1 00:00:05,000 --> 00:00:09,013 Alright, let's start with a little warm-up before we get into actually the new 2 00:00:09,013 --> 00:00:09,763 material. 3 00:00:20,120 --> 00:00:32,020 Find the equation of the tangent line at x = 0. 4 00:00:38,100 --> 00:00:39,560 X equals zero. 5 00:00:45,100 --> 00:00:51,560 To find the equation of the tangent line, we need its slope. 6 00:00:51,820 --> 00:00:54,940 To find the slope we need to find the derivative. 7 00:00:54,940 --> 00:01:00,620 We will use several derivative rules. 8 00:01:04,060 --> 00:01:05,520 We're not going to use the definition. 9 00:01:06,960 --> 00:01:08,060 What's the first rule? 10 00:01:09,640 --> 00:01:14,440 Since we have a sum of a bunch of power functions, we're going to use the... 11 00:01:15,520 --> 00:01:16,420 No. 12 00:01:16,820 --> 00:01:17,720 Sum of... 13 00:01:18,920 --> 00:01:19,800 The sum... 14 00:01:19,800 --> 00:01:20,700 Did we name it? 15 00:01:20,780 --> 00:01:21,680 Did I call it something? 16 00:01:21,820 --> 00:01:26,340 Did I write it down, but we didn't call it something? 17 00:01:28,660 --> 00:01:30,620 Use the sum and difference rule. 18 00:01:31,780 --> 00:01:36,340 The derivative of a sum is the sum of the derivatives. 19 00:01:37,220 --> 00:01:43,100 For each power function, apply the constant multiple rule and the power rule. The 20 00:01:43,100 --> 00:01:47,720 derivative of 5x cubed starts with 3 times 5, or 15. 21 00:01:49,140 --> 00:01:51,280 Subtract one from the exponent to get the second power. 22 00:01:52,780 --> 00:02:05,454 These are easy enough, hopefully we can just do it in our head, we don't have to 23 00:02:05,454 --> 00:02:06,200 write 24 00:02:06,200 --> 00:02:15,760 The derivative of 7x is 7, and the derivative of the constant is 0. 25 00:02:18,580 --> 00:02:21,951 The derivative of a cubic is quadratic; the degree decreases by one because of the 26 00:02:21,951 --> 00:02:22,400 power rule. 27 00:02:22,400 --> 00:02:29,860 1 because the power rule and okay here's the derivative at any value x so we want 28 00:02:29,860 --> 00:02:34,260 Evaluate the derivative at x = 0. 29 00:02:41,000 --> 00:02:48,140 The slope of the tangent line is 7. Now find its equation. 30 00:02:51,700 --> 00:03:00,780 The y-value is f(0) = -2. 31 00:03:00,780 --> 00:03:10,080 Use point-slope form: y minus y-one equals m times x minus x-one. 32 00:03:22,600 --> 00:03:25,400 The tangent line is y = 7x - 2. 33 00:03:27,620 --> 00:03:30,340 Okay, so the power rule is very nice, yes? 34 00:03:31,820 --> 00:03:33,100 It's a lot simpler than, 35 00:03:33,240 --> 00:03:34,900 we had to use the definition of the derivative 36 00:03:34,900 --> 00:03:41,840 on that thing and take us a year to do so. 37 00:03:42,600 --> 00:03:44,320 And of course the sum and difference rule 38 00:03:44,320 --> 00:03:45,440 and the constant multiple rule. 39 00:03:47,060 --> 00:03:49,740 Okay, so that was kind of review from Wednesday. 40 00:03:50,820 --> 00:03:52,280 Today, let's talk about, 41 00:03:57,500 --> 00:03:59,740 Now find the derivative of sine. 42 00:04:05,780 --> 00:04:08,020 How are we going to do this? 43 00:04:08,480 --> 00:04:10,380 Umm, there's rules. 44 00:04:11,020 --> 00:04:11,920 There's rules? 45 00:04:15,080 --> 00:04:16,140 I don't know of any rules. 46 00:04:17,800 --> 00:04:19,320 Negative sign? 47 00:04:20,420 --> 00:04:21,520 Oh my goodness. 