WEBVTT 00:00:05.000 --> 00:00:09.013 Alright, let's start with a little warm-up before we get into actually the new 00:00:09.013 --> 00:00:09.763 material. 00:00:20.120 --> 00:00:32.020 Find the equation of the tangent line at x = 0. 00:00:38.100 --> 00:00:39.560 X equals zero. 00:00:45.100 --> 00:00:51.560 To find the equation of the tangent line, we need its slope. 00:00:51.820 --> 00:00:54.940 To find the slope we need to find the derivative. 00:00:54.940 --> 00:01:00.620 We will use several derivative rules. 00:01:04.060 --> 00:01:05.520 We're not going to use the definition. 00:01:06.960 --> 00:01:08.060 What's the first rule? 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... 00:01:15.520 --> 00:01:16.420 No. 00:01:16.820 --> 00:01:17.720 Sum of... 00:01:18.920 --> 00:01:19.800 The sum... 00:01:19.800 --> 00:01:20.700 Did we name it? 00:01:20.780 --> 00:01:21.680 Did I call it something? 00:01:21.820 --> 00:01:26.340 Did I write it down, but we didn't call it something? 00:01:28.660 --> 00:01:30.620 Use the sum and difference rule. 00:01:31.780 --> 00:01:36.340 The derivative of a sum is the sum of the derivatives. 00:01:37.220 --> 00:01:43.100 For each power function, apply the constant multiple rule and the power rule. The 00:01:43.100 --> 00:01:47.720 derivative of 5x cubed starts with 3 times 5, or 15. 00:01:49.140 --> 00:01:51.280 Subtract one from the exponent to get the second power. 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 00:02:05.454 --> 00:02:06.200 write 00:02:06.200 --> 00:02:15.760 The derivative of 7x is 7, and the derivative of the constant is 0. 00:02:18.580 --> 00:02:21.951 The derivative of a cubic is quadratic; the degree decreases by one because of the 00:02:21.951 --> 00:02:22.400 power rule. 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 00:02:29.860 --> 00:02:34.260 Evaluate the derivative at x = 0. 00:02:41.000 --> 00:02:48.140 The slope of the tangent line is 7. Now find its equation. 00:02:51.700 --> 00:03:00.780 The y-value is f(0) = -2. 00:03:00.780 --> 00:03:10.080 Use point-slope form: y minus y-one equals m times x minus x-one. 00:03:22.600 --> 00:03:25.400 The tangent line is y = 7x - 2. 00:03:27.620 --> 00:03:30.340 Okay, so the power rule is very nice, yes? 00:03:31.820 --> 00:03:33.100 It's a lot simpler than, 00:03:33.240 --> 00:03:34.900 we had to use the definition of the derivative 00:03:34.900 --> 00:03:41.840 on that thing and take us a year to do so. 00:03:42.600 --> 00:03:44.320 And of course the sum and difference rule 00:03:44.320 --> 00:03:45.440 and the constant multiple rule. 00:03:47.060 --> 00:03:49.740 Okay, so that was kind of review from Wednesday. 00:03:50.820 --> 00:03:52.280 Today, let's talk about, 00:03:57.500 --> 00:03:59.740 Now find the derivative of sine. 00:04:05.780 --> 00:04:08.020 How are we going to do this? 00:04:08.480 --> 00:04:10.380 Umm, there's rules. 00:04:11.020 --> 00:04:11.920 There's rules? 00:04:15.080 --> 00:04:16.140 I don't know of any rules. 00:04:17.800 --> 00:04:19.320 Negative sign? 00:04:20.420 --> 00:04:21.520 Oh my goodness. 00:04:24.260 --> 00:04:25.640 Now we're going to do it. 00:04:26.080 --> 00:04:27.220 How are we going to find the derivative? 00:04:30.860 --> 00:04:31.760 Thank you. 00:04:32.320 --> 00:04:33.660 Use the definition of the derivative. 