1 00:00:05,000 --> 00:00:11,060 Oh, we know how to find the derivative of all polynomials. And why do we know how to 2 00:00:11,060 --> 00:00:19,797 find the derivative? Or what rules do we use today? Power rule and then some 3 00:00:19,797 --> 00:00:20,380 different 4 00:00:20,380 --> 00:00:24,220 It takes care of polynomials. 5 00:00:25,560 --> 00:00:28,240 And then rational functions. 6 00:00:31,660 --> 00:00:32,900 Why are we able to do 7 00:00:32,900 --> 00:00:36,420 rational functions now? Because it's a quotient rule. 8 00:00:38,220 --> 00:00:39,640 Radical functions. 9 00:00:42,380 --> 00:00:44,760 Like y equals the square root of 10 00:00:44,760 --> 00:00:46,660 x squared plus three or something. 11 00:00:47,660 --> 00:00:57,300 both by the power rule and then because of the chain rule: polynomials, rationals, 12 00:00:58,140 --> 00:01:10,480 radicals, exponentials, okay we proved it for the function e to the x and then we 13 00:01:10,480 --> 00:01:14,500 did something the other day because of the chain rule for any exponential 14 00:01:14,500 --> 00:01:17,740 of function. And of course, what's the derivative of this? 15 00:01:19,060 --> 00:01:20,900 b to the x. What is the derivative of this? 16 00:01:23,460 --> 00:01:26,200 B to the x times ln of the base. 17 00:01:28,460 --> 00:01:29,760 All trig functions. 18 00:01:31,280 --> 00:01:34,280 We prove for sine and cosine, and then we use 19 00:01:34,280 --> 00:01:38,720 quotient rule for tangent. C tan, cos c tan, 20 00:01:38,820 --> 00:01:43,280 c tan. What functions are we still missing? 21 00:01:43,840 --> 00:01:49,511 that we don't know how to find the derivative of. We kind of did piece-wise, we just 22 00:01:49,511 --> 00:01:52,180 have to break it up into different pieces. 23 00:01:53,780 --> 00:02:00,424 We have not done any logarithms. And then the only other type we're going to learn 24 00:02:00,424 --> 00:02:02,500 as well is inverse trig. 25 00:02:06,440 --> 00:02:09,000 So today we're talking about the derivative of logarithms. 26 00:02:14,200 --> 00:02:19,740 So let's talk about this function, our favorite logarithm. 27 00:02:21,440 --> 00:02:23,480 What is the base of this logarithm? 28 00:02:26,520 --> 00:02:27,060 No, no. 29 00:02:27,060 --> 00:02:28,260 e. 30 00:02:30,080 --> 00:02:30,980 Yes. 31 00:02:32,860 --> 00:02:37,200 If it's just log, what's the base? 10. 32 00:02:39,240 --> 00:02:42,820 ln, the natural log, is log base e of x. 33 00:02:42,820 --> 00:02:44,200 of x. 34 00:02:46,760 --> 00:02:49,200 If we start with the definition, 35 00:03:09,260 --> 00:03:13,040 Can we manipulate this at all? 36 00:03:22,240 --> 00:03:24,000 What can we do with this? 37 00:03:28,220 --> 00:03:29,820 Uh oh. 38 00:03:33,220 --> 00:03:35,140 What is the difference of two logarithms? 39 00:03:41,880 --> 00:03:45,840 It is one logarithm, and inside we divide. 40 00:03:45,840 --> 00:03:50,160 ln of x plus h over x. 41 00:03:58,300 --> 00:03:59,960 Is that going to help at all? 42 00:04:02,400 --> 00:04:09,700 We play in zero for h. We get ln of x over x, which is ln of one. 43 00:04:09,880 --> 00:04:10,780 What is ln of one? 44 00:04:14,040 --> 00:04:17,200 0 over 0. That didn't help much. 45 00:04:20,500 --> 00:04:22,880 We could do this. 46 00:04:35,420 --> 00:04:39,880 Now, what can I do with the coefficient of a logarithm? 47 00:04:47,900 --> 00:04:49,720 We can, because it is one over h. 48 00:04:49,840 --> 00:04:50,740 It's not a constant. 