1 00:00:05,220 --> 00:00:15,260 Okay. Tell me something, anything you know about inverse functions. 2 00:00:16,240 --> 00:00:17,060 Hold on. That's right. Yeah. 3 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 4 00:00:31,066 --> 00:00:31,900 what? 5 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 6 00:00:45,430 --> 00:00:46,200 ordered 7 00:00:46,200 --> 00:00:56,980 pairs. Somebody else tell me something. 8 00:01:05,240 --> 00:01:11,860 If F is a one to one function, 9 00:01:18,980 --> 00:01:32,420 then F inverse exists. Okay. So we're guaranteed to have inverse. 10 00:01:34,140 --> 00:01:37,080 If it is a one to one function, then what is a one to one function? 11 00:01:41,640 --> 00:01:45,620 Use the vertical and horizontal line tests. 12 00:01:49,320 --> 00:01:56,606 Every X corresponds only one Y. That makes it a function. And if every Y corresponds 13 00:01:56,606 --> 00:01:57,560 only one X, 14 00:01:57,560 --> 00:02:04,735 then it's a one to one function. Okay. Two Y values never get repeated. Okay. 15 00:02:04,735 --> 00:02:06,040 Somebody else. 16 00:02:10,880 --> 00:02:16,660 This is very, very important. F composed with F inverse 17 00:02:20,800 --> 00:02:25,080 gets you right back where we started and vice versa. 18 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 19 00:02:39,054 --> 00:02:39,840 inverse. 20 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. 21 00:02:50,640 --> 00:02:55,700 Very, very important. We're going to use that today. What else? 22 00:03:07,740 --> 00:03:11,440 Nothing else. How are their graphs related? 23 00:03:20,420 --> 00:03:28,903 We literally switch all of the X and Y coordinates and it gets reflected about the 24 00:03:28,903 --> 00:03:29,420 line. 25 00:03:29,420 --> 00:03:31,360 Y equals X. 26 00:03:39,960 --> 00:03:42,040 See if I can do this. 27 00:03:49,120 --> 00:03:49,780 Something like that. 28 00:03:53,120 --> 00:03:56,800 How are the domain and ranges related? 29 00:04:03,120 --> 00:04:06,180 Yeah. Opposites are flipped. It has everything to do with this, right? 30 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 31 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. 32 00:04:42,280 --> 00:04:47,300 That's pretty much it. 33 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. 34 00:05:12,660 --> 00:05:18,540 Our goal is to figure out F inverse prime, the derivative. 35 00:05:24,060 --> 00:05:25,580 How are we going to do this? 36 00:05:32,520 --> 00:05:34,180 We don't even know what the functions are. 37 00:05:42,660 --> 00:05:44,620 We're going to start with an equation that we know. 38 00:05:47,220 --> 00:05:49,100 We're going to start with one of these two equations. 39 00:05:52,320 --> 00:05:56,240 And we're going to differentiate it using what kind of differentiation? 40 00:05:57,780 --> 00:06:02,160 Implicit because we got functions within functions. 41 00:06:02,380 --> 00:06:03,760 Or we got functions on both sides. 42 00:06:05,020 --> 00:06:07,300 Are we going to use this one or are we going to use this one? 43 00:06:09,840 --> 00:06:10,800 Any thoughts? 44 00:06:13,200 --> 00:06:15,420 One will work. One's not going to work. 45 00:06:17,780 --> 00:06:18,880 Why a top one? 46 00:06:26,820 --> 00:06:33,980 Close. Let's start with the top one and then see what happens. 47 00:06:42,520 --> 00:06:43,620 You'll see why in a second. 48 00:06:45,620 --> 00:06:48,420 So we use implicit to differentiate the left-hand side. 49 00:06:49,380 --> 00:06:52,640 And since we have a function inside a function, what rule do we have to use? 50 00:06:54,580 --> 00:06:55,080 Chain rule. 51 00:06:56,500 --> 00:06:58,960 So we got F prime of the inside 52 00:07:04,100 --> 00:07:11,440 times the derivative of the inside, which is F inverse prime. 