1 00:00:05,040 --> 00:00:07,120 Somebody tell me where are these two things? 2 00:00:09,240 --> 00:00:15,693 Point-slope form. Okay, point-slope form. Yes, close. They are definitely point- 3 00:00:15,693 --> 00:00:16,580 slope form. 4 00:00:26,140 --> 00:00:37,578 But usually we don't put dy dx here. What do we normally put? M. M is the slope. 5 00:00:37,578 --> 00:00:47,300 Since I have dy dx or F prime, that's the slope of the tangent line. 6 00:00:47,300 --> 00:01:11,793 Okay. These are simply equations of the tangent line. Equations of the tangent line 7 00:01:11,793 --> 00:01:16,220 to y at x1, y1. 8 00:01:20,440 --> 00:01:27,600 So some function y. And why do I notice the function y for the top one? Because of 9 00:01:27,600 --> 00:01:34,760 this, is the derivative of this some function y, which is not the same function y. 10 00:01:35,800 --> 00:01:39,665 This notation is a little bit better on bottom because this is definitely the 11 00:01:39,665 --> 00:01:41,120 equation of the tangent line. 12 00:01:54,740 --> 00:02:07,820 To the function F at a comma F of a. 13 00:02:09,960 --> 00:02:15,549 It's just using two different notations. Both slopes are derivatives. So that's the 14 00:02:15,549 --> 00:02:20,600 slope of the tangent line. So these are the equations of the tangent lines. 15 00:02:25,220 --> 00:02:33,260 At either x1, y1 of this function y or a comma F of a. Same thing. 16 00:02:33,500 --> 00:02:35,660 What the function is this? 17 00:02:36,600 --> 00:02:37,600 Polynomial. 18 00:02:38,440 --> 00:02:39,440 Polynomial. 19 00:02:39,440 --> 00:02:41,200 Okay. Find the equation. 20 00:02:48,220 --> 00:02:49,260 The tangent line. 21 00:02:49,260 --> 00:02:49,320 Sub urban. 22 00:02:53,400 --> 00:02:54,040 One. 23 00:02:54,760 --> 00:02:58,280 X equals zero. 24 00:03:01,140 --> 00:03:01,320 Oh. 25 00:03:02,380 --> 00:03:09,580 It's all right. 26 00:03:16,040 --> 00:03:17,460 So what do we got to do first? 27 00:03:22,000 --> 00:03:29,319 So to find the equation of the tangent line, are we going to use this one or this 28 00:03:29,319 --> 00:03:29,680 one? 29 00:03:37,620 --> 00:03:40,120 The bottom, why the bottom? 30 00:03:44,300 --> 00:03:49,120 More importantly, this is not y equals, this is f of x equals. 31 00:03:51,340 --> 00:03:59,400 Okay, so we need to find f prime of a, the slope of the tangent line. 32 00:03:59,820 --> 00:04:02,040 To find f prime of a, we need to find what? 33 00:04:05,880 --> 00:04:08,380 f prime of x for any x value. 34 00:04:08,640 --> 00:04:17,936 Six x to the fifth minus thirty-five x to the fourth plus sixteen x cubed minus 35 00:04:17,936 --> 00:04:18,760 three x 36 00:04:18,760 --> 00:04:23,160 squared, plus six x plus two. 37 00:04:24,080 --> 00:04:24,420 There we go. 38 00:04:27,920 --> 00:04:33,760 Very simple, using a bunch of power rules and sum-and-difference rules. 39 00:04:34,760 --> 00:04:39,860 Okay, and then we need f prime of a, right? 40 00:04:40,580 --> 00:04:49,280 f prime of 0, 0 minus 0 plus 0 minus 0 plus 0 plus 2. 41 00:04:50,740 --> 00:04:52,420 f prime of 0 is 2. 42 00:04:56,560 --> 00:05:02,100 And the only other thing we need is f of 0. 43 00:05:02,980 --> 00:05:04,400 The y value. 44 00:05:06,960 --> 00:05:09,500 0, 0, 0, 0, 0 minus 8. 45 00:05:15,740 --> 00:05:24,420 So y minus f of 0 equals f prime of 0 times x minus 0. 46 00:05:29,980 --> 00:05:31,420 What was the slope? Two. 47 00:05:49,300 --> 00:05:55,000 It's kind of ugly. It looks like it only has two zeros, two real zeros. 