WEBVTT 00:00:05.220 --> 00:00:15.260 Okay. Tell me something, anything you know about inverse functions. 00:00:16.240 --> 00:00:17.060 Hold on. That's right. Yeah. 00:00:17.060 --> 00:00:31.066 Like the coordinates that look like flipped like that. So if a comma b is in F, then 00:00:31.066 --> 00:00:31.900 what? 00:00:36.300 --> 00:00:45.430 B comma a is in F inverse. That's literally the definition is you just flip all the 00:00:45.430 --> 00:00:46.200 ordered 00:00:46.200 --> 00:00:56.980 pairs. Somebody else tell me something. 00:01:05.240 --> 00:01:11.860 If F is a one to one function, 00:01:18.980 --> 00:01:32.420 then F inverse exists. Okay. So we're guaranteed to have inverse. 00:01:34.140 --> 00:01:37.080 If it is a one to one function, then what is a one to one function? 00:01:41.640 --> 00:01:45.620 Use the vertical and horizontal line tests. 00:01:49.320 --> 00:01:56.606 Every X corresponds only one Y. That makes it a function. And if every Y corresponds 00:01:56.606 --> 00:01:57.560 only one X, 00:01:57.560 --> 00:02:04.735 then it's a one to one function. Okay. Two Y values never get repeated. Okay. 00:02:04.735 --> 00:02:06.040 Somebody else. 00:02:10.880 --> 00:02:16.660 This is very, very important. F composed with F inverse 00:02:20.800 --> 00:02:25.080 gets you right back where we started and vice versa. 00:02:30.800 --> 00:02:39.054 I think we talked about that the other day. Right. E to the X and log base E of X or 00:02:39.054 --> 00:02:39.840 inverse. 00:02:40.900 --> 00:02:49.440 Because of that, we can plug them in. This equals X and LN of E to the X equals X. 00:02:50.640 --> 00:02:55.700 Very, very important. We're going to use that today. What else? 00:03:07.740 --> 00:03:11.440 Nothing else. How are their graphs related? 00:03:20.420 --> 00:03:28.903 We literally switch all of the X and Y coordinates and it gets reflected about the 00:03:28.903 --> 00:03:29.420 line. 00:03:29.420 --> 00:03:31.360 Y equals X. 00:03:39.960 --> 00:03:42.040 See if I can do this. 00:03:49.120 --> 00:03:49.780 Something like that. 00:03:53.120 --> 00:03:56.800 How are the domain and ranges related? 00:04:03.120 --> 00:04:06.180 Yeah. Opposites are flipped. It has everything to do with this, right? 00:04:06.740 --> 00:04:14.660 If b is in the range of f, then b is in the domain of f inverse. So the domain 00:04:18.200 --> 00:04:42.280 of F is the range of F inverse and the range of F is the domain of F inverse. 00:04:42.280 --> 00:04:47.300 That's pretty much it. 00:05:02.840 --> 00:05:10.940 So let's say we have some function, F of X and its inverse is F inverse of X. 00:05:12.660 --> 00:05:18.540 Our goal is to figure out F inverse prime, the derivative. 00:05:24.060 --> 00:05:25.580 How are we going to do this? 00:05:32.520 --> 00:05:34.180 We don't even know what the functions are. 00:05:42.660 --> 00:05:44.620 We're going to start with an equation that we know. 00:05:47.220 --> 00:05:49.100 We're going to start with one of these two equations. 00:05:52.320 --> 00:05:56.240 And we're going to differentiate it using what kind of differentiation? 00:05:57.780 --> 00:06:02.160 Implicit because we got functions within functions. 00:06:02.380 --> 00:06:03.760 Or we got functions on both sides. 