53 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. 54 00:07:27,040 --> 00:07:29,500 This is what we're looking for right here. 55 00:07:29,700 --> 00:07:30,800 The derivative of F inverse. 56 00:07:31,920 --> 00:07:36,453 That's why we chose the top function because we multiplied by the derivative of 57 00:07:36,453 --> 00:07:36,740 this. 58 00:07:37,440 --> 00:07:37,980 It gives us this. 59 00:07:38,260 --> 00:07:41,800 If we were to switch it, it wouldn't help us. 60 00:07:42,280 --> 00:07:47,360 And then we simply divide by that. 61 00:08:02,980 --> 00:08:07,000 OK, this is the derivative of the inverse of a function. 62 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 63 00:08:13,612 --> 00:08:14,320 function F, 64 00:08:15,880 --> 00:08:18,680 we plug into it the inverse. 65 00:08:19,040 --> 00:08:19,400 This is it. 66 00:08:20,600 --> 00:08:22,080 So let's do an example. 67 00:08:27,480 --> 00:08:31,380 We'll give you two functions that are inverses. 68 00:08:40,420 --> 00:08:40,940 What's that? 69 00:08:43,980 --> 00:08:44,760 I like that. 70 00:08:48,500 --> 00:08:49,520 Let's go with e to the x. 71 00:08:51,740 --> 00:08:55,400 We know the inverse is ln of X. 72 00:08:57,440 --> 00:08:59,840 We know the derivative of the inverse, right? 73 00:09:01,160 --> 00:09:02,460 We just did that last class. 74 00:09:03,760 --> 00:09:04,580 The derivative is what? 75 00:09:05,920 --> 00:09:07,020 One over X. 76 00:09:07,840 --> 00:09:09,680 But let's show it using that. 77 00:09:12,720 --> 00:09:16,480 That rule, F inverse. 78 00:09:16,920 --> 00:09:24,820 The derivative is one over F prime composed with the inverse. 79 00:09:33,300 --> 00:09:34,780 What is F prime of X? 80 00:09:36,280 --> 00:09:37,200 E to the X. 81 00:09:38,340 --> 00:09:42,980 So this is going to be one over E to the inverse. 82 00:09:47,520 --> 00:09:49,820 And the inverse is ln of X. 83 00:09:52,760 --> 00:09:55,080 And what is E to the ln of X? 84 00:09:57,920 --> 00:09:58,440 Just X. 85 00:09:58,440 --> 00:09:58,580 Just X. 86 00:09:59,320 --> 00:09:59,740 There it is. 87 00:10:00,400 --> 00:10:01,280 We got it. 88 00:10:02,020 --> 00:10:03,860 Even if we didn't know the derivative here. 89 00:10:07,240 --> 00:10:08,240 Let's try another one. 90 00:10:09,860 --> 00:10:13,120 How about X cubed? 91 00:10:18,640 --> 00:10:20,240 What's the inverse of X cubed? 92 00:10:23,780 --> 00:10:25,000 The cube root. 93 00:10:26,260 --> 00:10:27,240 Let's write it like this. 94 00:10:27,920 --> 00:10:28,480 One third power. 95 00:10:28,940 --> 00:10:29,820 That is the cube root. 96 00:10:33,440 --> 00:10:36,080 Obviously, we don't need that little rule. 97 00:10:36,080 --> 00:10:41,540 We know the derivative of this. 98 00:10:42,040 --> 00:10:45,540 One third x to the negative two-thirds power. 99 00:10:47,700 --> 00:10:48,900 Negative two thirds. 100 00:10:56,380 --> 00:10:58,500 But let's show it. 101 00:10:59,840 --> 00:11:04,600 F inverse prime is one over F prime. 102 00:11:06,460 --> 00:11:07,120 Of. 103 00:11:12,460 --> 00:11:14,840 Compose with F inverse. 104 00:11:24,940 --> 00:11:26,880 Deriv X cubed is three X squared. 105 00:11:30,840 --> 00:11:32,120 So it's one over. 106 00:11:34,960 --> 00:11:38,840 Three F inverse squared. 107 00:11:41,640 --> 00:11:46,780 Which is one over three times the cube root of X. 108 00:11:48,840 --> 00:11:49,700 Squared. 109 00:11:50,920 --> 00:11:52,240 Which is the same thing as. 110 00:11:59,240 --> 00:12:00,000 This makes sense. 111 00:12:00,000 --> 00:12:07,740 Okay, let's do some other examples. 112 00:12:08,160 --> 00:12:11,140 So we have no idea what the equation for F inverse is. 113 00:12:12,800 --> 00:12:16,200 We just give it a what six order pairs. 