48 00:05:56,040 --> 00:05:59,220 How many complex zeros does this have? 49 00:06:04,580 --> 00:06:07,220 What's the degree of this polynomial we wrote down? 50 00:06:07,220 --> 00:06:08,120 Six. 51 00:06:08,360 --> 00:06:08,500 Six. 52 00:06:11,980 --> 00:06:13,060 Guaranteed to have six. 53 00:06:14,320 --> 00:06:20,780 There's a maximum of five turning points or minimums and maximums. 54 00:06:23,820 --> 00:06:29,011 Anyways, we found the slope of the, or sorry, we found the equation of the tangent 55 00:06:29,011 --> 00:06:30,340 line, which was what? 56 00:06:30,340 --> 00:06:33,700 Y equals, say again? 57 00:06:35,440 --> 00:06:37,500 Two x minus six? 58 00:06:38,680 --> 00:06:39,740 Minus eight. 59 00:06:42,000 --> 00:06:43,840 Okay, we see it's tangent right there. 60 00:06:47,280 --> 00:06:48,920 If I zoom in, 61 00:06:53,440 --> 00:06:56,640 you can't even tell the difference between those right there, right? 62 00:06:56,640 --> 00:07:05,120 They only touch at one single part. Right there at zero negative 8. 63 00:07:08,960 --> 00:07:13,280 Nearby x equals zero, when I zoom in, we're only looking at very small x values. 64 00:07:15,660 --> 00:07:18,280 Nearby, the two functions look exactly the same. 65 00:07:19,240 --> 00:07:20,580 They're just slightly off. 66 00:07:20,580 --> 00:07:22,800 We agreed? Okay. 67 00:07:23,110 --> 00:07:25,910 So this is a crazy function. 68 00:07:42,770 --> 00:07:45,710 Near x equals zero. 69 00:07:48,110 --> 00:07:53,890 This crazy function is approximately this function. 70 00:07:56,830 --> 00:08:05,910 f of x is approximately 2x minus 8, which is a very simple linear function. 71 00:08:09,190 --> 00:08:10,210 Okay. 72 00:08:15,130 --> 00:08:15,410 So, 73 00:08:17,110 --> 00:08:19,530 if we want to approximate 74 00:08:24,970 --> 00:08:26,770 f of, let's say, 75 00:08:27,050 --> 00:08:28,050 point one. 76 00:08:31,790 --> 00:08:35,330 Actually, let's say we want to find f of point one. 77 00:08:37,970 --> 00:08:39,690 It would take a lot of work, right? 78 00:08:40,670 --> 00:08:47,081 Point one to the sixth power minus seven times point one to the fifth power plus 79 00:08:47,081 --> 00:08:52,050 four times point one to the fourth power minus point one cubed 80 00:08:52,050 --> 00:08:59,330 plus three times point one squared plus two times point one minus eight. 81 00:09:00,890 --> 00:09:04,430 We're going to take a lot of work and probably need a calculator. 82 00:09:05,510 --> 00:09:06,210 Yeah. Okay. 83 00:09:08,550 --> 00:09:13,730 We'll plug that in in a second, but the approximation is very simple. 84 00:09:15,950 --> 00:09:18,210 It's we plug it into this function. 85 00:09:18,610 --> 00:09:20,970 Two times point one minus eight, 86 00:09:22,710 --> 00:09:27,870 which is what point two minus eight, which is seven point. 87 00:09:31,390 --> 00:09:33,430 Negative seven point eight. 88 00:09:33,430 --> 00:09:35,370 Seven point eight. Is that right? Yeah. Okay. 89 00:09:38,470 --> 00:09:42,330 So if we want to find values of f 90 00:09:44,390 --> 00:09:48,510 nearby the x equals zero, like point one is pretty close, point zero. 91 00:09:50,250 --> 00:09:55,550 It's very simple. We find the equation of the tangent line. 92 00:09:57,170 --> 00:10:01,350 It's simply plug that value into the equation of the tangent line. 93 00:10:03,590 --> 00:10:07,850 So this is called the is called linear approximation. 