WEBVTT 00:00:05.020 --> 00:00:24.539 Okay, before we get started, what functions do we now know the derivative of very 00:00:24.539 --> 00:00:25.840 easily? 00:00:27.320 --> 00:00:29.220 Any polynomial? 00:00:39.920 --> 00:00:43.220 And why do we know the derivative of any polynomial? 00:00:46.560 --> 00:00:48.040 Power rule and? 00:00:52.580 --> 00:00:53.480 Sum and difference rule. 00:00:53.480 --> 00:00:58.540 So, the polynomial is just a sum of multiple power functions. 00:01:00.180 --> 00:01:04.679 And the sum and difference rules we separate by addition, we take the derivative 00:01:04.679 --> 00:01:05.429 separately. 00:01:07.840 --> 00:01:11.160 Just for funsies, what's the derivative of this? 00:01:11.160 --> 00:01:14.540 The function is x to the fifth plus 3x squared minus 7x plus 2. 00:01:14.540 --> 00:01:17.760 Its derivative is 5x to the fourth plus 6x minus 7. 00:01:17.760 --> 00:01:19.540 The derivative of the constant term is zero. 00:01:19.540 --> 00:01:21.960 ok, what other functions? 00:01:23.320 --> 00:01:24.920 Uh, okay, sorry, what else? 00:01:27.680 --> 00:01:28.580 What's that? 00:01:28.700 --> 00:01:29.600 Square root functions. 00:01:29.760 --> 00:01:31.760 Square root functions, that's true. 00:01:32.300 --> 00:01:33.440 Any kind of power function. 00:01:37.080 --> 00:01:41.160 Write the cube root of x, which is x to the one third, 00:01:41.160 --> 00:01:49.680 which is a power function the derivative is 1 3rd minus 1 is negative 2 3rds 00:01:49.680 --> 00:01:56.580 sounds good what other functions sine and cosine two of the trig functions 00:01:56.580 --> 00:02:06.240 derivative of sine is cosine and derivative of cosine is negative sine and then one 00:02:06.420 --> 00:02:17.040 What is the one other function? e to the x. What is its derivative? e to the x. 00:02:32.780 --> 00:02:41.340 So here's an example of the product of two functions. 00:02:41.340 --> 00:02:44.560 It's not the sum of it. 00:02:44.600 --> 00:02:46.440 If it were the sum of x to the fourth and sine x, we could differentiate each term. 00:02:46.480 --> 00:02:47.820 we could easily do it. 00:02:48.840 --> 00:02:52.200 But we have a product and it turns out it is not, 00:02:52.420 --> 00:02:54.040 the derivative, don't write this down. 00:02:55.200 --> 00:03:00.660 It is not 4x cubed times cosine x. 00:03:00.920 --> 00:03:02.200 not the derivative of one function 00:03:02.200 --> 00:03:03.100 times the derivative of the other. 00:03:04.580 --> 00:03:06.420 And let me just give you an example. 00:03:13.820 --> 00:03:19.080 X to the fifth, same thing as X squared times X cubed. 00:03:19.340 --> 00:03:20.240 We agree? 00:03:20.720 --> 00:03:21.620 Yeah. 00:03:21.860 --> 00:03:24.980 We know the derivative of X to the fifth, 00:03:25.060 --> 00:03:26.900 which is what, five X to the fourth. 00:03:30.480 --> 00:03:32.600 Derivative of X squared is two X, 00:03:33.560 --> 00:03:35.540 derivative of this is three X squared, 00:03:36.980 --> 00:03:40.080 which is six X cubed, which is definitely not this, right? 00:03:41.340 --> 00:03:55.320 There is a rule called the product rule. If y equals f of x times g of x, 00:03:55.320 --> 00:03:58.540 then its derivative has two terms. 00:04:06.440 --> 00:04:12.640 It turns out it's this. The derivative of the first function 00:04:12.640 --> 00:04:16.400 times the second plus the first function 00:04:17.980 --> 00:04:21.980 times the derivative of the second function. 00:04:53.260 --> 00:04:55.300 Apply the product rule to x squared times x cubed. 