114 00:12:18,060 --> 00:12:21,880 Okay, we can find F inverse. 115 00:12:25,780 --> 00:12:27,420 Sorry, I screwed this up. 116 00:12:28,900 --> 00:12:31,660 This should be the derivative of F not the inverse of F. 117 00:12:42,100 --> 00:12:45,500 Okay, I want to know F inverse prime. 118 00:12:48,120 --> 00:12:49,040 Of two. 119 00:12:56,980 --> 00:13:00,000 No, not up to. 120 00:13:16,700 --> 00:13:18,360 Hold on, let me take for a sec. 121 00:13:23,300 --> 00:13:24,100 Oh, yes, that is right. 122 00:13:26,920 --> 00:13:28,500 How long were we going to do this? 123 00:13:31,540 --> 00:13:38,960 F inverse prime of X is one over F prime of inverse. 124 00:13:41,200 --> 00:13:42,560 X. 125 00:13:53,140 --> 00:13:55,600 So by that is one over. 126 00:13:56,880 --> 00:13:59,440 F prime of F inverse. 127 00:14:01,220 --> 00:14:01,560 Of two. 128 00:14:14,400 --> 00:14:15,960 Now we got to figure out this number. 129 00:14:19,300 --> 00:14:21,360 I don't have an F inverse column. 130 00:14:27,400 --> 00:14:28,980 How do I figure this out? 131 00:14:34,960 --> 00:14:38,500 Okay, we do know three. 132 00:14:48,380 --> 00:14:49,540 It's not written down. 133 00:14:50,000 --> 00:14:51,640 We do know three order pairs of F inverse. 134 00:14:51,920 --> 00:14:52,860 What's three order pairs? 135 00:14:55,660 --> 00:15:02,220 Seven, five; three, two; and eight, three. 136 00:15:14,720 --> 00:15:16,200 Okay, we don't know this. 137 00:15:16,400 --> 00:15:18,020 Sorry, I did screw this up. 138 00:15:19,260 --> 00:15:22,540 Let's call this three. 139 00:15:24,120 --> 00:15:24,600 Sorry. 140 00:15:29,940 --> 00:15:31,820 We don't know what f inverse of two is. 141 00:15:31,980 --> 00:15:32,640 How do we get these? 142 00:15:35,940 --> 00:15:40,860 If five comma seven is in F, then seven comma five is in F inverse. 143 00:15:41,860 --> 00:15:50,140 Okay, now we know F inverse of three, which is two. 144 00:15:56,920 --> 00:15:59,800 And what is F prime of two? 145 00:16:04,260 --> 00:16:05,160 That's it. 146 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. 147 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 148 00:16:32,274 --> 00:16:33,260 this formula. 149 00:16:45,580 --> 00:16:51,220 Okay, what's the only functions that we don't know the derivative of yet? 150 00:16:56,480 --> 00:16:57,940 Inverse trig functions. 151 00:16:58,940 --> 00:17:00,400 We know the derivative of all other functions. 152 00:17:01,100 --> 00:17:04,920 Polynomial, rational, radicals, exponential, logarithm. 153 00:17:05,260 --> 00:17:07,900 We know the derivative of all six trig functions, 154 00:17:07,900 --> 00:17:10,240 but we haven't talked about the inverse of. 155 00:17:11,140 --> 00:17:12,740 The derivative of. 156 00:17:16,420 --> 00:17:16,880 So. 157 00:17:27,740 --> 00:17:30,860 What is the inverse of sine? 158 00:17:33,100 --> 00:17:35,280 We call this arcsine. 159 00:17:37,380 --> 00:17:39,500 Or we call it sine inverse. 160 00:17:42,580 --> 00:17:43,260 Okay. 161 00:17:45,800 --> 00:17:48,760 A quick review before we actually do this. 162 00:17:54,960 --> 00:17:56,820 This is sine, right? 163 00:17:58,540 --> 00:18:00,220 How do we find the inverses? 164 00:18:03,960 --> 00:18:11,620 The inverse relation is x equals sine of y. 165 00:18:12,640 --> 00:18:14,020 We just switch the order pair. 166 00:18:15,360 --> 00:18:17,300 And this function is our inverse. 167 00:18:18,820 --> 00:18:20,060 Okay, and then we graph it. 168 00:18:25,640 --> 00:18:27,180 It looks like. 169 00:18:40,000 --> 00:18:43,020 So, my God, except that's not a function. 170 00:18:43,460 --> 00:18:46,640 So what do we do? 171 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 172 00:18:55,471 --> 00:18:56,000 numbers. 