48 00:04:24,260 --> 00:04:25,640 Now we're going to do it. 49 00:04:26,080 --> 00:04:27,220 How are we going to find the derivative? 50 00:04:30,860 --> 00:04:31,760 Thank you. 51 00:04:32,320 --> 00:04:33,660 Use the definition of the derivative. 52 00:04:35,000 --> 00:04:35,900 Which is? 53 00:04:39,180 --> 00:04:40,580 Start with the limit as h approaches zero. 54 00:04:42,280 --> 00:04:44,000 As h approaches zero. 55 00:04:44,000 --> 00:04:51,360 Use f(x+h) minus f(x), all over h. 56 00:04:53,180 --> 00:05:09,340 This becomes sine of x plus h minus sine of x, all over h. 57 00:05:15,400 --> 00:05:17,020 Okay, now what? 58 00:05:18,480 --> 00:05:19,380 Distribute sine? 59 00:05:23,120 --> 00:05:26,780 What do you mean? 60 00:05:28,600 --> 00:05:29,580 Do something with this? 61 00:05:31,240 --> 00:05:32,140 What can we do with that? 62 00:05:33,540 --> 00:05:36,960 Sine of x plus h is not sine x plus sine h. 63 00:05:44,060 --> 00:05:45,940 Oh. Let's do an example. 64 00:05:47,940 --> 00:05:49,980 Pi over 2 plus pi over 2. 65 00:05:53,460 --> 00:05:55,420 What's pi over 2 plus pi over 2? 66 00:05:56,680 --> 00:05:58,160 Sine of pi. 67 00:05:59,740 --> 00:06:02,000 Is that sine of pi over 2? 68 00:06:02,000 --> 00:06:17,247 plus sine of pi over 2. Sine of pi is 0. What is sine of pi over 2? 1. Does 0 equal 69 00:06:17,247 --> 00:06:17,997 2? 70 00:06:18,180 --> 00:06:26,055 Definitely not. Okay, we definitely can't do that. Uh oh, we need something else. 71 00:06:26,055 --> 00:06:27,180 Any other 72 00:06:27,180 --> 00:06:28,080 What are ideas? 73 00:06:40,500 --> 00:06:42,640 Can I factor out a sign? 74 00:06:45,700 --> 00:06:48,000 You get x plus h minus x. 75 00:06:48,980 --> 00:06:50,100 Definitely can't do that. 76 00:06:50,720 --> 00:06:51,620 There's no such thing. 77 00:06:53,740 --> 00:06:55,260 Use a trigonometric identity. 78 00:06:55,500 --> 00:06:56,360 This is very important. 79 00:06:56,360 --> 00:06:57,260 Who said that? 80 00:06:57,340 --> 00:06:58,240 I said that one actually. 81 00:06:58,760 --> 00:06:59,660 You did? 82 00:06:59,760 --> 00:07:00,660 Good. 83 00:07:02,080 --> 00:07:03,480 Okay. 84 00:07:41,560 --> 00:07:43,900 I don't know, it's not an awesome thing. 85 00:07:44,160 --> 00:07:45,060 It's awesome. 86 00:07:46,300 --> 00:07:49,480 Okay, uh, so, we gotta... 87 00:07:49,480 --> 00:07:51,100 These are some of the trigonometric identities we will use. 88 00:07:52,240 --> 00:07:54,041 There are infinitely many trigonometric identities; these are the main ones for this 89 00:07:54,041 --> 00:07:54,720 lesson. 90 00:07:54,720 --> 00:07:55,740 These are the main ones. 91 00:07:56,400 --> 00:07:57,380 Which one are we gonna use? 92 00:07:58,660 --> 00:08:00,300 Use the sine sum identity. 93 00:08:01,340 --> 00:08:02,240 What'd it go? 94 00:08:04,760 --> 00:08:05,580 What? 95 00:08:05,580 --> 00:08:06,480 Uh oh. 96 00:08:07,280 --> 00:08:08,180 Let's go back here. 97 00:08:11,480 --> 00:08:16,180 The argument contains a sum, so use the sine addition identity. 98 00:08:17,460 --> 00:08:18,380 Sum a difference. 99 00:08:18,500 --> 00:08:19,720 Sum a difference. 100 00:08:19,720 --> 00:08:37,440 Use sine of alpha plus beta. 101 00:08:38,660 --> 00:08:46,000 That identity is sine alpha cosine beta plus cosine alpha sine beta. 102 00:08:46,000 --> 00:08:52,720 the plus and the minus means if it's plus then this is plus if it's minus then it's 103 00:08:53,560 --> 00:08:55,120 negative so 104 00:08:56,300 --> 00:08:57,200 Got 105 00:09:06,920 --> 00:09:11,600 With alpha = x and beta = h, it becomes sine x cosine h plus cosine x sine h. 106 00:09:13,320 --> 00:09:26,180 sine x cosine h plus cosine x sine h. 107 00:09:38,300 --> 00:09:41,400 Okay, now what? 108 00:09:47,700 --> 00:09:49,380 Any ideas? 