00:04:35.000 --> 00:04:35.900 Which is? 00:04:39.180 --> 00:04:40.580 Start with the limit as h approaches zero. 00:04:42.280 --> 00:04:44.000 As h approaches zero. 00:04:44.000 --> 00:04:51.360 Use f(x+h) minus f(x), all over h. 00:04:53.180 --> 00:05:09.340 This becomes sine of x plus h minus sine of x, all over h. 00:05:15.400 --> 00:05:17.020 Okay, now what? 00:05:18.480 --> 00:05:19.380 Distribute sine? 00:05:23.120 --> 00:05:26.780 What do you mean? 00:05:28.600 --> 00:05:29.580 Do something with this? 00:05:31.240 --> 00:05:32.140 What can we do with that? 00:05:33.540 --> 00:05:36.960 Sine of x plus h is not sine x plus sine h. 00:05:44.060 --> 00:05:45.940 Oh. Let's do an example. 00:05:47.940 --> 00:05:49.980 Pi over 2 plus pi over 2. 00:05:53.460 --> 00:05:55.420 What's pi over 2 plus pi over 2? 00:05:56.680 --> 00:05:58.160 Sine of pi. 00:05:59.740 --> 00:06:02.000 Is that sine of pi over 2? 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 00:06:17.247 --> 00:06:17.997 2? 00:06:18.180 --> 00:06:26.055 Definitely not. Okay, we definitely can't do that. Uh oh, we need something else. 00:06:26.055 --> 00:06:27.180 Any other 00:06:27.180 --> 00:06:28.080 What are ideas? 00:06:40.500 --> 00:06:42.640 Can I factor out a sign? 00:06:45.700 --> 00:06:48.000 You get x plus h minus x. 00:06:48.980 --> 00:06:50.100 Definitely can't do that. 00:06:50.720 --> 00:06:51.620 There's no such thing. 00:06:53.740 --> 00:06:55.260 Use a trigonometric identity. 00:06:55.500 --> 00:06:56.360 This is very important. 00:06:56.360 --> 00:06:57.260 Who said that? 00:06:57.340 --> 00:06:58.240 I said that one actually. 00:06:58.760 --> 00:06:59.660 You did? 00:06:59.760 --> 00:07:00.660 Good. 00:07:02.080 --> 00:07:03.480 Okay. 00:07:41.560 --> 00:07:43.900 I don't know, it's not an awesome thing. 00:07:44.160 --> 00:07:45.060 It's awesome. 00:07:46.300 --> 00:07:49.480 Okay, uh, so, we gotta... 00:07:49.480 --> 00:07:51.100 These are some of the trigonometric identities we will use. 00:07:52.240 --> 00:07:54.041 There are infinitely many trigonometric identities; these are the main ones for this 00:07:54.041 --> 00:07:54.720 lesson. 00:07:54.720 --> 00:07:55.740 These are the main ones. 00:07:56.400 --> 00:07:57.380 Which one are we gonna use? 00:07:58.660 --> 00:08:00.300 Use the sine sum identity. 00:08:01.340 --> 00:08:02.240 What'd it go? 00:08:04.760 --> 00:08:05.580 What? 00:08:05.580 --> 00:08:06.480 Uh oh. 00:08:07.280 --> 00:08:08.180 Let's go back here. 00:08:11.480 --> 00:08:16.180 The argument contains a sum, so use the sine addition identity. 00:08:17.460 --> 00:08:18.380 Sum a difference. 00:08:18.500 --> 00:08:19.720 Sum a difference. 00:08:19.720 --> 00:08:37.440 Use sine of alpha plus beta. 00:08:38.660 --> 00:08:46.000 That identity is sine alpha cosine beta plus cosine alpha sine beta. 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 00:08:53.560 --> 00:08:55.120 negative so 00:08:56.300 --> 00:08:57.200 Got 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. 00:09:13.320 --> 00:09:26.180 sine x cosine h plus cosine x sine h. 00:09:38.300 --> 00:09:41.400 Okay, now what? 00:09:47.700 --> 00:09:49.380 Any ideas? 00:09:51.960 --> 00:09:52.860 What? 00:10:00.260 --> 00:10:09.120 Group the sine x terms and the sine h terms. 00:10:09.120 --> 00:10:11.980 that so first let's rearrange it 00:10:22.900 --> 00:10:24.460 Rearrange the numerator. 00:10:26.140 --> 00:10:28.120 Factor out sine x and cosine x. 00:10:31.120 --> 00:10:32.020 This guy and this guy. 00:10:53.260 --> 00:10:55.300 I like that. 00:10:55.300 --> 00:10:58.680 Now what? 00:10:58.680 --> 00:11:07.480 Um, cosine h minus 1 is identity of something. 00:11:08.060 --> 00:11:08.960 That's it. 00:11:09.040 --> 00:11:10.080 That's cosine square. 