173 00:18:57,040 --> 00:19:01,180 But wherever it's one to one, it does have an inverse. 174 00:19:02,060 --> 00:19:03,180 So here's where we cut it off. 175 00:19:06,020 --> 00:19:09,700 We're going to go from here. 176 00:19:10,840 --> 00:19:12,300 This is going to be the domain. 177 00:19:14,600 --> 00:19:20,920 Okay, which is negative pi over two to pi over two. 178 00:19:26,000 --> 00:19:28,000 So we have to cut this off. 179 00:19:35,120 --> 00:19:36,720 It looks something like that. 180 00:19:37,400 --> 00:19:38,860 It's a very small function. 181 00:19:40,040 --> 00:19:44,900 The domain of this is the domain. 182 00:19:47,160 --> 00:19:54,520 We only plug in angles from negative pi over two to pi over two. 183 00:20:02,280 --> 00:20:03,360 No, I take this back. 184 00:20:12,960 --> 00:20:14,020 Let's go back to time. 185 00:20:14,420 --> 00:20:15,700 What is the domain of sine? 186 00:20:23,340 --> 00:20:25,400 The domain of sine is all real numbers. 187 00:20:27,920 --> 00:20:29,440 What is the range? 188 00:20:33,880 --> 00:20:34,840 From negative one to one. 189 00:20:35,420 --> 00:20:36,720 Sorry, I screwed this up. 190 00:20:38,100 --> 00:20:39,460 Actually, that was correct. 191 00:20:40,240 --> 00:20:44,160 Except I screwed up the. 192 00:20:48,900 --> 00:20:50,360 I screwed up the yellow part. 193 00:20:51,140 --> 00:20:51,920 We'll come back in a second. 194 00:20:52,580 --> 00:20:58,240 The range of sine becomes the domain of sine inverse. 195 00:21:01,100 --> 00:21:06,320 In arcsine, or sine inverse, we input a value 196 00:21:06,320 --> 00:21:10,360 right, as a y coordinate from negative one to one, and it outputs an angle. 197 00:21:12,720 --> 00:21:17,400 Some angle between negative pi over two to pi over two. 198 00:21:19,000 --> 00:21:22,560 Okay, so this is pi over two. 199 00:21:24,840 --> 00:21:26,680 This is negative pi over two. 200 00:21:28,740 --> 00:21:35,660 We cut it off right here at negative one and at one. 201 00:21:47,740 --> 00:21:50,740 Okay, that was just a review of what sign and arc sign is. 202 00:21:52,420 --> 00:21:54,480 Now let's find the derivative of it. 203 00:21:56,720 --> 00:21:58,960 We don't know it, so we can't do anything with this. 204 00:22:00,260 --> 00:22:01,040 But what do we know? 205 00:22:03,680 --> 00:22:16,060 Its derivative is one over f prime evaluated at f inverse of x. 206 00:22:19,140 --> 00:22:22,680 F prime evaluated at f inverse of x. 207 00:22:27,000 --> 00:22:30,340 Here, f of x is sine. 208 00:22:32,960 --> 00:22:37,500 F inverse is arcsine. 209 00:22:42,520 --> 00:22:44,060 What is the derivative of sine? 210 00:22:45,400 --> 00:22:46,120 Cosine. 211 00:22:48,300 --> 00:22:59,680 This is one over cosine of the inverse, which is arcsine x. 212 00:23:03,940 --> 00:23:07,440 Okay, this is the derivative. 213 00:23:08,520 --> 00:23:12,720 What do we call this circle? 214 00:23:16,220 --> 00:23:17,420 Uh-oh. 215 00:23:17,840 --> 00:23:19,120 Circle radius one is called. 216 00:23:22,220 --> 00:23:23,560 Oh boy, people. 217 00:23:30,400 --> 00:23:31,580 Let's try this one more time. 218 00:23:31,740 --> 00:23:34,660 Somebody tell me, what do we call a circle radius one? 219 00:23:36,540 --> 00:23:37,160 The unit circle. 220 00:23:37,720 --> 00:23:38,060 Thank you. 221 00:23:38,460 --> 00:23:38,900 Well done. 222 00:23:43,160 --> 00:23:46,540 Circle with the radius is one unit, whatever that might be. 223 00:23:46,780 --> 00:23:47,100 One point. 224 00:23:49,340 --> 00:23:50,800 Called the unit circle. 225 00:24:07,640 --> 00:24:08,900 Let's call this X. 226 00:24:13,360 --> 00:24:14,260 Let's call this Y. 227 00:24:24,620 --> 00:24:36,060 A and B. 228 00:24:43,020 --> 00:24:44,980 And let's call this theta. 