00:06:05.020 --> 00:06:07.300 Are we going to use this one or are we going to use this one? 00:06:09.840 --> 00:06:10.800 Any thoughts? 00:06:13.200 --> 00:06:15.420 One will work. One's not going to work. 00:06:17.780 --> 00:06:18.880 Why a top one? 00:06:26.820 --> 00:06:33.980 Close. Let's start with the top one and then see what happens. 00:06:42.520 --> 00:06:43.620 You'll see why in a second. 00:06:45.620 --> 00:06:48.420 So we use implicit to differentiate the left-hand side. 00:06:49.380 --> 00:06:52.640 And since we have a function inside a function, what rule do we have to use? 00:06:54.580 --> 00:06:55.080 Chain rule. 00:06:56.500 --> 00:06:58.960 So we got F prime of the inside 00:07:04.100 --> 00:07:11.440 times the derivative of the inside, which is F inverse prime. 00:07:16.860 --> 00:07:23.700 And then the derivative of X is with respect to X, which is one or DX DX. 00:07:27.040 --> 00:07:29.500 This is what we're looking for right here. 00:07:29.700 --> 00:07:30.800 The derivative of F inverse. 00:07:31.920 --> 00:07:36.453 That's why we chose the top function because we multiplied by the derivative of 00:07:36.453 --> 00:07:36.740 this. 00:07:37.440 --> 00:07:37.980 It gives us this. 00:07:38.260 --> 00:07:41.800 If we were to switch it, it wouldn't help us. 00:07:42.280 --> 00:07:47.360 And then we simply divide by that. 00:08:02.980 --> 00:08:07.000 OK, this is the derivative of the inverse of a function. 00:08:08.660 --> 00:08:13.612 Even if you don't know what the function is, if we know the derivative of our 00:08:13.612 --> 00:08:14.320 function F, 00:08:15.880 --> 00:08:18.680 we plug into it the inverse. 00:08:19.040 --> 00:08:19.400 This is it. 00:08:20.600 --> 00:08:22.080 So let's do an example. 00:08:27.480 --> 00:08:31.380 We'll give you two functions that are inverses. 00:08:40.420 --> 00:08:40.940 What's that? 00:08:43.980 --> 00:08:44.760 I like that. 00:08:48.500 --> 00:08:49.520 Let's go with e to the x. 00:08:51.740 --> 00:08:55.400 We know the inverse is ln of X. 00:08:57.440 --> 00:08:59.840 We know the derivative of the inverse, right? 00:09:01.160 --> 00:09:02.460 We just did that last class. 00:09:03.760 --> 00:09:04.580 The derivative is what? 00:09:05.920 --> 00:09:07.020 One over X. 00:09:07.840 --> 00:09:09.680 But let's show it using that. 00:09:12.720 --> 00:09:16.480 That rule, F inverse. 00:09:16.920 --> 00:09:24.820 The derivative is one over F prime composed with the inverse. 00:09:33.300 --> 00:09:34.780 What is F prime of X? 00:09:36.280 --> 00:09:37.200 E to the X. 00:09:38.340 --> 00:09:42.980 So this is going to be one over E to the inverse. 00:09:47.520 --> 00:09:49.820 And the inverse is ln of X. 00:09:52.760 --> 00:09:55.080 And what is E to the ln of X? 00:09:57.920 --> 00:09:58.440 Just X. 00:09:58.440 --> 00:09:58.580 Just X. 00:09:59.320 --> 00:09:59.740 There it is. 00:10:00.400 --> 00:10:01.280 We got it. 00:10:02.020 --> 00:10:03.860 Even if we didn't know the derivative here. 00:10:07.240 --> 00:10:08.240 Let's try another one. 00:10:09.860 --> 00:10:13.120 How about X cubed? 00:10:18.640 --> 00:10:20.240 What's the inverse of X cubed? 00:10:23.780 --> 00:10:25.000 The cube root. 00:10:26.260 --> 00:10:27.240 Let's write it like this. 00:10:27.920 --> 00:10:28.480 One third power. 