301 plus some constant. 47 00:02:18,301 --> 00:02:19,368 So this is what you would consider 48 00:02:19,368 --> 00:02:22,511 the antiderivative of 2X. 49 00:02:22,511 --> 00:02:25,072 Now that's all nice, but this is kind of 50 00:02:25,072 --> 00:02:27,279 clumsy to have to write the sentence like this 51 00:02:27,279 --> 00:02:29,523 so lets come up with some kind of notation 52 00:02:29,523 --> 00:02:30,969 for the antiderivative. 53 00:02:30,969 --> 00:02:33,100 And the convention here is to use 54 00:02:33,100 --> 00:02:35,412 kind of a strange-looking notation. 55 00:02:35,412 --> 00:02:38,012 Is to use a 56 00:02:38,012 --> 00:02:40,142 a big, elongated, S-looking thing like that 57 00:02:40,142 --> 00:02:43,844 and the dx around the function 58 00:02:43,844 --> 00:02:45,406 you're trying to take the antiderivative 59 00:02:45,406 --> 00:02:49,085 of. So in this case, it would look something like this. 60 00:02:49,085 --> 00:02:49,844 This is just 61 00:02:49,844 --> 00:02:51,076 saying: this is equal to the antiderivative of 2x. 62 00:02:51,076 --> 00:02:56,371 And the antiderivative of 2X, we have already seen, 63 00:02:56,371 --> 00:03:01,304 is X^2, is X^2 plus C. 64 00:03:01,304 --> 00:03:03,216 Now, you might be saying: 65 00:03:03,216 --> 00:03:04,689 Why do we use this type 66 00:03:04,689 --> 00:03:07,170 of crazy notation? 67 00:03:07,170 --> 00:03:08,519 It will become more obvious when we 68 00:03:08,519 --> 00:03:10,901 study the definitive integral, and areas under curves, 69 00:03:10,901 --> 00:03:12,363 and taking sums of 70 00:03:12,363 --> 00:03:16,185 rectangles, in order to approximate the area of the curve. 71 00:03:16,185 --> 00:03:19,500 Here, it really should just be viewed as a notation for 72 00:03:19,500 --> 00:03:21,839 antiderivatives. 73 00:03:21,839 --> 00:03:24,701 And, this notation right over here, 74 00:03:24,701 --> 00:03:26,160 this whole expression, is called: 75 00:03:26,160 --> 00:03:28,644 the indefinite integral. 76 00:03:28,644 --> 00:03:31,810 The indefinite...indefinite... 77 00:03:31,810 --> 00:03:36,825 indefinite integral of 2X 78 00:03:36,825 --> 00:03:37,845 which is another way of just saying: 79 00:03:37,845 --> 00:03:40,300 the antiderivative of 80 00:03:40,300 --> 00:03:42,300 2X.