229 00:24:58,980 --> 00:25:02,740 Okay, I changed it because this X was not the same thing as the other X. 230 00:25:03,960 --> 00:25:09,400 When you plug a number between negative one and positive one, 231 00:25:12,720 --> 00:25:17,980 it's this, it's this length here. 232 00:25:18,840 --> 00:25:19,640 Why is that? 233 00:25:25,400 --> 00:25:29,400 Arcsine gives you an angle 234 00:25:31,360 --> 00:25:36,340 such that the sine of the angle is this length. 235 00:25:37,940 --> 00:25:40,760 Right, what is sine of theta here? 236 00:25:43,360 --> 00:25:45,960 It's the, it's this value, B. 237 00:25:46,340 --> 00:25:46,620 We agree? 238 00:25:47,760 --> 00:25:48,000 Okay. 239 00:25:57,740 --> 00:26:00,440 So this is my B value. 240 00:26:03,040 --> 00:26:06,760 Okay, and this outputs an angle. 241 00:26:16,180 --> 00:26:18,040 This gives us our theta. 242 00:26:19,140 --> 00:26:20,080 This one right here. 243 00:26:21,740 --> 00:26:22,680 You with me so far? 244 00:26:29,100 --> 00:26:32,140 And what is cosine of that angle? 245 00:26:36,040 --> 00:26:36,580 It is A. 246 00:26:38,880 --> 00:26:39,480 Okay. 247 00:26:42,720 --> 00:26:44,580 Arcsine of b is the angle. 248 00:26:45,140 --> 00:26:46,400 This inner part is the angle. 249 00:26:51,860 --> 00:26:53,400 We're almost there. 250 00:26:59,300 --> 00:27:01,080 This was our input right here. 251 00:27:01,280 --> 00:27:02,660 It wasn't really X, it was B. 252 00:27:03,300 --> 00:27:04,480 Let's call B for a second. 253 00:27:07,640 --> 00:27:10,580 The derivative is one over A. 254 00:27:11,820 --> 00:27:14,220 But we can put that in terms of B. 255 00:27:15,360 --> 00:27:17,860 I should have called that X, but I screwed up. 256 00:27:19,540 --> 00:27:23,440 Okay, what is this in terms of our number B? 257 00:27:29,000 --> 00:27:30,260 What's this length right here? 258 00:27:33,400 --> 00:27:36,500 Because we're on the unit circle, it's of length one. 259 00:27:36,820 --> 00:27:37,000 Thank you. 260 00:27:38,020 --> 00:27:40,500 Okay, the same magic people. 261 00:27:43,480 --> 00:27:45,300 And then we can set up the Pythagorean theorem. 262 00:27:46,180 --> 00:27:46,240 Yeah? 263 00:27:47,900 --> 00:27:50,820 A squared plus B squared equals one squared. 264 00:27:50,980 --> 00:27:51,340 We agree? 265 00:27:52,220 --> 00:27:59,200 And if we solve for A here, A squared is one minus B squared. 266 00:28:01,900 --> 00:28:05,340 And then A will either equal the positive or the negative. 267 00:28:08,080 --> 00:28:10,140 Square root of one minus B squared. 268 00:28:11,460 --> 00:28:12,080 Which one is it? 269 00:28:12,160 --> 00:28:13,100 Is it a positive or negative? 270 00:28:15,100 --> 00:28:15,980 Why positive? 271 00:28:18,140 --> 00:28:19,740 We're in the first quadrant. 272 00:28:23,780 --> 00:28:26,600 And even if our angle is down here, 273 00:28:28,820 --> 00:28:31,140 which would be, no, it would be this angle. 274 00:28:32,700 --> 00:28:34,760 Okay, the A value is still positive. 275 00:28:35,720 --> 00:28:40,920 Because cosine of any angle in quadrant four is still the positive A. 276 00:28:43,620 --> 00:28:45,640 Because all students take calculus. 277 00:28:48,120 --> 00:28:49,240 So long story short, 278 00:28:50,820 --> 00:28:55,600 this is the square root of one minus B squared. 279 00:28:57,660 --> 00:29:01,580 Which really was our input here. 280 00:29:01,720 --> 00:29:02,760 I should have put a B here. 281 00:29:05,600 --> 00:29:11,000 Okay, so that was the proof that the derivative, that was the proof, 282 00:29:12,900 --> 00:29:14,600 of arc sine. 283 00:29:16,900 --> 00:29:20,960 This inverse trig function turns out to be this algebraic function, 284 00:29:21,280 --> 00:29:23,280 one over the square root of one minus x squared. 