00:10:28.940 --> 00:10:29.820 That is the cube root. 00:10:33.440 --> 00:10:36.080 Obviously, we don't need that little rule. 00:10:36.080 --> 00:10:41.540 We know the derivative of this. 00:10:42.040 --> 00:10:45.540 One third x to the negative two-thirds power. 00:10:47.700 --> 00:10:48.900 Negative two thirds. 00:10:56.380 --> 00:10:58.500 But let's show it. 00:10:59.840 --> 00:11:04.600 F inverse prime is one over F prime. 00:11:06.460 --> 00:11:07.120 Of. 00:11:12.460 --> 00:11:14.840 Compose with F inverse. 00:11:24.940 --> 00:11:26.880 Deriv X cubed is three X squared. 00:11:30.840 --> 00:11:32.120 So it's one over. 00:11:34.960 --> 00:11:38.840 Three F inverse squared. 00:11:41.640 --> 00:11:46.780 Which is one over three times the cube root of X. 00:11:48.840 --> 00:11:49.700 Squared. 00:11:50.920 --> 00:11:52.240 Which is the same thing as. 00:11:59.240 --> 00:12:00.000 This makes sense. 00:12:00.000 --> 00:12:07.740 Okay, let's do some other examples. 00:12:08.160 --> 00:12:11.140 So we have no idea what the equation for F inverse is. 00:12:12.800 --> 00:12:16.200 We just give it a what six order pairs. 00:12:18.060 --> 00:12:21.880 Okay, we can find F inverse. 00:12:25.780 --> 00:12:27.420 Sorry, I screwed this up. 00:12:28.900 --> 00:12:31.660 This should be the derivative of F not the inverse of F. 00:12:42.100 --> 00:12:45.500 Okay, I want to know F inverse prime. 00:12:48.120 --> 00:12:49.040 Of two. 00:12:56.980 --> 00:13:00.000 No, not up to. 00:13:16.700 --> 00:13:18.360 Hold on, let me take for a sec. 00:13:23.300 --> 00:13:24.100 Oh, yes, that is right. 00:13:26.920 --> 00:13:28.500 How long were we going to do this? 00:13:31.540 --> 00:13:38.960 F inverse prime of X is one over F prime of inverse. 00:13:41.200 --> 00:13:42.560 X. 00:13:53.140 --> 00:13:55.600 So by that is one over. 00:13:56.880 --> 00:13:59.440 F prime of F inverse. 00:14:01.220 --> 00:14:01.560 Of two. 00:14:14.400 --> 00:14:15.960 Now we got to figure out this number. 00:14:19.300 --> 00:14:21.360 I don't have an F inverse column. 00:14:27.400 --> 00:14:28.980 How do I figure this out? 00:14:34.960 --> 00:14:38.500 Okay, we do know three. 00:14:48.380 --> 00:14:49.540 It's not written down. 00:14:50.000 --> 00:14:51.640 We do know three order pairs of F inverse. 00:14:51.920 --> 00:14:52.860 What's three order pairs? 00:14:55.660 --> 00:15:02.220 Seven, five; three, two; and eight, three. 00:15:14.720 --> 00:15:16.200 Okay, we don't know this. 00:15:16.400 --> 00:15:18.020 Sorry, I did screw this up. 00:15:19.260 --> 00:15:22.540 Let's call this three. 00:15:24.120 --> 00:15:24.600 Sorry. 00:15:29.940 --> 00:15:31.820 We don't know what f inverse of two is. 00:15:31.980 --> 00:15:32.640 How do we get these? 00:15:35.940 --> 00:15:40.860 If five comma seven is in F, then seven comma five is in F inverse. 00:15:41.860 --> 00:15:50.140 Okay, now we know F inverse of three, which is two. 00:15:56.920 --> 00:15:59.800 And what is F prime of two? 00:16:04.260 --> 00:16:05.160 That's it. 00:16:21.520 --> 00:16:25.780 Okay, so we know the slope of the tangent line of F inverse at X equals three. 00:16:25.980 --> 00:16:32.274 We have no idea what the function looks like, but we do not need it because we have 00:16:32.274 --> 00:16:33.260 this formula. 