00:04:55.300 --> 00:05:06.640 The derivative of the first function is 2x, times the second function, 00:05:06.640 --> 00:05:19.500 plus the first function times the derivative of the second function. 00:05:19.500 --> 00:05:34.780 This is 2x to the fourth plus 3x to the fourth, which equals 5x to the fourth. 00:05:34.780 --> 00:05:41.580 showing an example that works out is not proof that's always true how do we know 00:05:41.580 --> 00:06:04.100 this is always true let's prove it and let me call this a H of X just for a 00:06:05.380 --> 00:06:14.000 Let m of x equal f of x times g of x. 00:06:19.360 --> 00:06:21.300 And how are we going to prove it? 00:06:24.420 --> 00:06:25.880 The definition of the derivative. 00:06:28.000 --> 00:06:35.973 m prime of x is the limit as h approaches zero of m of x plus h minus m of x, all 00:06:35.973 --> 00:06:37.060 divided by h. 00:06:37.940 --> 00:06:40.940 I shouldn't use h's here, but that's okay. 00:06:45.280 --> 00:06:49.520 Smaller h. Aye yi yi, too many h's. Let's call this something else. 00:06:54.380 --> 00:06:59.880 Call the product m of x. 00:07:15.460 --> 00:07:19.820 For m of x plus h, substitute x plus h into both factors. 00:07:19.820 --> 00:07:20.720 X. 00:07:30.820 --> 00:07:40.162 The numerator is f of x plus h times g of x plus h minus f of x times g of x, all 00:07:40.162 --> 00:07:41.380 divided by h. 00:08:00.100 --> 00:08:01.940 Now what? 00:08:04.580 --> 00:08:06.200 Any ideas? 00:08:14.480 --> 00:08:16.140 No ideas? 00:08:19.720 --> 00:08:29.433 Should we try multiplying by the conjugate? No, it's not gonna help. There is a 00:08:29.433 --> 00:08:30.183 trick. 00:09:03.220 --> 00:09:06.200 All I do is separate those two terms. 00:09:12.820 --> 00:09:14.260 We're going to add something here. 00:09:15.580 --> 00:09:17.320 Anybody want to take a guess what we're going to add? 00:09:21.700 --> 00:09:23.980 The only thing we're allowed to do is add zero. 00:09:25.460 --> 00:09:26.640 It doesn't change anything, right? 00:09:27.780 --> 00:09:30.180 We're going to add something and then we're going to subtract something. 00:09:32.880 --> 00:09:34.020 H plus h. 00:09:35.180 --> 00:09:37.280 I like your idea, but I don't think that's going to work. 00:09:38.920 --> 00:09:43.240 We're going to add something so then we could factor out something in common. 00:09:50.900 --> 00:09:53.160 We're going to add negative. 00:10:27.440 --> 00:10:30.040 Let me think for a second what it is. 00:10:47.160 --> 00:10:52.220 So we're going to add... 00:10:52.220 --> 00:10:52.960 Let's try this. 00:10:52.960 --> 00:10:54.120 It might not work out. 00:10:55.100 --> 00:10:56.000 I think this is it. 00:10:56.380 --> 00:11:03.937 So if we add a negative f of x, g of x plus h, we're going to have to add a positive 00:11:03.937 --> 00:11:04.280 f 00:11:04.280 --> 00:11:07.880 of x times g of x plus h. 00:11:11.820 --> 00:11:15.640 All right, so all I did was add the opposites, which is just 0. 00:11:24.680 --> 00:11:26.960 Now what can I do with these first two terms? 00:11:33.200 --> 00:11:34.360 What can I factor out? 00:11:34.360 --> 00:11:37.280 g of x plus h. 00:11:42.540 --> 00:11:45.020 And I'm going to change this into two fractions. 00:11:46.600 --> 00:11:48.260 This and this. 00:11:48.640 --> 00:11:52.260 So g of x plus h. 00:12:10.260 --> 00:12:12.660 Let me change it to two different limits. 00:12:14.260 --> 00:12:16.540 From the second pair of terms, what can I factor out? 00:12:16.540 --> 00:12:20.220 F of X. 00:12:36.080 --> 00:12:41.500 And we're very close now. 00:12:46.820 --> 00:12:50.060 This product here, I'm going to change it to two different limits. 