109 00:09:51,960 --> 00:09:52,860 What? 110 00:10:00,260 --> 00:10:09,120 Group the sine x terms and the sine h terms. 111 00:10:09,120 --> 00:10:11,980 that so first let's rearrange it 112 00:10:22,900 --> 00:10:24,460 Rearrange the numerator. 113 00:10:26,140 --> 00:10:28,120 Factor out sine x and cosine x. 114 00:10:31,120 --> 00:10:32,020 This guy and this guy. 115 00:10:53,260 --> 00:10:55,300 I like that. 116 00:10:55,300 --> 00:10:58,680 Now what? 117 00:10:58,680 --> 00:11:07,480 Um, cosine h minus 1 is identity of something. 118 00:11:08,060 --> 00:11:08,960 That's it. 119 00:11:09,040 --> 00:11:10,080 That's cosine square. 120 00:11:10,080 --> 00:11:10,980 I forgot what these are. 121 00:11:11,240 --> 00:11:12,440 They have to be cosine squared. 122 00:11:12,600 --> 00:11:13,520 Oh, that's square. 123 00:11:13,740 --> 00:11:14,000 They're not. 124 00:11:14,000 --> 00:11:14,320 They're not. 125 00:11:14,320 --> 00:11:16,580 We have to be cosine squared minus 1. 126 00:11:17,040 --> 00:11:17,940 Cosine square. 127 00:11:20,440 --> 00:11:23,040 Uh, any other ideas? 128 00:11:23,040 --> 00:11:23,940 this. 129 00:11:35,920 --> 00:11:37,240 Split the expression into two limits. 130 00:11:37,580 --> 00:11:38,600 This plus that. 131 00:11:39,980 --> 00:11:53,473 This becomes the limit of sine x times cosine h minus one, over h, plus the limit of 132 00:11:53,473 --> 00:11:58,720 cosine x times sine h, over h. 133 00:11:58,720 --> 00:12:05,920 The first special limit equals zero. 134 00:12:07,320 --> 00:12:12,360 Sine x is constant with respect to h, so it may be moved outside the limit. 135 00:12:12,520 --> 00:12:16,380 You don't have to do this, but you could bring it outside the limit. 136 00:12:16,380 --> 00:12:21,280 The cosine-minus-one limit equals zero. 137 00:12:23,720 --> 00:12:27,140 The sine-over-h limit equals one. 138 00:12:28,420 --> 00:12:36,520 The result is sine x times zero plus cosine x times one. 139 00:12:37,560 --> 00:12:38,740 And there it is! 140 00:12:40,340 --> 00:12:42,360 The derivative of sine x is cosine x. 141 00:12:43,080 --> 00:12:43,980 Beautiful. 142 00:12:49,140 --> 00:12:51,420 That is math has finds. 143 00:12:51,660 --> 00:12:53,340 Please prove that derivative of sine is cosine. 144 00:12:55,920 --> 00:12:58,640 So are we gonna do this every single time? 145 00:13:00,060 --> 00:13:00,960 Definitely not. 146 00:13:01,660 --> 00:13:04,340 At this point, we just memorize. 147 00:13:06,560 --> 00:13:07,880 Derivative of sine. 148 00:13:10,220 --> 00:13:11,120 Okay next one. 149 00:13:12,600 --> 00:13:17,720 Now find the derivative of cosine. 150 00:13:30,720 --> 00:13:32,840 Let's work it out people! 151 00:13:33,760 --> 00:13:35,200 How are we going to start? 152 00:13:36,620 --> 00:13:40,740 Begin with the limit as h approaches zero of f(x+h) minus f(x), all over h. 153 00:13:40,740 --> 00:13:46,960 f of x plus h minus f of x all over h. 154 00:13:59,720 --> 00:14:02,820 Okay. I bet you could do it. 155 00:14:03,580 --> 00:14:06,000 Finish the proof or finish the limit. 156 00:15:59,580 --> 00:16:02,040 What's the next step? 157 00:16:06,560 --> 00:16:09,040 Let's switch these guys. 158 00:16:46,260 --> 00:16:50,440 Separate the expression into two limits. 