49 00:04:51,580 --> 00:04:57,600 But it is a coefficient of a logarithm. 50 00:04:57,600 --> 00:05:23,600 This can become the power of the inside. 51 00:05:25,760 --> 00:05:27,940 Did this help at all? 52 00:05:39,140 --> 00:05:40,420 Any ideas now? 53 00:05:40,960 --> 00:05:42,260 Now what if we plug in zero? 54 00:05:46,380 --> 00:05:47,340 Wait, is this an x? 55 00:05:47,640 --> 00:05:51,360 Is this an x? Sorry, is that what you were talking about? Thanks. 56 00:05:55,160 --> 00:06:02,920 As h goes to 0, this goes to x over x which is 1. 57 00:06:04,660 --> 00:06:07,600 And what does that go to? 58 00:06:07,600 --> 00:06:18,582 1 over 0 goes to either infinity or negative infinity. What is 1 to the infinity or 59 00:06:18,582 --> 00:06:20,520 1 to the 60 00:06:20,520 --> 00:06:31,793 negative infinity? This is an indeterminate form. Okay, long story short, this is 61 00:06:31,793 --> 00:06:32,660 not 62 00:06:32,660 --> 00:06:39,720 gonna work but it was a good review of rules of logarithms. Difference of logs 63 00:06:39,720 --> 00:06:47,000 is one log with quotient coefficients to become exponents but anyways long search 64 00:06:47,000 --> 00:07:02,100 yeah it's not gonna work so we gotta try something else let's go back to the 65 00:07:04,920 --> 00:07:11,062 So, before you learned about logarithms, what did logarithms, when did we come up 66 00:07:11,062 --> 00:07:11,940 with logarithms? 67 00:07:11,940 --> 00:07:13,380 Logarithms are inverses. 68 00:07:15,260 --> 00:07:18,920 We knew any exponential looks like this, right? 69 00:07:19,980 --> 00:07:21,940 Some would depending on the base. 70 00:07:27,020 --> 00:07:29,160 This was a one to one function. 71 00:07:29,420 --> 00:07:30,600 What's a one to one function? 72 00:07:34,180 --> 00:07:38,980 It's a function. How do we know that there's a function? 73 00:07:40,620 --> 00:07:44,580 It's a vertical line test. Then it's a one to one 74 00:07:44,580 --> 00:07:48,060 function if it passed the vertical line test and it passes the horizontal. 75 00:07:50,400 --> 00:07:57,540 There's only going to be for every input there's only one output and never 76 00:07:57,540 --> 00:08:07,340 repeats the same y-value. Okay in every inverse function sorry I just gave away 77 00:08:07,340 --> 00:08:13,300 the answer. Every one to one function is guaranteed to have an inverse. 78 00:08:17,000 --> 00:08:23,878 If we reflect about the line it's this. This is the inverse of B to the X. And we 79 00:08:23,878 --> 00:08:24,240 just 80 00:08:24,240 --> 00:08:33,794 call this function a log with the base, wherever the base B is. Does this ring a 81 00:08:33,794 --> 00:08:35,480 bell? I don't 82 00:08:35,480 --> 00:08:38,860 No. What is an inverse functional? 83 00:08:44,200 --> 00:08:52,480 It is simply if we have some ordered pair in F, what is in F inverse? 84 00:08:54,680 --> 00:08:57,060 B, A. It's literally just all the ordered pairs. 85 00:08:57,060 --> 00:09:07,920 way. Okay. And that's why it was a reflection about the line y equals x. Okay 86 00:09:07,920 --> 00:09:13,780 and then inverses have a very very important property. What's the most 87 00:09:13,780 --> 00:09:24,980 important property of inverses? You compose a function and its inverse and 88 00:09:24,980 --> 00:09:31,320 And you end up with x whatever you started with there and vice versa 89 00:09:37,180 --> 00:09:38,080 Okay 90 00:09:42,420 --> 00:09:44,880 This is very important if 91 00:09:45,700 --> 00:09:47,100 We have to inverse something 92 00:09:48,320 --> 00:09:53,020 Okay, so our function e to the x and 93 00:09:53,140 --> 00:09:56,620 ln of x are inverse functions. 