00:13:50.060 --> 00:13:52.300 And what is this guy right here? 00:13:56.780 --> 00:14:02.620 That is the definition of the derivative of f of x. 00:14:02.740 --> 00:14:04.020 That's f prime of x. 00:14:07.120 --> 00:14:11.460 And as h goes to 0 here, this goes to g of x. 00:14:17.300 --> 00:14:20.480 There is no H's, so this is just F of X. 00:14:25.860 --> 00:14:31.220 And this right here is the definition of the derivative of G. 00:14:33.180 --> 00:14:34.960 And let me just switch the order 00:14:44.960 --> 00:14:48.320 And there it is we just proved the product rule 00:14:51.640 --> 00:14:57.920 The first person to prove this was clever; that first move is tricky. 00:15:00.060 --> 00:15:01.200 But there it is. 00:15:02.980 --> 00:15:08.560 Whenever we differentiate a product of two functions, we use the product rule. 00:15:08.800 --> 00:15:10.720 We have to use the product rule. 00:15:10.920 --> 00:15:11.820 Let's do some examples. 00:15:17.840 --> 00:15:20.120 Oh actually, let's do that example I had up here. 00:15:20.220 --> 00:15:21.040 What was it? 00:15:21.040 --> 00:15:26.440 x to the fourth times sine x. 00:15:42.500 --> 00:15:44.240 So the derivative 00:15:56.000 --> 00:16:03.803 So it's the derivative of our first function times our second function plus the 00:16:03.803 --> 00:16:04.553 derivative, 00:16:04.900 --> 00:16:08.220 sorry, first function times the derivative of the second function. 00:16:19.580 --> 00:16:21.020 So we'll just write this down. 00:16:31.000 --> 00:16:33.240 And the derivative of x to the fourth is? 00:16:33.640 --> 00:16:34.540 4x cubed. 00:16:34.560 --> 00:16:41.340 4x cubed times sine x plus x to the fourth times cosine x. 00:16:42.600 --> 00:16:43.880 Derivative of sine of x? 00:16:44.140 --> 00:16:45.040 Cosine x. 00:16:48.380 --> 00:16:49.600 And that's it. 00:16:56.180 --> 00:16:57.220 Easy peasy. 00:16:59.860 --> 00:17:03.960 Please do it in this order where it's the derivative of the first function times the 00:17:03.960 --> 00:17:06.580 second plus the first times the derivative of the second. 00:17:07.400 --> 00:17:12.351 Either way, I like it this way. It's going to be very important for the quotient 00:17:12.351 --> 00:17:13.101 rule. 00:17:17.140 --> 00:17:18.840 Okay, let's do another one. 00:17:41.700 --> 00:17:43.340 Uh oh, we got a problem here. 00:17:44.800 --> 00:17:45.700 What's the problem? 00:17:46.800 --> 00:17:47.700 It's three terms. 00:17:48.760 --> 00:17:49.680 It's not three terms. 00:17:51.460 --> 00:17:53.200 Terms are things separated by addition. 00:17:53.460 --> 00:17:55.640 There are three... 00:17:55.640 --> 00:17:57.580 Or it's a product of three things. 00:17:58.320 --> 00:17:59.640 There are three factors. 00:18:00.680 --> 00:18:02.360 Okay, it's not just a product of two functions. 00:18:02.600 --> 00:18:04.340 It's the product of three functions. 00:18:04.980 --> 00:18:11.860 So now what? What's the idea? 00:18:16.360 --> 00:18:18.600 Okay, I like that. 00:18:20.500 --> 00:18:23.820 So let's, which should we pair up? Let's call these two. 00:18:24.680 --> 00:18:25.580 One function. 