159 00:16:55,320 --> 00:16:58,920 This goes to zero. 160 00:17:00,840 --> 00:17:10,040 The cosine-minus-one limit is zero and the sine-over-h limit is one, giving cosine x 161 00:17:10,040 --> 00:17:14,640 times zero minus sine x times one. 162 00:17:14,640 --> 00:17:15,540 The result is negative sine x. 163 00:17:16,920 --> 00:17:22,501 Okay, and I think when we went over graphing derivatives we graphed cosine, we 164 00:17:22,501 --> 00:17:22,900 graphed 165 00:17:22,900 --> 00:17:27,920 When we graphed the derivative of cosine, it also matched negative sine x. 166 00:17:27,920 --> 00:17:29,700 We proved it. Beautiful. 167 00:17:30,940 --> 00:17:32,880 Okay, so we we memorize this one 168 00:17:34,300 --> 00:17:35,820 at this point 169 00:17:37,560 --> 00:17:39,560 It's negative sine of x 170 00:17:40,760 --> 00:17:45,320 The derivative of sine is cosine, and the derivative of cosine is negative sine. 171 00:17:46,880 --> 00:17:50,600 All right, now we can do any polynomial. 172 00:17:51,540 --> 00:17:54,240 Any multiple of sine, any multiple of cosine. 173 00:17:55,900 --> 00:17:58,020 Now consider tangent. 174 00:18:40,680 --> 00:18:48,360 If we substitute this in, it's going to be a really ugly mess. 175 00:18:48,360 --> 00:18:50,780 The definition with tangent of x plus h would be a complicated expression. 176 00:18:52,200 --> 00:18:54,300 Okay, so we're not going to do this 177 00:18:56,760 --> 00:18:59,240 This right here is going to be real messy 178 00:19:04,620 --> 00:19:09,260 Rewrite tangent as sine x divided by cosine x. 179 00:19:15,220 --> 00:19:16,120 Now what 180 00:19:19,520 --> 00:19:20,940 That's as far as I got. 181 00:19:21,860 --> 00:19:22,760 I like this. 182 00:19:26,240 --> 00:19:30,080 The derivative of a sum is the sum of derivatives. 183 00:19:39,640 --> 00:19:41,520 Is that another rule? 184 00:19:45,900 --> 00:19:47,340 It's definitely not. 185 00:19:49,220 --> 00:19:50,120 Okay. 186 00:19:51,000 --> 00:19:54,420 The derivative of a quotient is not the quotient of the derivatives. 187 00:19:56,280 --> 00:19:59,520 It's true for sums and differences, not true for quotients. 188 00:20:00,980 --> 00:20:02,420 Okay. I don't know what I just did here. 189 00:20:03,380 --> 00:20:06,340 Long story short, we have no idea what the derivative is. 190 00:20:06,340 --> 00:20:12,620 We will learn the quotient rule after the test. 191 00:20:14,040 --> 00:20:15,920 We haven't proved that yet, right? 192 00:20:20,840 --> 00:20:24,500 So that's for next week after the test. 193 00:20:29,720 --> 00:20:32,020 We cannot finish tangent until we have the quotient rule. 194 00:20:34,180 --> 00:20:36,000 We're going to do one more. 195 00:20:39,320 --> 00:20:40,220 No more trig. 196 00:20:40,400 --> 00:20:41,960 The rest are all quotients. 197 00:20:42,140 --> 00:20:43,260 We don't have any more quotients. 198 00:20:43,260 --> 00:20:53,680 Right, uh, cosecant is one over sine, which is another quotient. 199 00:20:55,740 --> 00:20:57,140 We don't have a quotient rule yet. 200 00:20:59,220 --> 00:21:01,880 We're going to do e to the x. 201 00:21:15,740 --> 00:21:18,140 Before we start, what is our number e? 202 00:21:18,140 --> 00:21:39,240 The number e is approximately 2.71828; it is irrational. 