00:11:10.080 --> 00:11:10.980 I forgot what these are. 00:11:11.240 --> 00:11:12.440 They have to be cosine squared. 00:11:12.600 --> 00:11:13.520 Oh, that's square. 00:11:13.740 --> 00:11:14.000 They're not. 00:11:14.000 --> 00:11:14.320 They're not. 00:11:14.320 --> 00:11:16.580 We have to be cosine squared minus 1. 00:11:17.040 --> 00:11:17.940 Cosine square. 00:11:20.440 --> 00:11:23.040 Uh, any other ideas? 00:11:23.040 --> 00:11:23.940 this. 00:11:35.920 --> 00:11:37.240 Split the expression into two limits. 00:11:37.580 --> 00:11:38.600 This plus that. 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 00:11:53.473 --> 00:11:58.720 cosine x times sine h, over h. 00:11:58.720 --> 00:12:05.920 The first special limit equals zero. 00:12:07.320 --> 00:12:12.360 Sine x is constant with respect to h, so it may be moved outside the limit. 00:12:12.520 --> 00:12:16.380 You don't have to do this, but you could bring it outside the limit. 00:12:16.380 --> 00:12:21.280 The cosine-minus-one limit equals zero. 00:12:23.720 --> 00:12:27.140 The sine-over-h limit equals one. 00:12:28.420 --> 00:12:36.520 The result is sine x times zero plus cosine x times one. 00:12:37.560 --> 00:12:38.740 And there it is! 00:12:40.340 --> 00:12:42.360 The derivative of sine x is cosine x. 00:12:43.080 --> 00:12:43.980 Beautiful. 00:12:49.140 --> 00:12:51.420 That is math has finds. 00:12:51.660 --> 00:12:53.340 Please prove that derivative of sine is cosine. 00:12:55.920 --> 00:12:58.640 So are we gonna do this every single time? 00:13:00.060 --> 00:13:00.960 Definitely not. 00:13:01.660 --> 00:13:04.340 At this point, we just memorize. 00:13:06.560 --> 00:13:07.880 Derivative of sine. 00:13:10.220 --> 00:13:11.120 Okay next one. 00:13:12.600 --> 00:13:17.720 Now find the derivative of cosine. 00:13:30.720 --> 00:13:32.840 Let's work it out people! 00:13:33.760 --> 00:13:35.200 How are we going to start? 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. 00:13:40.740 --> 00:13:46.960 f of x plus h minus f of x all over h. 00:13:59.720 --> 00:14:02.820 Okay. I bet you could do it. 00:14:03.580 --> 00:14:06.000 Finish the proof or finish the limit. 00:15:59.580 --> 00:16:02.040 What's the next step? 00:16:06.560 --> 00:16:09.040 Let's switch these guys. 00:16:46.260 --> 00:16:50.440 Separate the expression into two limits. 00:16:55.320 --> 00:16:58.920 This goes to zero. 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 00:17:10.040 --> 00:17:14.640 times zero minus sine x times one. 00:17:14.640 --> 00:17:15.540 The result is negative sine x. 00:17:16.920 --> 00:17:22.501 Okay, and I think when we went over graphing derivatives we graphed cosine, we 00:17:22.501 --> 00:17:22.900 graphed 00:17:22.900 --> 00:17:27.920 When we graphed the derivative of cosine, it also matched negative sine x. 00:17:27.920 --> 00:17:29.700 We proved it. Beautiful. 00:17:30.940 --> 00:17:32.880 Okay, so we we memorize this one 00:17:34.300 --> 00:17:35.820 at this point 00:17:37.560 --> 00:17:39.560 It's negative sine of x 00:17:40.760 --> 00:17:45.320 The derivative of sine is cosine, and the derivative of cosine is negative sine. 00:17:46.880 --> 00:17:50.600 All right, now we can do any polynomial. 00:17:51.540 --> 00:17:54.240 Any multiple of sine, any multiple of cosine. 00:17:55.900 --> 00:17:58.020 Now consider tangent. 00:18:40.680 --> 00:18:48.360 If we substitute this in, it's going to be a really ugly mess. 00:18:48.360 --> 00:18:50.780 The definition with tangent of x plus h would be a complicated expression. 