94 00:10:09,330 --> 00:10:11,450 Actually, it's called three different things. 95 00:10:13,090 --> 00:10:14,090 Linear approximation. 96 00:10:19,130 --> 00:10:24,730 Okay. We approximated some very complicated function with a simple linear function. 97 00:10:25,190 --> 00:10:27,110 That's why it's called linear approximation. 98 00:10:27,890 --> 00:10:33,610 Or it's called linearization. 99 00:10:38,010 --> 00:10:40,170 Why is it called that? 100 00:10:42,690 --> 00:10:47,290 We turn a nonlinear function into a linear function. 101 00:10:48,890 --> 00:10:52,770 So it's the linearization of the function. We turn it to a linear function. 102 00:10:54,070 --> 00:11:02,330 Okay. We linearize it. It's also called the tangent line approximation. 103 00:11:06,350 --> 00:11:08,450 Why is it called that? 104 00:11:09,110 --> 00:11:10,410 It's the tangent line. 105 00:11:11,370 --> 00:11:15,135 We're approximated with the tangent slope, or sorry, the equation of the tangent 106 00:11:15,135 --> 00:11:15,370 line. 107 00:11:16,510 --> 00:11:18,530 It all means the same thing. 108 00:11:25,050 --> 00:11:30,550 So this is it. Any function. 109 00:11:36,130 --> 00:11:42,530 Well, actually before I write that down, if we take this thing right here 110 00:11:46,110 --> 00:11:49,490 and solve for this guy, which is our linear equation, 111 00:11:51,610 --> 00:11:57,810 we get f prime of a times x minus a plus f of a. 112 00:11:59,470 --> 00:12:02,770 This is the equation of the tangent line. 113 00:12:03,870 --> 00:12:09,890 And this linear function here, y, is our linear function. 114 00:12:11,150 --> 00:12:23,710 Okay. So it is f of x is approximately f prime of a times x minus a plus f of a. 115 00:12:30,530 --> 00:12:34,750 This is known as the linearization of a function or linear approximation. 116 00:12:34,750 --> 00:12:44,590 Let's just call it linear approximation of f. 117 00:12:45,070 --> 00:12:52,750 But it's only the linear approximation of f at x equals whatever number a is. 118 00:12:53,010 --> 00:13:02,550 Let's find the linear approximation for the same function at x equals 1. 119 00:13:05,710 --> 00:13:19,185 Okay. At x equals 1, f of x is approximately f prime of 1 times x minus 1 plus f of 120 00:13:19,185 --> 00:13:19,510 1. 121 00:13:20,610 --> 00:13:27,717 And we just have to find two things, the derivative at 1 and the function value at 122 00:13:27,717 --> 00:13:27,890 1. 123 00:13:28,550 --> 00:13:31,030 It's only different than the previous problem. 124 00:13:35,010 --> 00:13:40,710 So where we're at? Let's do that up here real quick. 125 00:13:44,630 --> 00:13:46,710 This is going to take a little bit more work. 126 00:13:50,850 --> 00:14:00,110 f prime of 1 is 6 minus 35 plus 16 minus 3 plus 6 plus 2. 127 00:14:00,450 --> 00:14:01,710 Could somebody do that work? 128 00:14:02,310 --> 00:14:07,450 So f of 1, what did we say it was? 129 00:14:07,670 --> 00:14:08,090 Negative 6. 130 00:14:08,370 --> 00:14:10,790 f prime of 1 is? 131 00:14:12,210 --> 00:14:12,530 Negative 8. 132 00:14:12,750 --> 00:14:13,290 Negative 8. 133 00:14:13,830 --> 00:14:25,430 So our function is approximately negative 8 times x minus 1 plus minus 8 minus 6. 134 00:14:30,670 --> 00:14:33,630 You just want to keep it real simple. 135 00:14:33,970 --> 00:14:37,130 Negative eight x plus two. 136 00:14:56,790 --> 00:14:59,830 And let's approximate at.9. 137 00:14:59,970 --> 00:15:01,130 Is that close to 1? 138 00:15:06,330 --> 00:15:07,290 Oh. 139 00:15:08,410 --> 00:15:19,770 What is that negative.72 plus 2, which is 1 point? 