94 00:10:02,660 --> 00:10:12,660 That tells us e composed with ln of x is equal to x, 95 00:10:15,880 --> 00:10:19,460 ln of e to the x is equal to x. 96 00:10:23,140 --> 00:10:24,300 We agree with that 97 00:10:26,340 --> 00:10:27,240 Okay 98 00:10:34,920 --> 00:10:37,060 So let's go back to this 99 00:10:43,020 --> 00:10:48,420 I'm gonna plug both sides of this equation into our function e to the x 100 00:10:49,980 --> 00:10:52,560 So the left-hand side becomes e 101 00:10:53,140 --> 00:10:57,280 to the y and the right hand side is 102 00:11:01,400 --> 00:11:04,280 e to the ln of x 103 00:11:08,640 --> 00:11:13,880 And e to the ln of x is x because they're inverse functions 104 00:11:17,040 --> 00:11:22,927 Okay, sometimes we call this the exponential form and call this the logarithmic 105 00:11:22,927 --> 00:11:23,677 form. 106 00:11:25,280 --> 00:11:33,544 Right, it's literally if y equals e to the x, remember this is how we found 107 00:11:33,544 --> 00:11:34,120 inverses, 108 00:11:34,120 --> 00:11:40,334 you just switch the ordered pairs, x equals e to the y, which is the same thing 109 00:11:40,334 --> 00:11:40,700 here. 110 00:11:40,700 --> 00:11:51,640 Okay, so this is exactly the same thing as this. 111 00:11:51,660 --> 00:11:53,300 Just written two different ways. 112 00:11:55,460 --> 00:12:07,799 Okay, now we're going to take the derivative of this using what kind of 113 00:12:07,799 --> 00:12:08,680 differentiation? 114 00:12:11,840 --> 00:12:16,973 implicit because now we have functions on both sides of the equation. We're using 115 00:12:16,973 --> 00:12:17,340 implicit 116 00:12:17,340 --> 00:12:24,409 differentiation. And we are looking right we're looking for the derivative of this 117 00:12:24,409 --> 00:12:26,040 which is dy 118 00:12:26,040 --> 00:12:41,780 dx. So we're going to take the derivative with respect to x here. So implicit with 119 00:12:41,780 --> 00:12:57,968 respect x. And the derivative of e to the y is e to the y times the derivative of 120 00:12:57,968 --> 00:12:58,820 our 121 00:12:58,840 --> 00:13:01,520 inside function y, what is the derivative of y? 122 00:13:04,800 --> 00:13:06,000 dy dx. 123 00:13:07,780 --> 00:13:09,900 Okay, that's the derivative of the left side. 124 00:13:12,080 --> 00:13:19,040 Derivative of x is either dx dx or just the number one. 125 00:13:24,520 --> 00:13:26,300 And then I solve for dy dx. 126 00:13:27,320 --> 00:13:40,660 I get 1 over e to the y and what is oops 127 00:13:40,660 --> 00:14:05,360 What is e to the y equal to? x. dy over dx is 1 over x. 128 00:14:05,360 --> 00:14:11,000 derivative of the natural log turns out to be this rational function. 129 00:14:13,800 --> 00:14:19,120 Okay, this is very important we need to memorize it. 130 00:14:35,760 --> 00:14:39,979 Okay, so now we know the derivative of the natural log. What about for any 131 00:14:39,979 --> 00:14:40,729 logarithm? 132 00:14:42,840 --> 00:14:58,400 Let's say y equals log base b of x. We're gonna do the same exact process, except 133 00:14:58,400 --> 00:15:04,680 without ease. What do we do first? 134 00:15:13,160 --> 00:15:20,880 Yeah we need to change it from logarithmic form to exponential form. So 135 00:15:20,880 --> 00:15:25,760 So I'm going to go B to both sides. 