203 00:21:39,240 --> 00:21:51,460 is the real definition of E? Where did it come from? Natural log is the inverse of E 204 00:21:51,460 --> 00:21:59,421 to the x. So we first had to find what E to the x was for natural log. Anybody 205 00:21:59,421 --> 00:21:59,840 remember 206 00:21:59,840 --> 00:22:03,920 when we first talked about it? I know we did it at Math 2.3 several years ago. 207 00:22:12,080 --> 00:22:15,520 Where do we use this in math 2.3? 208 00:22:19,780 --> 00:22:20,680 Compound interest. 209 00:22:23,160 --> 00:22:24,060 Okay. 210 00:22:24,080 --> 00:22:25,634 Compound interest uses A = P times the quantity 1 plus r over n, raised to the nt 211 00:22:25,634 --> 00:22:25,720 power. 212 00:22:25,720 --> 00:22:32,880 It's a principle, A equals principle. 213 00:22:33,360 --> 00:22:37,740 Principle times E. Let's do compound interest first. 214 00:22:41,020 --> 00:22:47,320 The factor is 1 plus r over n, raised to nt. 215 00:22:47,340 --> 00:22:48,420 Oh boy, people. 216 00:22:49,640 --> 00:22:50,840 N plus E plus. 217 00:22:51,060 --> 00:22:51,700 Yeah, that's fine. 218 00:22:51,700 --> 00:22:52,900 R is your interest rate. 219 00:22:52,900 --> 00:23:06,400 The variable n is the number of compounding periods per year. 220 00:23:06,400 --> 00:23:15,700 larger we got the more it just kept getting closer to this one number and 221 00:23:15,920 --> 00:23:21,400 And then we came up this formula for continuously compounded interest 222 00:23:22,800 --> 00:23:24,540 E to the RT, right? 223 00:23:25,660 --> 00:23:30,380 Okay, it has everything to do with this the real definition of E 224 00:23:32,840 --> 00:23:35,920 The number e is the limit as n approaches infinity. 225 00:23:41,460 --> 00:23:44,160 X goes to infinity 226 00:23:46,960 --> 00:23:52,060 Use the quantity 1 plus 1 over n, raised to the nth power. 227 00:23:56,100 --> 00:23:58,860 which is just that compound 228 00:23:58,860 --> 00:24:02,240 This is the compound-interest limit with the extra rate and time parameters removed. 229 00:24:04,400 --> 00:24:07,460 As you go to infinity, this gets very, 230 00:24:07,640 --> 00:24:09,360 very small, but then the power gets very, 231 00:24:09,360 --> 00:24:11,660 very big at the same rate. 232 00:24:12,600 --> 00:24:13,940 We're at the same time, 233 00:24:13,940 --> 00:24:20,340 It turns out this is exactly the definition of E. This is where the E comes from. 234 00:24:20,940 --> 00:24:22,940 There is an equivalent definition. 235 00:24:27,140 --> 00:24:39,000 As h approaches zero, use the quantity 1 plus h, raised to the 1 over h power. 236 00:24:40,460 --> 00:24:43,340 So, 1 over x. 237 00:24:47,760 --> 00:24:50,160 These two definitions are equivalent. 238 00:24:51,720 --> 00:24:54,560 As x goes to infinity, 1 over x goes to 0. 239 00:24:56,500 --> 00:25:00,740 And as x goes to infinity, this goes to infinity, 1 over x goes to infinity. 240 00:25:00,740 --> 00:25:03,580 sorry, one over zero goes to infinity. 241 00:25:04,840 --> 00:25:08,000 Okay, we're gonna use, well, we need this, 242 00:25:09,660 --> 00:25:12,060 except we're gonna call it this instead. 243 00:25:13,840 --> 00:25:15,460 So keep this in mind. 244 00:25:17,120 --> 00:25:19,700 Doesn't matter what the variable is, same exact thing. 245 00:25:23,120 --> 00:25:24,700 Okay, we'll come back to that. 246 00:25:26,840 --> 00:25:30,560 Now let f(x) = e to the x. 247 00:25:31,900 --> 00:25:35,540 The derivative is the limit as h approaches zero. 248 00:25:37,160 --> 00:25:38,700 Use f(x+h) minus f(x), all over h. 249 00:25:38,700 --> 00:25:42,740 minus f of x all over h. 250 00:26:00,740 --> 00:26:02,420 Okay, now what? 251 00:26:13,940 --> 00:26:15,560 Factor the numerator after using exponent rules. 