00:16:45.580 --> 00:16:51.220 Okay, what's the only functions that we don't know the derivative of yet? 00:16:56.480 --> 00:16:57.940 Inverse trig functions. 00:16:58.940 --> 00:17:00.400 We know the derivative of all other functions. 00:17:01.100 --> 00:17:04.920 Polynomial, rational, radicals, exponential, logarithm. 00:17:05.260 --> 00:17:07.900 We know the derivative of all six trig functions, 00:17:07.900 --> 00:17:10.240 but we haven't talked about the inverse of. 00:17:11.140 --> 00:17:12.740 The derivative of. 00:17:16.420 --> 00:17:16.880 So. 00:17:27.740 --> 00:17:30.860 What is the inverse of sine? 00:17:33.100 --> 00:17:35.280 We call this arcsine. 00:17:37.380 --> 00:17:39.500 Or we call it sine inverse. 00:17:42.580 --> 00:17:43.260 Okay. 00:17:45.800 --> 00:17:48.760 A quick review before we actually do this. 00:17:54.960 --> 00:17:56.820 This is sine, right? 00:17:58.540 --> 00:18:00.220 How do we find the inverses? 00:18:03.960 --> 00:18:11.620 The inverse relation is x equals sine of y. 00:18:12.640 --> 00:18:14.020 We just switch the order pair. 00:18:15.360 --> 00:18:17.300 And this function is our inverse. 00:18:18.820 --> 00:18:20.060 Okay, and then we graph it. 00:18:25.640 --> 00:18:27.180 It looks like. 00:18:40.000 --> 00:18:43.020 So, my God, except that's not a function. 00:18:43.460 --> 00:18:46.640 So what do we do? 00:18:49.920 --> 00:18:55.471 Because sine is not a one-to-one function, it does not have an inverse over all real 00:18:55.471 --> 00:18:56.000 numbers. 00:18:57.040 --> 00:19:01.180 But wherever it's one to one, it does have an inverse. 00:19:02.060 --> 00:19:03.180 So here's where we cut it off. 00:19:06.020 --> 00:19:09.700 We're going to go from here. 00:19:10.840 --> 00:19:12.300 This is going to be the domain. 00:19:14.600 --> 00:19:20.920 Okay, which is negative pi over two to pi over two. 00:19:26.000 --> 00:19:28.000 So we have to cut this off. 00:19:35.120 --> 00:19:36.720 It looks something like that. 00:19:37.400 --> 00:19:38.860 It's a very small function. 00:19:40.040 --> 00:19:44.900 The domain of this is the domain. 00:19:47.160 --> 00:19:54.520 We only plug in angles from negative pi over two to pi over two. 00:20:02.280 --> 00:20:03.360 No, I take this back. 00:20:12.960 --> 00:20:14.020 Let's go back to time. 00:20:14.420 --> 00:20:15.700 What is the domain of sine? 00:20:23.340 --> 00:20:25.400 The domain of sine is all real numbers. 00:20:27.920 --> 00:20:29.440 What is the range? 00:20:33.880 --> 00:20:34.840 From negative one to one. 00:20:35.420 --> 00:20:36.720 Sorry, I screwed this up. 00:20:38.100 --> 00:20:39.460 Actually, that was correct. 00:20:40.240 --> 00:20:44.160 Except I screwed up the. 00:20:48.900 --> 00:20:50.360 I screwed up the yellow part. 00:20:51.140 --> 00:20:51.920 We'll come back in a second. 00:20:52.580 --> 00:20:58.240 The range of sine becomes the domain of sine inverse. 00:21:01.100 --> 00:21:06.320 In arcsine, or sine inverse, we input a value 00:21:06.320 --> 00:21:10.360 right, as a y coordinate from negative one to one, and it outputs an angle. 