00:18:29.340 --> 00:18:31.600 We're going to use the product rule right here. 00:18:35.900 --> 00:18:44.319 So it's the derivative of the first times the second plus the first times the 00:18:44.319 --> 00:18:44.880 derivative 00:18:44.880 --> 00:18:45.780 of all the second 00:19:17.520 --> 00:19:24.060 x to the fifth times cosine x times e to the x. 00:19:26.720 --> 00:19:29.260 Uh oh, then what are we doing when we get to here? 00:19:29.400 --> 00:19:34.376 Yeah, we got, again, we're finding the derivative of a product, so we got another 00:19:34.376 --> 00:19:35.126 product rule. 00:19:39.160 --> 00:19:40.560 Oops. 00:20:05.460 --> 00:20:10.180 Deriv of cosine is negative sine. 00:20:16.040 --> 00:20:20.800 Deriv of e to the x is e to the x. 00:20:24.820 --> 00:20:27.500 And then let me distribute this. 00:20:50.720 --> 00:20:55.360 There it is. 00:20:55.540 --> 00:20:57.480 We just gotta use the product rule twice. 00:21:02.720 --> 00:21:05.360 Is there any shortcut? 00:21:08.600 --> 00:21:12.080 Instead of using the product rule once and using it twice? 00:21:20.700 --> 00:21:22.560 So tell me what to do. 00:21:22.560 --> 00:21:27.460 First, differentiate the first function and multiply by the other two. 00:21:27.460 --> 00:21:28.560 That gives 5x to the fourth times cosine x times e to the x. 00:21:29.120 --> 00:21:33.700 And it would be 5x to the fourth times cosine times x. 00:21:35.300 --> 00:21:38.880 Then differentiate the second function and multiply by the other two. 00:21:39.560 --> 00:21:40.460 And then what? 00:21:40.660 --> 00:21:43.560 Differentiate cosine x. 00:21:43.980 --> 00:21:49.000 This gives x to the fifth times negative sine x times e to the x. 00:21:49.000 --> 00:21:50.260 times x to the x. 00:21:53.920 --> 00:21:54.820 And then? 00:21:55.020 --> 00:21:56.820 Finally, differentiate the third function. 00:21:57.400 --> 00:22:02.040 This gives x to the fifth times cosine x times e to the x. 00:22:06.060 --> 00:22:13.110 For three factors, differentiate one factor at a time and multiply by the other two, 00:22:13.110 --> 00:22:15.460 then add all three terms. 00:22:18.160 --> 00:22:23.860 And that is exactly what we have here, derivative of the first function times the 00:22:23.860 --> 00:22:24.620 other two, 00:22:31.760 --> 00:22:39.059 plus the derivative of the second function times the other two, plus the derivative 00:22:39.059 --> 00:22:39.580 of 00:22:39.580 --> 00:22:45.909 The derivative of the third function may look unchanged because e to the x is its 00:22:45.909 --> 00:22:46.700 own derivative. 00:22:46.700 --> 00:22:58.700 always gonna work? Definitely. Okay we could do it with f of x, g of x, and h of 00:22:58.700 --> 00:23:04.068 x. Just use the product rule without these actual functions. It will always work. So 00:23:04.068 --> 00:23:05.500 if we have four 00:23:05.500 --> 00:23:15.140 functions let's say this was a times sine of X or something we just have what 00:23:15.140 --> 00:23:24.100 we have to have sine of X in here sine of X sine of X and then plus the first 00:23:24.100 --> 00:23:30.560 three functions times the derivative of sine of X okay so we can have a product 00:23:30.560 --> 00:23:35.605 to more than just two it's just the derivative of the first function times the other 00:23:35.605 --> 00:23:36.355 two 00:23:36.680 --> 00:23:43.489 plus the derivative of the second function times the other two and so on. Okay not 00:23:43.489 --> 00:23:44.340 too bad. 00:24:14.940 --> 00:24:17.260 Can we use the product rule here? 00:24:19.200 --> 00:24:20.480 Could we change this into a product? 00:24:22.580 --> 00:24:23.880 I could change this into a product. 00:24:25.120 --> 00:24:26.420 That's x to the cubed times what? 00:24:26.940 --> 00:24:27.940 One over sine x. 00:24:29.040 --> 00:24:36.380 true, 1 over sine of x, sine of x to the negative 1 power. 