00:18:52.200 --> 00:18:54.300 Okay, so we're not going to do this 00:18:56.760 --> 00:18:59.240 This right here is going to be real messy 00:19:04.620 --> 00:19:09.260 Rewrite tangent as sine x divided by cosine x. 00:19:15.220 --> 00:19:16.120 Now what 00:19:19.520 --> 00:19:20.940 That's as far as I got. 00:19:21.860 --> 00:19:22.760 I like this. 00:19:26.240 --> 00:19:30.080 The derivative of a sum is the sum of derivatives. 00:19:39.640 --> 00:19:41.520 Is that another rule? 00:19:45.900 --> 00:19:47.340 It's definitely not. 00:19:49.220 --> 00:19:50.120 Okay. 00:19:51.000 --> 00:19:54.420 The derivative of a quotient is not the quotient of the derivatives. 00:19:56.280 --> 00:19:59.520 It's true for sums and differences, not true for quotients. 00:20:00.980 --> 00:20:02.420 Okay. I don't know what I just did here. 00:20:03.380 --> 00:20:06.340 Long story short, we have no idea what the derivative is. 00:20:06.340 --> 00:20:12.620 We will learn the quotient rule after the test. 00:20:14.040 --> 00:20:15.920 We haven't proved that yet, right? 00:20:20.840 --> 00:20:24.500 So that's for next week after the test. 00:20:29.720 --> 00:20:32.020 We cannot finish tangent until we have the quotient rule. 00:20:34.180 --> 00:20:36.000 We're going to do one more. 00:20:39.320 --> 00:20:40.220 No more trig. 00:20:40.400 --> 00:20:41.960 The rest are all quotients. 00:20:42.140 --> 00:20:43.260 We don't have any more quotients. 00:20:43.260 --> 00:20:53.680 Right, uh, cosecant is one over sine, which is another quotient. 00:20:55.740 --> 00:20:57.140 We don't have a quotient rule yet. 00:20:59.220 --> 00:21:01.880 We're going to do e to the x. 00:21:15.740 --> 00:21:18.140 Before we start, what is our number e? 00:21:18.140 --> 00:21:39.240 The number e is approximately 2.71828; it is irrational. 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 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 00:21:59.421 --> 00:21:59.840 remember 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. 00:22:12.080 --> 00:22:15.520 Where do we use this in math 2.3? 00:22:19.780 --> 00:22:20.680 Compound interest. 00:22:23.160 --> 00:22:24.060 Okay. 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 00:22:25.634 --> 00:22:25.720 power. 00:22:25.720 --> 00:22:32.880 It's a principle, A equals principle. 00:22:33.360 --> 00:22:37.740 Principle times E. Let's do compound interest first. 00:22:41.020 --> 00:22:47.320 The factor is 1 plus r over n, raised to nt. 00:22:47.340 --> 00:22:48.420 Oh boy, people. 00:22:49.640 --> 00:22:50.840 N plus E plus. 00:22:51.060 --> 00:22:51.700 Yeah, that's fine. 00:22:51.700 --> 00:22:52.900 R is your interest rate. 00:22:52.900 --> 00:23:06.400 The variable n is the number of compounding periods per year. 00:23:06.400 --> 00:23:15.700 larger we got the more it just kept getting closer to this one number and 00:23:15.920 --> 00:23:21.400 And then we came up this formula for continuously compounded interest 00:23:22.800 --> 00:23:24.540 E to the RT, right? 00:23:25.660 --> 00:23:30.380 Okay, it has everything to do with this the real definition of E 00:23:32.840 --> 00:23:35.920 The number e is the limit as n approaches infinity. 00:23:41.460 --> 00:23:44.160 X goes to infinity 00:23:46.960 --> 00:23:52.060 Use the quantity 1 plus 1 over n, raised to the nth power. 00:23:56.100 --> 00:23:58.860 which is just that compound 00:23:58.860 --> 00:24:02.240 This is the compound-interest limit with the extra rate and time parameters removed. 00:24:04.400 --> 00:24:07.460 As you go to infinity, this gets very, 00:24:07.640 --> 00:24:09.360 very small, but then the power gets very, 00:24:09.360 --> 00:24:11.660 very big at the same rate. 00:24:12.600 --> 00:24:13.940 We're at the same time, 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. 