140 00:15:26,430 --> 00:15:27,770 1.28, is that right? 141 00:15:39,270 --> 00:15:42,890 Can somebody confirm my basics arithmetic? 142 00:15:46,110 --> 00:15:49,090 Is that negative 7.2? 143 00:15:49,630 --> 00:15:49,850 Yes. 144 00:15:51,130 --> 00:16:00,890 Is that negative 5.2? 145 00:16:02,070 --> 00:16:03,170 There we go. 146 00:16:04,760 --> 00:16:10,160 Sorry, it was y minus, or sorry, y equals, what did we get? 147 00:16:11,540 --> 00:16:12,440 Negative 8. 148 00:16:22,440 --> 00:16:22,960 Okay. 149 00:16:33,840 --> 00:16:36,220 So this is the approximation right here. 150 00:16:36,780 --> 00:16:37,700 Let me turn off the old one. 151 00:16:39,300 --> 00:16:39,820 Okay. 152 00:16:39,970 --> 00:16:43,170 Again, it's only the approximation near x equals 1. 153 00:16:44,450 --> 00:16:44,850 Okay. 154 00:16:45,490 --> 00:16:49,568 Every different x value, we have a different linear equation, different change in 155 00:16:49,568 --> 00:16:51,230 line, different linear equations. 156 00:16:54,890 --> 00:16:55,270 Oops. 157 00:17:00,510 --> 00:17:00,530 Okay. 158 00:17:00,730 --> 00:17:01,670 Let's do some more examples. 159 00:17:01,830 --> 00:17:03,770 Let's keep it simple. 160 00:17:04,990 --> 00:17:08,290 Again, our function. 161 00:17:10,510 --> 00:17:18,493 Just the point slope formula, solve for the y, it's f prime of a times x minus a 162 00:17:18,493 --> 00:17:19,690 plus f of a. 163 00:17:22,410 --> 00:17:23,970 This should say approximation. 164 00:17:34,270 --> 00:17:37,070 What is f prime? 165 00:18:04,250 --> 00:18:07,050 What is f prime? 166 00:18:12,790 --> 00:18:29,410 f of x is approximately one times x minus zero plus zero. 167 00:18:29,410 --> 00:18:36,670 Sorry, let's just say f of x is approximately x. 168 00:18:38,870 --> 00:18:41,750 This is our linear function. 169 00:18:42,950 --> 00:18:53,430 So sine of x is approximately just our function x near x equals 0. 170 00:18:54,450 --> 00:18:56,490 In fact, let's look at that real quick. 171 00:18:56,550 --> 00:18:57,830 This is kind of handy. 172 00:18:59,750 --> 00:19:04,190 For example, we're going to find sine of 0.2. 173 00:19:05,110 --> 00:19:06,670 We definitely need a calculator for that, right? 174 00:19:08,310 --> 00:19:10,850 Can somebody take out the calculator and tell me what that's equal to? 175 00:19:13,330 --> 00:19:15,850 It's approximately 0.2. 176 00:19:16,030 --> 00:19:18,690 Oh, got to make sure we're in what mode? 177 00:19:19,430 --> 00:19:22,470 It has to be in radians. Let's see what this is. 178 00:19:22,850 --> 00:19:23,110 Oh, it is. 179 00:19:26,670 --> 00:19:29,750 So sine of 0.2 is about 0.198. 180 00:19:32,150 --> 00:19:32,850 Slightly less. 181 00:19:37,550 --> 00:19:39,730 But 0.2 is pretty good approximation. 182 00:19:40,730 --> 00:19:44,070 It's kind of cool. 183 00:19:44,230 --> 00:19:45,990 It's a complicated function, right? 184 00:19:45,990 --> 00:19:47,370 Well, it's not that complicated. 185 00:19:51,830 --> 00:19:52,910 What's that like that? 186 00:19:54,290 --> 00:19:55,330 See if you can do this. 187 00:19:55,990 --> 00:19:58,490 Find the equation. 188 00:20:00,390 --> 00:20:12,110 Find the linear approximation at x equals 1. 189 00:20:27,750 --> 00:20:29,150 Okay. 190 00:20:51,590 --> 00:20:53,790 What's the derivative of the natural? 191 00:20:58,310 --> 00:20:58,830 Okay. 192 00:20:58,970 --> 00:21:01,750 Which is just one. That's right. 