00:21:12.720 --> 00:21:17.400 Some angle between negative pi over two to pi over two. 00:21:19.000 --> 00:21:22.560 Okay, so this is pi over two. 00:21:24.840 --> 00:21:26.680 This is negative pi over two. 00:21:28.740 --> 00:21:35.660 We cut it off right here at negative one and at one. 00:21:47.740 --> 00:21:50.740 Okay, that was just a review of what sign and arc sign is. 00:21:52.420 --> 00:21:54.480 Now let's find the derivative of it. 00:21:56.720 --> 00:21:58.960 We don't know it, so we can't do anything with this. 00:22:00.260 --> 00:22:01.040 But what do we know? 00:22:03.680 --> 00:22:16.060 Its derivative is one over f prime evaluated at f inverse of x. 00:22:19.140 --> 00:22:22.680 F prime evaluated at f inverse of x. 00:22:27.000 --> 00:22:30.340 Here, f of x is sine. 00:22:32.960 --> 00:22:37.500 F inverse is arcsine. 00:22:42.520 --> 00:22:44.060 What is the derivative of sine? 00:22:45.400 --> 00:22:46.120 Cosine. 00:22:48.300 --> 00:22:59.680 This is one over cosine of the inverse, which is arcsine x. 00:23:03.940 --> 00:23:07.440 Okay, this is the derivative. 00:23:08.520 --> 00:23:12.720 What do we call this circle? 00:23:16.220 --> 00:23:17.420 Uh-oh. 00:23:17.840 --> 00:23:19.120 Circle radius one is called. 00:23:22.220 --> 00:23:23.560 Oh boy, people. 00:23:30.400 --> 00:23:31.580 Let's try this one more time. 00:23:31.740 --> 00:23:34.660 Somebody tell me, what do we call a circle radius one? 00:23:36.540 --> 00:23:37.160 The unit circle. 00:23:37.720 --> 00:23:38.060 Thank you. 00:23:38.460 --> 00:23:38.900 Well done. 00:23:43.160 --> 00:23:46.540 Circle with the radius is one unit, whatever that might be. 00:23:46.780 --> 00:23:47.100 One point. 00:23:49.340 --> 00:23:50.800 Called the unit circle. 00:24:07.640 --> 00:24:08.900 Let's call this X. 00:24:13.360 --> 00:24:14.260 Let's call this Y. 00:24:24.620 --> 00:24:36.060 A and B. 00:24:43.020 --> 00:24:44.980 And let's call this theta. 00:24:58.980 --> 00:25:02.740 Okay, I changed it because this X was not the same thing as the other X. 00:25:03.960 --> 00:25:09.400 When you plug a number between negative one and positive one, 00:25:12.720 --> 00:25:17.980 it's this, it's this length here. 00:25:18.840 --> 00:25:19.640 Why is that? 00:25:25.400 --> 00:25:29.400 Arcsine gives you an angle 00:25:31.360 --> 00:25:36.340 such that the sine of the angle is this length. 00:25:37.940 --> 00:25:40.760 Right, what is sine of theta here? 00:25:43.360 --> 00:25:45.960 It's the, it's this value, B. 00:25:46.340 --> 00:25:46.620 We agree? 00:25:47.760 --> 00:25:48.000 Okay. 00:25:57.740 --> 00:26:00.440 So this is my B value. 00:26:03.040 --> 00:26:06.760 Okay, and this outputs an angle. 00:26:16.180 --> 00:26:18.040 This gives us our theta. 00:26:19.140 --> 00:26:20.080 This one right here. 00:26:21.740 --> 00:26:22.680 You with me so far? 00:26:29.100 --> 00:26:32.140 And what is cosine of that angle? 00:26:36.040 --> 00:26:36.580 It is A. 00:26:38.880 --> 00:26:39.480 Okay. 00:26:42.720 --> 00:26:44.580 Arcsine of b is the angle. 00:26:45.140 --> 00:26:46.400 This inner part is the angle. 00:26:51.860 --> 00:26:53.400 We're almost there. 00:26:59.300 --> 00:27:01.080 This was our input right here. 00:27:01.280 --> 00:27:02.660 It wasn't really X, it was B. 00:27:03.300 --> 00:27:04.480 Let's call B for a second. 