285 00:29:32,260 --> 00:29:33,280 It's kind of cool. 286 00:29:34,020 --> 00:29:35,140 It's kind of weird. 287 00:29:35,880 --> 00:29:39,100 The derivative of some trig function, well it's an inverse trig, 288 00:29:39,320 --> 00:29:42,320 turns out to be this radical function. 289 00:29:43,920 --> 00:29:45,020 But there it is. 290 00:29:46,660 --> 00:29:49,560 Let's do the same thing with arccosine. 291 00:29:53,940 --> 00:29:54,940 Let's call it f. 292 00:30:09,400 --> 00:30:10,100 So 293 00:30:25,680 --> 00:30:29,300 we're going to have to memorize all of them. 294 00:30:34,140 --> 00:30:35,420 We never proved it. 295 00:30:36,480 --> 00:30:37,260 Now you guys know it. 296 00:30:49,400 --> 00:30:51,660 If you ever forget, just prove it. 297 00:30:52,480 --> 00:30:53,640 Exactly, I like this. 298 00:30:59,300 --> 00:31:01,000 Okay, this one's going to work out with x's. 299 00:31:01,040 --> 00:31:01,640 It'll be much better. 300 00:31:03,940 --> 00:31:07,500 Okay, so this is one over. 301 00:31:08,960 --> 00:31:20,320 F prime is negative sine, evaluated at the inverse, arccosine x. 302 00:31:22,660 --> 00:31:24,160 Again, this is the derivative. 303 00:31:24,380 --> 00:31:27,100 We could stop here, but it will, turns out it simplifies. 304 00:31:29,440 --> 00:31:30,400 Let's see if we can do this. 305 00:31:33,280 --> 00:31:42,740 Correctly this time. 306 00:31:54,640 --> 00:31:58,340 Okay, now x will work out quite nicely. 307 00:32:14,900 --> 00:32:20,860 So, again, we're inputting some length between negative one and positive one. 308 00:32:23,100 --> 00:32:24,400 So arc cosine. 309 00:32:31,020 --> 00:32:33,300 We're inputting this length x. 310 00:32:36,340 --> 00:32:38,060 And it outputs an angle. 311 00:32:38,980 --> 00:32:42,180 It outputs this angle, theta. 312 00:32:50,120 --> 00:32:52,120 And what is sine of this angle? 313 00:32:57,420 --> 00:32:57,940 Y. 314 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. 315 00:33:12,040 --> 00:33:12,900 What's the hypotenuse? 316 00:33:14,040 --> 00:33:14,060 One. 317 00:33:14,140 --> 00:33:14,880 One, thank you. 318 00:33:16,880 --> 00:33:18,060 Because it's the unit circle. 319 00:33:19,020 --> 00:33:22,460 So we got x squared plus y squared equals one squared. 320 00:33:24,900 --> 00:33:28,600 Y squared equals one minus x squared. 321 00:33:29,940 --> 00:33:39,840 Y equals plus or minus the square root of one minus x squared. 322 00:33:46,160 --> 00:33:47,700 Is it the plus or the minus? 323 00:33:54,400 --> 00:33:55,220 Y plus. 324 00:33:56,860 --> 00:33:58,280 Because we're in quadrant one. 325 00:34:00,720 --> 00:34:11,942 If we were in this quadrant, okay, because the range of arc cosine is between zero 326 00:34:11,942 --> 00:34:12,900 and pi. 327 00:34:15,100 --> 00:34:20,280 Okay, this would be my x, this would be my y. 328 00:34:21,040 --> 00:34:23,520 The y is still positive in quadrant two, we agree? 329 00:34:24,160 --> 00:34:25,140 So it has to be positive. 330 00:34:27,280 --> 00:34:28,460 So we can get rid of this. 331 00:34:30,200 --> 00:34:33,640 And so it is the square root of one minus x squared. 332 00:34:39,960 --> 00:34:47,420 Okay, so long story short, the derivative of arc cosine, 333 00:34:49,360 --> 00:34:55,060 We normally also call it cosine inverse, but it's nice to distinguish it here. 334 00:34:56,360 --> 00:35:01,540 is simply negative one over the square root of one minus x squared. 335 00:35:03,440 --> 00:35:07,960 Which is the same as derivative of sine inverse, except it's negative. 336 00:35:15,280 --> 00:35:18,220 So should we do tangents? 