193 00:21:06,390 --> 00:21:08,010 And so that's the derivative. 194 00:21:10,110 --> 00:21:12,870 So f prime of one is simply one over one. 195 00:21:15,310 --> 00:21:18,270 And what is the natural log of one, everybody? 196 00:21:18,270 --> 00:21:19,410 Zero. 197 00:21:23,270 --> 00:21:33,290 So our function, ln x, is approximately one times 198 00:21:33,890 --> 00:21:35,990 X minus one plus zero. 199 00:21:41,190 --> 00:21:43,490 There is x minus one. 200 00:21:43,850 --> 00:21:47,210 Which is that line right there. 201 00:21:54,350 --> 00:21:59,850 So let's approximate ln. What's the number near one? 202 00:22:01,290 --> 00:22:02,830 One point two. 203 00:22:08,210 --> 00:22:12,390 That is 1.2 minus one. 204 00:22:13,590 --> 00:22:18,030 Let's see what ln of 1.2 is. 205 00:22:28,950 --> 00:22:31,250 Sorry, ln of one point two. 206 00:22:36,570 --> 00:22:39,930 So we got point two. It's really point one eight. 207 00:22:40,710 --> 00:22:41,750 It's a pretty good approximation. 208 00:22:42,450 --> 00:22:46,370 The farther we get away from our a value, in this case it was one. 209 00:22:46,910 --> 00:22:47,710 So the worse approximation. 210 00:22:49,330 --> 00:22:51,190 The closer we get, the better approximation. 211 00:22:52,250 --> 00:22:54,010 There is a way to calculate the error. 212 00:22:54,750 --> 00:22:58,870 We don't need to talk about that, but you don't need that for this. 213 00:23:01,350 --> 00:23:03,770 Actually going back here. 214 00:23:08,430 --> 00:23:14,843 Without actually plugging in the calculator, how can we tell if this is going to be 215 00:23:14,843 --> 00:23:19,170 an overestimate or an underestimate of the actual value? 216 00:23:20,530 --> 00:23:21,790 We should discuss that. 217 00:23:22,270 --> 00:23:25,957 Looking at the graph, how can we tell if it's going to be an overestimate or an 218 00:23:25,957 --> 00:23:26,610 underestimate? 219 00:23:29,850 --> 00:23:30,410 What's that? 220 00:23:32,530 --> 00:23:37,890 The tangent line is higher, greater than the function. 221 00:23:38,090 --> 00:23:40,430 So it's definitely going to be an overestimate. 222 00:23:40,810 --> 00:23:44,170 Let's say this is one point two. 223 00:23:44,170 --> 00:23:51,690 And we're going to learn later, because this is curving down, called concave down. 224 00:23:54,590 --> 00:23:58,190 We're going to be able to tell whether it's an overestimate or an underestimate. 225 00:23:58,630 --> 00:24:00,570 Next question, approximate. 226 00:24:06,470 --> 00:24:12,710 The square root of nine point three. 227 00:24:15,910 --> 00:24:17,430 Here, I just give you a value. 228 00:24:21,930 --> 00:24:23,750 So we're going to have to come up with two things. 229 00:24:23,910 --> 00:24:30,470 We're going to have to come up with a function and an A value where we can easily 230 00:24:30,470 --> 00:24:33,710 find the function value at that A value. 231 00:24:37,430 --> 00:24:39,310 So what function are we going to come up with? 232 00:24:43,730 --> 00:24:45,510 Let's go with the square root of X. 233 00:24:47,810 --> 00:24:48,950 And what A value? 234 00:24:49,110 --> 00:24:52,450 What value do we know the square root of nearby? 235 00:24:52,990 --> 00:24:53,490 Nine point three? 236 00:24:54,210 --> 00:24:54,730 Nine. 237 00:24:56,490 --> 00:25:02,950 So F prime of X is, what is that? 238 00:25:03,230 --> 00:25:07,370 One half X to negative one half. 239 00:25:08,430 --> 00:25:11,210 One over two square root x. 240 00:25:15,730 --> 00:25:17,750 F prime of nine. 241 00:25:21,770 --> 00:25:24,090 Is that one sixth? 