00:24:38.660 --> 00:24:41.323 We know the derivative of this, do we know the derivative of sine of x to the 00:24:41.323 --> 00:24:41.480 negative 00:24:41.480 --> 00:24:42.380 1 power? 00:24:45.000 --> 00:24:45.900 We don't. 00:24:49.840 --> 00:24:52.520 We know the derivative of sine of x, but not sine of x to the negative 1. 00:24:52.820 --> 00:24:55.140 This is actually the composition of two functions. 00:24:55.140 --> 00:25:00.900 This is sine of x inside this function or... 00:25:03.400 --> 00:25:09.725 Okay, it's a function within a function. We need it, so yeah, so that's not gonna 00:25:09.725 --> 00:25:10.475 work. 00:25:12.100 --> 00:25:17.640 So what are we gonna do here? There's another rule called the quotient rule. 00:25:25.640 --> 00:25:50.200 let me just tell you what it is we have a quotient of two functions turns out 00:25:50.200 --> 00:26:05.637 the derivative. The numerator is exactly like the product rule, f prime times second 00:26:05.637 --> 00:26:06.740 function, 00:26:06.900 --> 00:26:10.444 but it's really the denominator function, except we have a minus here instead of a 00:26:10.444 --> 00:26:11.194 plus. 00:26:18.620 --> 00:26:23.280 And then the denominator turns out to be g of x squared. 00:26:32.620 --> 00:26:39.260 The derivative of a quotient resembles the product rule in the numerator, 00:26:39.260 --> 00:26:45.620 except it uses subtraction instead of addition, and the denominator is squared. 00:26:55.980 --> 00:27:05.940 To prove it, we would start with the definition of the derivative. 00:27:06.920 --> 00:27:11.320 And if you can prove it by the end of the class, we'll have time at the end of the 00:27:11.320 --> 00:27:12.070 class. 00:27:12.480 --> 00:27:13.380 You get extra credit. 00:27:14.980 --> 00:27:17.120 I rarely ever offer extra credit. 00:27:33.420 --> 00:27:36.380 Is it going to be similar to the proof of the product rule? 00:27:36.680 --> 00:27:38.680 It's going to be very similar. 00:27:39.180 --> 00:27:41.920 We would add and subtract a carefully chosen expression. 00:27:42.820 --> 00:27:45.160 But then it's a little bit more complicated, now we got fractions. 00:27:47.420 --> 00:27:48.820 But let's not do it right now. 00:27:50.140 --> 00:27:51.700 We'll have plenty of time at the end of class. 00:27:52.920 --> 00:27:54.980 Let's just use it. 00:27:56.640 --> 00:28:00.240 So f of x, what was it? x cubed over sine of x? 00:28:19.300 --> 00:28:19.660 I 00:28:19.660 --> 00:28:21.480 Often just set up parentheses like that 00:28:23.280 --> 00:28:26.380 You can do it in your head if you want and just go right to the answer 00:28:26.380 --> 00:28:30.100 but it's the derivative of the numerator, the first, I call it the first function, 00:28:32.480 --> 00:28:35.040 times the denominator minus 00:28:38.640 --> 00:28:41.020 first function times the derivative of the second, 00:28:42.780 --> 00:28:48.841 just like the product rule but with a subtraction, and then we simply divide by the 00:28:48.841 --> 00:28:49.220 denominator 00:28:49.220 --> 00:28:50.120 squared. 00:28:58.060 --> 00:29:09.640 The derivative of x cubed is 3x squared by the power rule. 00:29:14.200 --> 00:29:24.760 This gives 3x squared sine x minus x cubed cosine x, all over sine squared x. 00:29:30.440 --> 00:29:31.100 That's it. 00:29:31.100 --> 00:29:32.060 Easy peasy. 