337 00:35:24,280 --> 00:35:25,680 Let's do tangents for fun. 338 00:35:27,620 --> 00:35:29,800 For fun. Definitely fun. 339 00:35:50,580 --> 00:35:53,800 I'm using a different letter because it won't work out that nice. 340 00:36:05,320 --> 00:36:07,180 Arc tan, same thing as tan inverse. 341 00:36:09,640 --> 00:36:16,620 A, A, A, A. It doesn't matter. Let's go A. Why not? 342 00:36:27,700 --> 00:36:29,000 Sorry, what is the derivative of tan? 343 00:36:29,000 --> 00:36:30,420 Secant squared. 344 00:36:31,900 --> 00:36:32,440 Secant squared. 345 00:36:33,580 --> 00:36:36,120 Let me write it like that, secant squared. 346 00:36:42,640 --> 00:36:51,380 F inverse prime is one over F prime composed with the inverse. 347 00:36:54,800 --> 00:37:14,180 This is going to be one over secant squared composed with tan inverse. 348 00:37:19,640 --> 00:37:21,460 We proved that? 349 00:37:22,080 --> 00:37:22,640 I do. 350 00:37:22,860 --> 00:37:24,040 Okay. 351 00:37:38,500 --> 00:37:40,860 Uh oh. 352 00:38:07,040 --> 00:38:10,280 Go and look back. 353 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 354 00:38:17,857 --> 00:38:18,040 x. 355 00:38:18,040 --> 00:38:31,060 the ratio y over x okay wherever it is and this outputs an angle between 356 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 357 00:38:44,600 --> 00:39:04,720 and what is secant of that angle secant is 1 over cosine okay so 1 over 358 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 359 00:40:17,000 --> 00:40:31,520 okay so a is y over x so x is y over a 360 00:40:54,220 --> 00:41:03,660 X squared plus y squared is one. We get 361 00:41:03,660 --> 00:41:15,220 that by the Pythagorean theorem x squared equals 1 minus y squared 362 00:41:22,240 --> 00:41:38,060 X equals the square root of one minus y squared. Why is it guaranteed to be 363 00:41:38,060 --> 00:42:02,316 be positive actually it might not be guaranteed in the first quadrant is but I guess 364 00:42:02,316 --> 00:42:03,760 my we 365 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 366 00:42:32,876 --> 00:42:33,740 doesn't matter 367 00:43:07,840 --> 00:43:09,880 give me one more sec we're almost done 368 00:43:13,100 --> 00:43:26,020 if I substitute in a here okay which 369 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 370 00:43:42,244 --> 00:43:43,700 squared 371 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 372 00:44:01,126 --> 00:44:03,200 the square 373 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 374 00:44:13,941 --> 00:44:15,120 time this 375 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 376 00:44:28,666 --> 00:44:30,440 square root 377 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 378 00:44:54,317 --> 00:44:55,940 and then 379 00:44:55,940 --> 00:45:15,920 that simplifies to one over one plus a squared. That was messy because 380 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 381 00:45:26,100 --> 00:45:26,980 arc tan 382 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 383 00:45:39,470 --> 00:45:41,500 function okay 384 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 385 00:45:48,748 --> 00:45:49,070 blue 386 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 387 00:46:08,753 --> 00:46:12,490 function inside 388 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 389 00:46:22,420 --> 00:46:24,530 putting the chain 390 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 391 00:46:33,526 --> 00:46:35,690 three they're messy 392 00:46:36,370 --> 00:46:41,742 the other three involve these absolute values, and we don't need them this time. We 393 00:46:41,742 --> 00:46:43,490 need to memorize the three: 394 00:46:43,510 --> 00:46:51,757 arcsine arc cosine and arc tangent we wrote those three down already yeah okay I 395 00:46:51,757 --> 00:46:53,510 think we're gonna 396 00:46:55,890 --> 00:46:57,750 stop there