242 00:25:28,290 --> 00:25:31,830 And F of nine. 243 00:25:33,510 --> 00:25:35,450 Square root of nine is three. 244 00:25:39,790 --> 00:25:42,430 So our function, square root x, is approximately 245 00:25:43,650 --> 00:25:55,590 f prime of nine times x minus nine plus f of nine. 246 00:26:05,070 --> 00:26:07,810 There's our linear approximation. 247 00:26:09,290 --> 00:26:10,370 Our linearization. 248 00:26:12,270 --> 00:26:13,490 Tangent line approximation. 249 00:26:14,670 --> 00:26:15,750 Square root of X. 250 00:26:18,430 --> 00:26:20,550 And now we plug in nine point three. 251 00:26:22,630 --> 00:26:24,830 Is that what I said nine point three? 252 00:26:24,850 --> 00:26:25,210 Yes. 253 00:26:25,210 --> 00:26:25,510 Okay. 254 00:26:46,550 --> 00:26:48,310 Point three. 255 00:26:50,510 --> 00:26:53,710 What is that? 256 00:26:54,410 --> 00:26:55,630 Tangent line. 257 00:26:59,550 --> 00:27:03,390 Three over sixty. 258 00:27:07,790 --> 00:27:08,990 One twentieth. Is that right? 259 00:27:14,570 --> 00:27:18,230 Change that to decimals. That's three point oh five. 260 00:27:23,870 --> 00:27:25,830 So that's a pretty good approximate. 261 00:27:25,970 --> 00:27:33,750 Well, I don't know how good it is, but we'll do that with some very simple math. 262 00:27:33,750 --> 00:27:39,950 Obviously all we got to know what derivatives are. All our rules. 263 00:27:40,510 --> 00:27:46,570 At least the power rule. Let's see, what is the square root of 9.3? 264 00:27:51,570 --> 00:27:54,370 What do we get? Three point oh five. 265 00:27:55,210 --> 00:27:57,130 Three point oh four nine. 266 00:27:57,970 --> 00:27:59,210 Pretty good. 267 00:28:05,390 --> 00:28:09,606 So linearization has everything to do with finding the equation of the tangent line 268 00:28:09,606 --> 00:28:10,470 to some function. 269 00:28:11,510 --> 00:28:14,350 Finding the derivative and plugging in the values. 270 00:28:14,530 --> 00:28:19,690 Let me draw you something real quick and then I'll explain it. 271 00:28:28,450 --> 00:28:29,810 This is some function. 272 00:28:34,490 --> 00:28:35,970 And it's some X value. 273 00:28:39,750 --> 00:28:45,330 This would be the tangent line. 274 00:28:51,690 --> 00:28:52,290 Okay. 275 00:28:54,950 --> 00:28:58,230 And if you have some other X value. 276 00:29:21,550 --> 00:29:24,130 Let me do different colors here. 277 00:29:24,690 --> 00:29:28,230 This right here is what's known as delta X. 278 00:29:28,370 --> 00:29:29,550 The change in X from A. 279 00:29:30,290 --> 00:29:31,590 This is our delta X right here. 280 00:29:34,270 --> 00:29:36,490 And this is our actual change in Y. 281 00:29:37,930 --> 00:29:38,510 Delta Y. 282 00:29:40,490 --> 00:29:40,930 Okay. 283 00:29:41,610 --> 00:29:42,570 Does this make sense so far? 284 00:29:43,650 --> 00:29:43,870 Okay. 285 00:29:44,410 --> 00:29:45,310 We have two different different directions. 286 00:29:45,310 --> 00:29:52,450 One is called dy and the other is called dx. 287 00:29:56,390 --> 00:30:00,270 This delta X is the same as our DX. 288 00:30:03,110 --> 00:30:11,510 Our differential DY is this distance right here. 289 00:30:13,030 --> 00:30:13,610 DY. 290 00:30:15,430 --> 00:30:23,793 It's the change from our equation of the tangent line as opposed to just the actual 291 00:30:23,793 --> 00:30:26,110 change in the function. 