252 00:26:16,100 --> 00:26:16,560 That's awesome. 253 00:26:16,560 --> 00:26:19,200 Okay, I like that except what do we got to do with this first? 254 00:26:26,700 --> 00:26:28,660 Rewrite e to the x plus h as e to the x times e to the h. 255 00:26:29,620 --> 00:26:35,240 You have addition in the exponent. We can change this to e to the x times e to the H 256 00:26:42,300 --> 00:26:44,680 Factor out e to the x. 257 00:27:03,120 --> 00:27:03,560 I 258 00:27:03,560 --> 00:27:04,780 Like it now what? 259 00:27:08,900 --> 00:27:11,700 Replace e using its limit definition. 260 00:27:13,280 --> 00:27:22,060 Use e as the limit of the quantity 1 plus h, raised to the 1 over h power. 261 00:27:22,060 --> 00:27:35,020 zero, one plus H to the one over H. What we just talked about. 262 00:27:38,180 --> 00:27:41,000 Can we have limits inside limits? 263 00:27:42,840 --> 00:27:42,880 Absolutely. 264 00:27:42,880 --> 00:27:43,780 be. 265 00:28:12,880 --> 00:28:14,420 Okay, now what? 266 00:28:21,800 --> 00:28:24,240 Use the power law for limits to bring the exponent into the inner expression. 267 00:28:26,660 --> 00:28:31,540 Is a limit of uh, let me just get the law 268 00:28:36,240 --> 00:28:41,620 I think this was the limit law the limit is whatever x approaches any number 269 00:28:44,020 --> 00:28:54,240 f of x to the any number n is n to the limit. 270 00:28:57,560 --> 00:28:59,000 This was one of the limit laws. 271 00:29:00,280 --> 00:29:03,974 So basically we're going to bring this inside the limit and put it over the 272 00:29:03,974 --> 00:29:04,724 function. 273 00:29:37,880 --> 00:29:43,440 Ok, now what? 274 00:29:48,620 --> 00:29:52,180 For a power raised to a power, multiply the exponents. 275 00:29:52,500 --> 00:29:53,400 What can we do? 276 00:29:53,880 --> 00:29:54,600 We can't solve it. 277 00:29:54,600 --> 00:29:55,380 We can't solve it. 278 00:29:55,380 --> 00:29:58,520 We can't solve it. 279 00:29:58,520 --> 00:30:04,624 Rule of exponents any base with an exponent all raised to an exponent. What do we do 280 00:30:04,624 --> 00:30:06,060 the exponents times them? 281 00:30:06,620 --> 00:30:06,700 multiply 282 00:30:06,700 --> 00:30:33,840 The exponent 1 over h times h simplifies to 1. 283 00:30:39,380 --> 00:30:40,460 Now what? 284 00:30:45,780 --> 00:30:48,960 The limit of the constant 1 is 1. 285 00:30:51,400 --> 00:30:55,840 Use the sum and difference laws for limits. 286 00:30:55,840 --> 00:30:59,280 so we can bring this negative one inside here. 287 00:31:12,040 --> 00:31:14,600 One plus h minus one simplifies to h. 288 00:31:20,740 --> 00:31:21,640 Okay. 289 00:31:23,440 --> 00:31:26,200 Now these ones add to be zero. 290 00:31:28,240 --> 00:31:29,140 Okay. 291 00:31:30,160 --> 00:31:31,720 And let me bring this guy out here. 292 00:31:31,720 --> 00:31:36,920 Factor e to the x outside the remaining limit. 293 00:31:37,540 --> 00:31:43,840 And this is the limit as h approaches 0. 294 00:31:49,620 --> 00:31:53,320 The ratio h over h equals 1. 295 00:31:53,320 --> 00:31:54,220 1. 296 00:31:54,660 --> 00:31:55,560 1. 297 00:31:56,340 --> 00:31:57,920 This is 1. 298 00:31:58,500 --> 00:31:59,760 This is 1. 299 00:32:00,360 --> 00:32:02,940 We get e to the x times 1. 