00:27:07.640 --> 00:27:10.580 The derivative is one over A. 00:27:11.820 --> 00:27:14.220 But we can put that in terms of B. 00:27:15.360 --> 00:27:17.860 I should have called that X, but I screwed up. 00:27:19.540 --> 00:27:23.440 Okay, what is this in terms of our number B? 00:27:29.000 --> 00:27:30.260 What's this length right here? 00:27:33.400 --> 00:27:36.500 Because we're on the unit circle, it's of length one. 00:27:36.820 --> 00:27:37.000 Thank you. 00:27:38.020 --> 00:27:40.500 Okay, the same magic people. 00:27:43.480 --> 00:27:45.300 And then we can set up the Pythagorean theorem. 00:27:46.180 --> 00:27:46.240 Yeah? 00:27:47.900 --> 00:27:50.820 A squared plus B squared equals one squared. 00:27:50.980 --> 00:27:51.340 We agree? 00:27:52.220 --> 00:27:59.200 And if we solve for A here, A squared is one minus B squared. 00:28:01.900 --> 00:28:05.340 And then A will either equal the positive or the negative. 00:28:08.080 --> 00:28:10.140 Square root of one minus B squared. 00:28:11.460 --> 00:28:12.080 Which one is it? 00:28:12.160 --> 00:28:13.100 Is it a positive or negative? 00:28:15.100 --> 00:28:15.980 Why positive? 00:28:18.140 --> 00:28:19.740 We're in the first quadrant. 00:28:23.780 --> 00:28:26.600 And even if our angle is down here, 00:28:28.820 --> 00:28:31.140 which would be, no, it would be this angle. 00:28:32.700 --> 00:28:34.760 Okay, the A value is still positive. 00:28:35.720 --> 00:28:40.920 Because cosine of any angle in quadrant four is still the positive A. 00:28:43.620 --> 00:28:45.640 Because all students take calculus. 00:28:48.120 --> 00:28:49.240 So long story short, 00:28:50.820 --> 00:28:55.600 this is the square root of one minus B squared. 00:28:57.660 --> 00:29:01.580 Which really was our input here. 00:29:01.720 --> 00:29:02.760 I should have put a B here. 00:29:05.600 --> 00:29:11.000 Okay, so that was the proof that the derivative, that was the proof, 00:29:12.900 --> 00:29:14.600 of arc sine. 00:29:16.900 --> 00:29:20.960 This inverse trig function turns out to be this algebraic function, 00:29:21.280 --> 00:29:23.280 one over the square root of one minus x squared. 00:29:32.260 --> 00:29:33.280 It's kind of cool. 00:29:34.020 --> 00:29:35.140 It's kind of weird. 00:29:35.880 --> 00:29:39.100 The derivative of some trig function, well it's an inverse trig, 00:29:39.320 --> 00:29:42.320 turns out to be this radical function. 00:29:43.920 --> 00:29:45.020 But there it is. 00:29:46.660 --> 00:29:49.560 Let's do the same thing with arccosine. 00:29:53.940 --> 00:29:54.940 Let's call it f. 00:30:09.400 --> 00:30:10.100 So 00:30:25.680 --> 00:30:29.300 we're going to have to memorize all of them. 00:30:34.140 --> 00:30:35.420 We never proved it. 00:30:36.480 --> 00:30:37.260 Now you guys know it. 00:30:49.400 --> 00:30:51.660 If you ever forget, just prove it. 00:30:52.480 --> 00:30:53.640 Exactly, I like this. 00:30:59.300 --> 00:31:01.000 Okay, this one's going to work out with x's. 00:31:01.040 --> 00:31:01.640 It'll be much better. 00:31:03.940 --> 00:31:07.500 Okay, so this is one over. 00:31:08.960 --> 00:31:20.320 F prime is negative sine, evaluated at the inverse, arccosine x. 00:31:22.660 --> 00:31:24.160 Again, this is the derivative. 00:31:24.380 --> 00:31:27.100 We could stop here, but it will, turns out it simplifies. 00:31:29.440 --> 00:31:30.400 Let's see if we can do this. 00:31:33.280 --> 00:31:42.740 Correctly this time. 00:31:54.640 --> 00:31:58.340 Okay, now x will work out quite nicely. 