292 00:30:27,770 --> 00:30:29,510 That's what the differentials are. 293 00:30:30,270 --> 00:30:31,730 You don't really need to know that. 294 00:30:33,090 --> 00:30:33,230 Okay. 295 00:30:33,310 --> 00:30:36,110 But this was just a brief explanation. 296 00:30:36,950 --> 00:30:38,190 You don't have to memorize that. 297 00:30:38,190 --> 00:30:40,910 But that's what they are. 298 00:30:43,850 --> 00:30:45,170 And it's very simple. 299 00:30:46,690 --> 00:30:52,490 dy over dx equals f prime of x. 300 00:30:52,570 --> 00:30:53,730 Those mean the same thing, right? 301 00:30:55,570 --> 00:30:55,890 Yes. 302 00:30:59,710 --> 00:31:03,350 This is not a fraction even though it looks like a fraction, right? 303 00:31:04,770 --> 00:31:05,770 What is this? 304 00:31:05,770 --> 00:31:09,670 This is the derivative of this function Y with respect to X. 305 00:31:10,650 --> 00:31:16,510 But all you have to do is pretend it is a fraction and you multiply. 306 00:31:16,850 --> 00:31:18,090 And that's not really what's happening. 307 00:31:18,910 --> 00:31:20,350 Multiply both sides by DX. 308 00:31:22,650 --> 00:31:24,030 It looks like these cancel. 309 00:31:26,950 --> 00:31:27,890 And we get this. 310 00:31:32,570 --> 00:31:33,630 And this is it. 311 00:31:33,630 --> 00:31:41,610 Our differential dy is equal to the derivative times the differential dx. 312 00:31:49,530 --> 00:31:51,930 And we just have to know this. 313 00:31:52,110 --> 00:31:53,350 This is going to be very important later. 314 00:31:55,270 --> 00:31:55,350 Okay. 315 00:31:55,830 --> 00:31:57,170 So quick example. 316 00:32:04,470 --> 00:32:05,750 Sine of x. 317 00:32:06,610 --> 00:32:11,570 Find the differential dy. 318 00:32:14,130 --> 00:32:22,450 All we do is we take the derivative, which is cosine of X, right? 319 00:32:25,910 --> 00:32:29,750 And the derivative is the same thing as DY DX. 320 00:32:29,750 --> 00:32:32,150 We set it equal to cosine of x. 321 00:32:35,110 --> 00:32:37,810 Just pretend that you're multiplying by DX. 322 00:32:41,170 --> 00:32:42,450 And we get our differentials. 323 00:32:42,670 --> 00:32:46,290 dy is cosine of x times dx. 324 00:32:51,210 --> 00:32:57,170 dy is the derivative of the function times our differential dx. 325 00:32:58,070 --> 00:33:00,430 That's all you got to know how to do. 326 00:33:01,910 --> 00:33:08,470 We could find this DY here for that function we just did. 327 00:33:08,930 --> 00:33:14,150 dy for this function is cosine of x times dx. 328 00:33:16,230 --> 00:33:20,610 And that will give you the change right there. 329 00:33:21,430 --> 00:33:22,070 You don't really need that. 330 00:33:22,070 --> 00:33:28,635 All you got to do is you find the derivative, which is DY DX, multiplied by DX, even 331 00:33:28,635 --> 00:33:31,370 though that's not what's happening. 332 00:33:32,710 --> 00:33:33,410 And that's your answer. 333 00:33:34,990 --> 00:33:35,470 Okay. 334 00:33:36,370 --> 00:33:37,910 What if I want to find DX? 335 00:33:38,270 --> 00:33:39,110 What is it? 336 00:33:44,530 --> 00:33:46,070 We divide both sides by cosine. 337 00:33:48,970 --> 00:33:52,430 One over cosine times DY. 338 00:33:56,630 --> 00:33:58,610 And sometimes we write it like this. 339 00:33:58,750 --> 00:34:00,850 dy over cosine of x. 340 00:34:02,010 --> 00:34:02,330 Okay. 341 00:34:02,670 --> 00:34:04,030 This is going to be very important later. 342 00:34:05,570 --> 00:34:09,590 All you need to know is dy equals the derivative times dx. 343 00:34:10,590 --> 00:34:10,770 Okay.