00:24:20.940 --> 00:24:22.940 There is an equivalent definition. 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. 00:24:40.460 --> 00:24:43.340 So, 1 over x. 00:24:47.760 --> 00:24:50.160 These two definitions are equivalent. 00:24:51.720 --> 00:24:54.560 As x goes to infinity, 1 over x goes to 0. 00:24:56.500 --> 00:25:00.740 And as x goes to infinity, this goes to infinity, 1 over x goes to infinity. 00:25:00.740 --> 00:25:03.580 sorry, one over zero goes to infinity. 00:25:04.840 --> 00:25:08.000 Okay, we're gonna use, well, we need this, 00:25:09.660 --> 00:25:12.060 except we're gonna call it this instead. 00:25:13.840 --> 00:25:15.460 So keep this in mind. 00:25:17.120 --> 00:25:19.700 Doesn't matter what the variable is, same exact thing. 00:25:23.120 --> 00:25:24.700 Okay, we'll come back to that. 00:25:26.840 --> 00:25:30.560 Now let f(x) = e to the x. 00:25:31.900 --> 00:25:35.540 The derivative is the limit as h approaches zero. 00:25:37.160 --> 00:25:38.700 Use f(x+h) minus f(x), all over h. 00:25:38.700 --> 00:25:42.740 minus f of x all over h. 00:26:00.740 --> 00:26:02.420 Okay, now what? 00:26:13.940 --> 00:26:15.560 Factor the numerator after using exponent rules. 00:26:16.100 --> 00:26:16.560 That's awesome. 00:26:16.560 --> 00:26:19.200 Okay, I like that except what do we got to do with this first? 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. 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 00:26:42.300 --> 00:26:44.680 Factor out e to the x. 00:27:03.120 --> 00:27:03.560 I 00:27:03.560 --> 00:27:04.780 Like it now what? 00:27:08.900 --> 00:27:11.700 Replace e using its limit definition. 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. 00:27:22.060 --> 00:27:35.020 zero, one plus H to the one over H. What we just talked about. 00:27:38.180 --> 00:27:41.000 Can we have limits inside limits? 00:27:42.840 --> 00:27:42.880 Absolutely. 00:27:42.880 --> 00:27:43.780 be. 00:28:12.880 --> 00:28:14.420 Okay, now what? 00:28:21.800 --> 00:28:24.240 Use the power law for limits to bring the exponent into the inner expression. 00:28:26.660 --> 00:28:31.540 Is a limit of uh, let me just get the law 00:28:36.240 --> 00:28:41.620 I think this was the limit law the limit is whatever x approaches any number 00:28:44.020 --> 00:28:54.240 f of x to the any number n is n to the limit. 00:28:57.560 --> 00:28:59.000 This was one of the limit laws. 00:29:00.280 --> 00:29:03.974 So basically we're going to bring this inside the limit and put it over the 00:29:03.974 --> 00:29:04.724 function. 00:29:37.880 --> 00:29:43.440 Ok, now what? 00:29:48.620 --> 00:29:52.180 For a power raised to a power, multiply the exponents. 00:29:52.500 --> 00:29:53.400 What can we do? 00:29:53.880 --> 00:29:54.600 We can't solve it. 00:29:54.600 --> 00:29:55.380 We can't solve it. 00:29:55.380 --> 00:29:58.520 We can't solve it. 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 00:30:04.624 --> 00:30:06.060 the exponents times them? 00:30:06.620 --> 00:30:06.700 multiply 00:30:06.700 --> 00:30:33.840 The exponent 1 over h times h simplifies to 1. 00:30:39.380 --> 00:30:40.460 Now what? 00:30:45.780 --> 00:30:48.960 The limit of the constant 1 is 1. 00:30:51.400 --> 00:30:55.840 Use the sum and difference laws for limits. 