1 00:00:00,012 --> 00:00:03,446 [MUSIC]. You've heard of the, ask not what your 2 00:00:03,446 --> 00:00:08,139 country can do for you, but what you can do for your country rule. 3 00:00:08,139 --> 00:00:13,187 These chiastic rules for limits. The limit's the sum of the limits. 4 00:00:13,187 --> 00:00:17,400 Same thing is true for derivatives. Let's go to the board. 5 00:00:17,400 --> 00:00:21,772 Here's the rule for derivatives. The derivative of f + g. 6 00:00:21,772 --> 00:00:24,963 Is the derivative of f + the derivative of g. 7 00:00:24,963 --> 00:00:29,712 In short, the derivative of the sum is the sum of the derivatives. 8 00:00:29,712 --> 00:00:35,522 Why does this make sense? Well, think back to what the derivative is measuring. 9 00:00:35,522 --> 00:00:41,162 The derivative is measuring how changing the input affects the output. 10 00:00:41,162 --> 00:00:47,166 In this case, I want to know how changing x affects the sum of f of x and g of x. 11 00:00:47,166 --> 00:00:51,229 Well, the sum is affected by the sum of the effects. 12 00:00:51,229 --> 00:00:55,622 The sum of the derivative of f and the derivative of g. 13 00:00:55,622 --> 00:01:00,860 Let' see this in a specific case. Here's a specific case. 14 00:01:00,860 --> 00:01:06,148 The function f(x) = x^3 + x^2. Let's differentiate this. 15 00:01:06,148 --> 00:01:08,703 I'm going to calculate d / dx (x^3+3=x^2). 16 00:01:09,722 --> 00:01:15,812 Now this is a derivative of a sum, which is the sum of derivatives. 17 00:01:15,812 --> 00:01:21,476 Now we have to figure out what's the derivative of x ^ 3 and what's the 18 00:01:21,476 --> 00:01:25,254 derivative of x^2. That's the power law. 19 00:01:25,254 --> 00:01:30,266 derivative of x^3 is 3x^2, and the derivative of x^2 is 2x. 20 00:01:30,266 --> 00:01:35,736 And now there's no more d/dx's. We've calculated the derivative. 21 00:01:35,736 --> 00:01:41,491 The derivative of x^3+x^2+2x. Once we know this, we can figure out 22 00:01:41,491 --> 00:01:45,318 where the derivative is positive and where it's negative. 23 00:01:45,318 --> 00:01:50,659 So, the derivative was 3x^3+2x and I want to know where that's positive and 24 00:01:50,659 --> 00:01:55,537 negative, which values of x make that bigger than 0, which values of x make 25 00:01:55,537 --> 00:02:00,785 that less than 0. one approach to thinking about this is to 26 00:02:00,785 --> 00:02:05,666 factor through the x^2+2x. I can write that as x(3x+2). 27 00:02:05,666 --> 00:02:12,031 And once I factor it like this I can figure out the SIGN of this by figuring 28 00:02:12,031 --> 00:02:15,872 out the SIGN of these two terms separately. 29 00:02:15,872 --> 00:02:19,673 visualize this as a direct whole number line. 30 00:02:19,673 --> 00:02:24,570 So x, here's a number line. X is positive when it is bigger than 0 31 00:02:24,570 --> 00:02:29,392 and negative when X is less than 0. That's not too complicated. 32 00:02:29,392 --> 00:02:33,909 Well look 3x+2. Well, 3x+2, I draw a number line for 33 00:02:33,909 --> 00:02:37,691 that. The exciting point is -2/3. 34 00:02:37,691 --> 00:02:42,671 When x is less than -2/3, 3x+2 is negative. 35 00:02:42,671 --> 00:02:47,852 And when x is bigger than -2/3's, then 3x+2. 36 00:02:47,852 --> 00:02:51,255 Is positive. Now, I really don't care about x and 3x+2 37 00:02:51,255 --> 00:02:54,274 separately. I want to put them together right. 38 00:02:54,274 --> 00:02:57,886 I want to know when their product is positive or negative. 39 00:02:57,886 --> 00:03:02,142 So, I write down the product x * 3x+2 make a new number line here. 40 00:03:02,142 --> 00:03:08,027 I'll record both of these points -2/3 and 0 then I can think about what happens. 41 00:03:08,027 --> 00:03:12,820 When x is less than -2/3 then x is negative and 3x+2 is negative. 42 00:03:12,820 --> 00:03:17,452 So the product is a negative * a negative, which is positive. 43 00:03:17,452 --> 00:03:23,734 When x is between -2/3 and 0, then x is negative but 3x+2 is positive and a 44 00:03:23,734 --> 00:03:27,756 negative times a positive number is negative. 45 00:03:27,756 --> 00:03:34,357 And finally, when x is bigger than 0, well then x is positive and also 3x+2 is 46 00:03:34,357 --> 00:03:36,841 positive. So the Is positive. 47 00:03:36,841 --> 00:03:42,508 So, here on this number line, I've recorded the information about when 3x^2 48 00:03:42,508 --> 00:03:47,503 + 2x is positive or negative. Now we can use this information to say 49 00:03:47,503 --> 00:03:52,772 something about the graph. Here's the graph of the function x^3+x^2. 50 00:03:52,772 --> 00:03:58,081 Goes up, down and up, And that's exactly what you'd expect from the derivative, 51 00:03:58,081 --> 00:04:03,069 right? We calculated before that if you're standing to the left of - 2/3's, 52 00:04:03,069 --> 00:04:07,732 then the derivative was positive, and indeed the functions going up. 53 00:04:07,732 --> 00:04:11,319 Up. Now once you get to -2/3, the derivative 54 00:04:11,319 --> 00:04:17,816 is 0 but then over here, between -2/3 and 0 the derivative is negative and indeed 55 00:04:17,816 --> 00:04:24,405 the graph is moving down until you get to 0 when the derivative of this function is 56 00:04:24,405 --> 00:04:30,512 positive again and the graph is going up. Look, the sign of the derivative 57 00:04:30,512 --> 00:04:35,218 Positive, negative, positive is reflected in the direction that this graph is 58 00:04:35,218 --> 00:04:38,058 moving. Increasing, decreasing, increasing. 59 00:04:38,058 --> 00:04:41,303 Incredible. By being able to differentiate x^2+x^3, 60 00:04:41,303 --> 00:04:45,060 we're able to gain real insight into the graph of the function. 61 00:04:45,060 --> 00:04:49,775 We're not just plotting a whole bunch of points and hoping that we can fill it in 62 00:04:49,775 --> 00:04:55,655 with a straight line. By looking at the derivative, we know 63 00:04:55,655 --> 00:05:01,308 that the function is increasing and decreasing. 64 00:05:01,308 --> 00:05:06,733 We're able to say something, for sure. [MUSIC]