300 00:32:05,060 --> 00:32:08,400 Therefore, the derivative of e to the x is e to the x. 301 00:32:11,860 --> 00:32:14,120 All this for this? 302 00:32:14,280 --> 00:32:15,880 Yes, it was awesome. 303 00:32:17,240 --> 00:32:20,360 We just proved the derivative of e to the x is e to the x. 304 00:32:22,160 --> 00:32:24,380 It's the easiest derivative to find. 305 00:32:27,560 --> 00:32:32,960 The definition-based proof depends on the limit definition of the number e. 306 00:32:32,960 --> 00:32:43,340 definition of the number E. Okay and we probably really didn't tell you that 307 00:32:43,340 --> 00:32:46,924 that's the definition because you didn't know what limits were back a couple years 308 00:32:46,924 --> 00:32:47,420 ago. 309 00:32:47,420 --> 00:32:53,200 But now we do. That is the definite. So long story short, we've memorized this 310 00:32:53,200 --> 00:33:07,940 From now on, memorize that the derivative of e to the x is e to the x. 311 00:33:07,940 --> 00:33:11,035 Any constant multiple of e to the x differentiates to the same constant multiple of 312 00:33:11,035 --> 00:33:11,860 e to the x. 313 00:33:14,260 --> 00:33:15,160 Same seed. 314 00:33:16,880 --> 00:33:19,960 Now that we have proved the derivative of e to the x, we can use the rule directly. 315 00:33:21,080 --> 00:33:29,620 We now know derivatives of polynomials, sine, cosine, and e to the x. 316 00:33:30,620 --> 00:33:33,040 Or the sum of any combination of those. 317 00:33:34,140 --> 00:33:37,080 Apply the rules to a mixed example. 318 00:33:55,480 --> 00:33:57,920 This is some crazy function. 319 00:33:57,920 --> 00:34:18,800 who knows what it looks like so how are we gonna find the derivative here so 320 00:34:18,800 --> 00:34:24,425 The derivative of a sum is the sum of the derivatives, so differentiate each term 321 00:34:24,425 --> 00:34:24,800 separately. 322 00:34:24,800 --> 00:34:37,440 Differentiate each term separately, beginning with the power rule. 323 00:34:37,440 --> 00:34:51,084 The derivative of 5x to the fourth is 20x cubed. The derivative of 2 sine x is 2 324 00:34:51,084 --> 00:34:52,600 cosine x. 325 00:34:53,640 --> 00:34:59,920 The derivative of negative 4 cosine x is positive 4 sine x. 326 00:35:04,640 --> 00:35:09,900 The derivative of 8e to the x is 8e to the x. 327 00:35:19,140 --> 00:35:25,720 The derivative is 20x cubed plus 2 cosine x plus 4 sine x plus 8e to the x. 328 00:35:27,140 --> 00:35:27,660 There it is. 329 00:35:27,660 --> 00:35:33,320 This gives the slope of the tangent line at any value of x. 330 00:35:35,660 --> 00:35:37,060 Okay. 331 00:35:47,240 --> 00:35:47,940 Okay. 332 00:35:47,940 --> 00:35:48,840 from? 333 00:35:50,220 --> 00:35:51,620 Okay. 334 00:36:10,980 --> 00:36:14,700 We know the derivatives of e to the x and sine x, but a product needs another rule. 335 00:36:15,400 --> 00:36:18,740 Do we know the derivative of a product? 336 00:36:20,560 --> 00:36:21,840 What's the product rule? 337 00:36:36,920 --> 00:36:48,700 What's the product rule? 338 00:36:48,700 --> 00:37:03,940 this is why in the world is it that we got to prove it maybe it's something 339 00:37:03,940 --> 00:37:12,040 We need the product rule for this expression and the quotient rule for tangent. 340 00:37:12,040 --> 00:37:20,140 Save those rules for next week after the test. 341 00:37:20,140 --> 00:37:27,600 The rules from today are enough for the assigned Section 2.2 problems. 342 00:37:28,140 --> 00:37:28,960 We will stop here.