00:29:33.500 --> 00:29:34.400 Okay. 00:29:37.020 --> 00:29:38.860 Could we simplify this at all? 00:29:40.960 --> 00:29:48.110 I mean we could manipulate it, have two different fractions, but if there was like 00:29:48.110 --> 00:29:49.540 an x squared 00:29:49.540 --> 00:29:55.345 here or something like that, then we could possibly simplify some x's, but we're 00:29:55.345 --> 00:29:55.760 done 00:29:55.760 --> 00:29:56.660 here. 00:30:07.860 --> 00:30:08.760 Okay. 00:30:18.720 --> 00:30:22.380 We talked about this one last week, something like that. 00:30:22.380 --> 00:30:25.020 And we weren't able to find the derivative 00:30:29.700 --> 00:30:31.820 But how can I rewrite tangent 00:30:33.500 --> 00:30:37.640 Sine over cosine and now we have a quotient rule 00:30:39.740 --> 00:30:42.460 So we're gonna find the derivative by the quotient rule 00:31:54.520 --> 00:31:59.100 The derivative of sine is cosine. 00:32:09.380 --> 00:32:11.480 The derivative of cosine is negative sine. 00:32:45.360 --> 00:32:47.480 Can we simplify this at all? 00:32:49.380 --> 00:32:51.120 How can we simplify this? 00:32:53.380 --> 00:32:54.300 Can I do this? 00:32:56.120 --> 00:32:57.020 Definitely not! 00:32:57.240 --> 00:32:58.740 You go to math jail, you do that. 00:32:59.940 --> 00:33:04.760 You can't simplify one term with a factor. 00:33:04.900 --> 00:33:05.800 It has to be a common factor. 00:33:08.200 --> 00:33:10.300 But this simplifies to what? 00:33:11.340 --> 00:33:13.300 Sine squared plus cosine squared equals one. 00:33:13.440 --> 00:33:14.740 The Pythagorean identity. 00:33:22.220 --> 00:33:25.740 That is the derivative. 00:33:26.080 --> 00:33:29.700 However, what is one over cosine? 00:33:35.720 --> 00:33:38.520 Is secant. 00:33:42.720 --> 00:33:45.740 And so we normally write it as secant squared. 00:34:00.340 --> 00:34:03.560 We do want to just memorize this from now on. 00:34:03.560 --> 00:34:15.089 The derivative of tangent is secant squared. Now we know the derivatives of sine, 00:34:15.089 --> 00:34:17.560 cosine, and tangent. 00:34:26.380 --> 00:34:31.060 Let's do one more and we'll start our homework. 00:34:37.160 --> 00:34:39.120 Let's find the derivative of secant x. 00:34:43.420 --> 00:34:44.420 How am I going to start this? 00:34:44.420 --> 00:34:46.200 Rewrite secant x as one over cosine x. 00:34:47.060 --> 00:34:50.620 Change it to 1 over cosine. 00:34:52.460 --> 00:34:53.660 And what rule are we going to use? 00:34:54.300 --> 00:34:55.160 Quotient rule. 00:34:55.160 --> 00:34:56.060 Quotient rule. 00:35:08.920 --> 00:35:10.900 Go ahead and see if you can do this one on your own. 00:36:16.280 --> 00:36:22.451 Zero times cosine x minus one times negative sine x, over cosine squared x, gives 00:36:22.451 --> 00:36:24.920 sine x over cosine squared x. 00:36:26.480 --> 00:36:28.520 That is the derivative of secant x. 00:36:29.840 --> 00:36:33.280 However, we normally don't write it like that. 00:36:34.240 --> 00:36:40.280 we normally change this to... 00:36:50.920 --> 00:36:52.800 to this... 00:36:54.080 --> 00:36:56.020 and then what is one over cosine? 00:36:57.860 --> 00:37:00.080 that is secant... 00:37:00.080 --> 00:37:04.240 And sine over cosine is tangent 00:37:06.260 --> 00:37:12.080 Okay, that's how we normally write it that's the one we just memorize from now on 00:37:19.020 --> 00:37:21.200 Derivative of secant 00:37:31.440 --> 00:37:36.200 There's two more we're gonna do next week. 00:37:36.640 --> 00:37:39.557 Cotangent and cosecant are next; then we will have all six trigonometric 00:37:39.557 --> 00:37:40.307 derivatives. 00:37:42.300 --> 00:37:43.880 Okay, any questions on this? 00:37:44.580 --> 00:37:44.900 We're gonna stop here.