00:30:55.840 --> 00:30:59.280 so we can bring this negative one inside here. 00:31:12.040 --> 00:31:14.600 One plus h minus one simplifies to h. 00:31:20.740 --> 00:31:21.640 Okay. 00:31:23.440 --> 00:31:26.200 Now these ones add to be zero. 00:31:28.240 --> 00:31:29.140 Okay. 00:31:30.160 --> 00:31:31.720 And let me bring this guy out here. 00:31:31.720 --> 00:31:36.920 Factor e to the x outside the remaining limit. 00:31:37.540 --> 00:31:43.840 And this is the limit as h approaches 0. 00:31:49.620 --> 00:31:53.320 The ratio h over h equals 1. 00:31:53.320 --> 00:31:54.220 1. 00:31:54.660 --> 00:31:55.560 1. 00:31:56.340 --> 00:31:57.920 This is 1. 00:31:58.500 --> 00:31:59.760 This is 1. 00:32:00.360 --> 00:32:02.940 We get e to the x times 1. 00:32:05.060 --> 00:32:08.400 Therefore, the derivative of e to the x is e to the x. 00:32:11.860 --> 00:32:14.120 All this for this? 00:32:14.280 --> 00:32:15.880 Yes, it was awesome. 00:32:17.240 --> 00:32:20.360 We just proved the derivative of e to the x is e to the x. 00:32:22.160 --> 00:32:24.380 It's the easiest derivative to find. 00:32:27.560 --> 00:32:32.960 The definition-based proof depends on the limit definition of the number e. 00:32:32.960 --> 00:32:43.340 definition of the number E. Okay and we probably really didn't tell you that 00:32:43.340 --> 00:32:46.924 that's the definition because you didn't know what limits were back a couple years 00:32:46.924 --> 00:32:47.420 ago. 00:32:47.420 --> 00:32:53.200 But now we do. That is the definite. So long story short, we've memorized this 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. 00:33:07.940 --> 00:33:11.035 Any constant multiple of e to the x differentiates to the same constant multiple of 00:33:11.035 --> 00:33:11.860 e to the x. 00:33:14.260 --> 00:33:15.160 Same seed. 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. 00:33:21.080 --> 00:33:29.620 We now know derivatives of polynomials, sine, cosine, and e to the x. 00:33:30.620 --> 00:33:33.040 Or the sum of any combination of those. 00:33:34.140 --> 00:33:37.080 Apply the rules to a mixed example. 00:33:55.480 --> 00:33:57.920 This is some crazy function. 00:33:57.920 --> 00:34:18.800 who knows what it looks like so how are we gonna find the derivative here so 00:34:18.800 --> 00:34:24.425 The derivative of a sum is the sum of the derivatives, so differentiate each term 00:34:24.425 --> 00:34:24.800 separately. 00:34:24.800 --> 00:34:37.440 Differentiate each term separately, beginning with the power rule. 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 00:34:51.084 --> 00:34:52.600 cosine x. 00:34:53.640 --> 00:34:59.920 The derivative of negative 4 cosine x is positive 4 sine x. 00:35:04.640 --> 00:35:09.900 The derivative of 8e to the x is 8e to the x. 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. 00:35:27.140 --> 00:35:27.660 There it is. 00:35:27.660 --> 00:35:33.320 This gives the slope of the tangent line at any value of x. 00:35:35.660 --> 00:35:37.060 Okay. 00:35:47.240 --> 00:35:47.940 Okay. 00:35:47.940 --> 00:35:48.840 from? 00:35:50.220 --> 00:35:51.620 Okay. 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. 00:36:15.400 --> 00:36:18.740 Do we know the derivative of a product? 00:36:20.560 --> 00:36:21.840 What's the product rule? 00:36:36.920 --> 00:36:48.700 What's the product rule? 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 00:37:03.940 --> 00:37:12.040 We need the product rule for this expression and the quotient rule for tangent. 00:37:12.040 --> 00:37:20.140 Save those rules for next week after the test. 00:37:20.140 --> 00:37:27.600 The rules from today are enough for the assigned Section 2.2 problems. 00:37:28.140 --> 00:37:28.960 We will stop here.