00:32:14.900 --> 00:32:20.860 So, again, we're inputting some length between negative one and positive one. 00:32:23.100 --> 00:32:24.400 So arc cosine. 00:32:31.020 --> 00:32:33.300 We're inputting this length x. 00:32:36.340 --> 00:32:38.060 And it outputs an angle. 00:32:38.980 --> 00:32:42.180 It outputs this angle, theta. 00:32:50.120 --> 00:32:52.120 And what is sine of this angle? 00:32:57.420 --> 00:32:57.940 Y. 00:33:02.520 --> 00:33:08.680 Okay, and then the last thing we need to do is change this in terms of x. 00:33:12.040 --> 00:33:12.900 What's the hypotenuse? 00:33:14.040 --> 00:33:14.060 One. 00:33:14.140 --> 00:33:14.880 One, thank you. 00:33:16.880 --> 00:33:18.060 Because it's the unit circle. 00:33:19.020 --> 00:33:22.460 So we got x squared plus y squared equals one squared. 00:33:24.900 --> 00:33:28.600 Y squared equals one minus x squared. 00:33:29.940 --> 00:33:39.840 Y equals plus or minus the square root of one minus x squared. 00:33:46.160 --> 00:33:47.700 Is it the plus or the minus? 00:33:54.400 --> 00:33:55.220 Y plus. 00:33:56.860 --> 00:33:58.280 Because we're in quadrant one. 00:34:00.720 --> 00:34:11.942 If we were in this quadrant, okay, because the range of arc cosine is between zero 00:34:11.942 --> 00:34:12.900 and pi. 00:34:15.100 --> 00:34:20.280 Okay, this would be my x, this would be my y. 00:34:21.040 --> 00:34:23.520 The y is still positive in quadrant two, we agree? 00:34:24.160 --> 00:34:25.140 So it has to be positive. 00:34:27.280 --> 00:34:28.460 So we can get rid of this. 00:34:30.200 --> 00:34:33.640 And so it is the square root of one minus x squared. 00:34:39.960 --> 00:34:47.420 Okay, so long story short, the derivative of arc cosine, 00:34:49.360 --> 00:34:55.060 We normally also call it cosine inverse, but it's nice to distinguish it here. 00:34:56.360 --> 00:35:01.540 is simply negative one over the square root of one minus x squared. 00:35:03.440 --> 00:35:07.960 Which is the same as derivative of sine inverse, except it's negative. 00:35:15.280 --> 00:35:18.220 So should we do tangents? 00:35:24.280 --> 00:35:25.680 Let's do tangents for fun. 00:35:27.620 --> 00:35:29.800 For fun. Definitely fun. 00:35:50.580 --> 00:35:53.800 I'm using a different letter because it won't work out that nice. 00:36:05.320 --> 00:36:07.180 Arc tan, same thing as tan inverse. 00:36:09.640 --> 00:36:16.620 A, A, A, A. It doesn't matter. Let's go A. Why not? 00:36:27.700 --> 00:36:29.000 Sorry, what is the derivative of tan? 00:36:29.000 --> 00:36:30.420 Secant squared. 00:36:31.900 --> 00:36:32.440 Secant squared. 00:36:33.580 --> 00:36:36.120 Let me write it like that, secant squared. 00:36:42.640 --> 00:36:51.380 F inverse prime is one over F prime composed with the inverse. 00:36:54.800 --> 00:37:14.180 This is going to be one over secant squared composed with tan inverse. 00:37:19.640 --> 00:37:21.460 We proved that? 00:37:22.080 --> 00:37:22.640 I do. 00:37:22.860 --> 00:37:24.040 Okay. 00:37:38.500 --> 00:37:40.860 Uh oh. 00:38:07.040 --> 00:38:10.280 Go and look back. 00:38:10.280 --> 00:38:17.857 We are not plugging in just a y value or just an x value; we are plugging in y over 00:38:17.857 --> 00:38:18.040 x. 00:38:18.040 --> 00:38:31.060 the ratio y over x okay wherever it is and this outputs an angle between 00:38:31.060 --> 00:38:36.920 negative pi over 2 pi over 2 okay and that angle would be theta whatever it is 00:38:44.600 --> 00:39:04.720 and what is secant of that angle secant is 1 over cosine okay so 1 over 00:39:43.940 --> 00:40:06.760 a was we agree it's y over x okay so we need to put in terms of a not just x 00:40:17.000 --> 00:40:31.520 okay so a is y over x so x is y over a 00:40:54.220 --> 00:41:03.660 X squared plus y squared is one. We get 00:41:03.660 --> 00:41:15.220 that by the Pythagorean theorem x squared equals 1 minus y squared 00:41:22.240 --> 00:41:38.060 X equals the square root of one minus y squared. Why is it guaranteed to be 00:41:38.060 --> 00:42:02.316 be positive actually it might not be guaranteed in the first quadrant is but I guess 00:42:02.316 --> 00:42:03.760 my we 00:42:27.880 --> 00:42:32.876 didn't have to have an angle in the first one it could have been down here but it 00:42:32.876 --> 00:42:33.740 doesn't matter 00:43:07.840 --> 00:43:09.880 give me one more sec we're almost done 00:43:13.100 --> 00:43:26.020 if I substitute in a here okay which 00:43:26.020 --> 00:43:42.244 becomes x squared plus a squared x squared equals 1 we agree which is 1 plus a 00:43:42.244 --> 00:43:43.700 squared 00:43:43.700 --> 00:44:01.126 times x squared so x squared is 1 over 1 plus a squared so x is either plus or minus 00:44:01.126 --> 00:44:03.200 the square 00:44:03.200 --> 00:44:13.941 root of 1 over let's do this 1 plus a squared okay this is where sorry took a long 00:44:13.941 --> 00:44:15.120 time this 00:44:15.120 --> 00:44:28.666 is what we're gonna plug in for x right here so we got 1 over 1 over where is it the 00:44:28.666 --> 00:44:30.440 square root 00:44:37.880 --> 00:44:54.317 what is 1 over 1 over it's just becomes this in the numerator it took a long time 00:44:54.317 --> 00:44:55.940 and then 00:44:55.940 --> 00:45:15.920 that simplifies to one over one plus a squared. That was messy because 00:45:15.920 --> 00:45:26.100 it's a ratio it's not just the x to the y coordinate so what is the derivative of 00:45:26.100 --> 00:45:26.980 arc tan 00:45:26.980 --> 00:45:39.470 or tan inverse is simply 1 over 1 plus x squared which is a nice little rational 00:45:39.470 --> 00:45:41.500 function okay 00:45:42.070 --> 00:45:48.748 okay these three are the only three we need for this class I don't know why the the 00:45:48.748 --> 00:45:49.070 blue 00:45:49.070 --> 00:46:08.753 book we use they like to put in use wait this is just saying if there's another 00:46:08.753 --> 00:46:12.490 function inside 00:46:12.490 --> 00:46:22.420 it's not just x it's 1 over it times the derivative what our u is okay it's just 00:46:22.420 --> 00:46:24.530 putting the chain 00:46:24.530 --> 00:46:33.526 rule in there if there's something else inside of it okay so here are the other 00:46:33.526 --> 00:46:35.690 three they're messy 00:46:36.370 --> 00:46:41.742 the other three involve these absolute values, and we don't need them this time. We 00:46:41.742 --> 00:46:43.490 need to memorize the three: 00:46:43.510 --> 00:46:51.757 arcsine arc cosine and arc tangent we wrote those three down already yeah okay I 00:46:51.757 --> 00:46:53.510 think we're gonna 00:46:55.890 --> 00:46:57.750 stop there