136 00:15:38,700 --> 00:15:43,200 B to the log base B of X is X, 137 00:15:43,400 --> 00:15:45,040 because they're inverse functions, 138 00:15:45,300 --> 00:15:46,200 composed through inverse. 139 00:15:47,060 --> 00:15:49,680 So this is the same function. 140 00:15:50,880 --> 00:15:55,340 As this so now we're going to differentiate using implicit differentiation 141 00:16:03,720 --> 00:16:10,580 Derivative of B to the Y is B to the Y times 142 00:16:13,500 --> 00:16:20,140 Ln of the base I think we talked about that at the very top 143 00:16:24,100 --> 00:16:34,983 derivative of any exponential function is the same function times ln of the base. 144 00:16:34,983 --> 00:16:35,760 Then 145 00:16:35,760 --> 00:16:41,536 we multiply by the chain rule by the derivative of the inside. The derivative of y 146 00:16:41,536 --> 00:16:42,980 is dy over dx. 147 00:16:45,460 --> 00:16:47,260 So that's the derivative of the left side. 148 00:16:47,700 --> 00:16:52,020 The derivative of the right side is 1 or dx over dx. 149 00:16:53,160 --> 00:16:55,760 And we solve for this. 150 00:17:10,780 --> 00:17:20,140 And then, B to the Y is simply a... 151 00:17:30,280 --> 00:17:32,200 And there it is. 152 00:17:33,540 --> 00:17:50,040 The derivative of any logarithm with any base is simply 1 over x times 153 00:17:50,040 --> 00:17:59,960 the natural log of the base. Which really this is the coefficient of the x but we 154 00:17:59,960 --> 00:18:02,840 usually write it like that. 155 00:18:04,560 --> 00:18:07,480 of any logarithm. 156 00:18:16,280 --> 00:18:28,600 So what is the derivative of log of x? 157 00:18:28,600 --> 00:18:40,240 What's the base? There's no base here. 10. It's gonna be 1 over x times the 158 00:18:40,240 --> 00:18:42,680 natural log of 10. Easy. 159 00:18:55,380 --> 00:18:56,760 Okay, let's do some more 160 00:19:18,640 --> 00:19:24,340 Okay, what role do we have these? 161 00:19:24,600 --> 00:19:30,260 We're going to use lots of the big ideas chain rule. 162 00:19:30,420 --> 00:19:34,380 We got a function sine of x plus x cubed inside another function log base 4. 163 00:19:35,460 --> 00:19:37,120 So this is my u. 164 00:19:41,440 --> 00:19:44,340 What is the derivative of log base 4 of u? 165 00:19:49,920 --> 00:19:59,120 1 over u times ln of 4, but what is u equal to? 166 00:20:05,820 --> 00:20:11,200 sine of x plus x cubed times 167 00:20:13,000 --> 00:20:14,880 the natural logarithm of 4. 168 00:20:16,440 --> 00:20:19,520 Now we've got to multiply by the derivative of this. 169 00:20:24,660 --> 00:20:39,000 By the chain rule, the derivative is cosine of x plus 3x squared. 170 00:20:39,000 --> 00:20:40,540 We write it as one fraction. 171 00:21:04,160 --> 00:21:05,560 So 172 00:21:06,720 --> 00:21:13,358 Again, so if there's anything inside the logarithm, that's our U. It's 1 over U 173 00:21:13,358 --> 00:21:13,800 times 174 00:21:13,800 --> 00:21:19,687 ln of the base times the derivative of the inside U. By the chain rule. Let's try 175 00:21:19,687 --> 00:21:20,437 some more. 176 00:21:44,100 --> 00:21:46,880 So we got a function inside a function. 177 00:21:48,140 --> 00:21:50,340 What is the derivative of ln of u? 178 00:21:52,680 --> 00:21:53,920 One over u. 179 00:21:56,720 --> 00:21:58,400 Times the derivative of u. 180 00:21:58,400 --> 00:22:05,400 We have x to the 7th is 7x to the 6th. 181 00:22:09,380 --> 00:22:14,920 And what does that simplify to? 7 over x. 182 00:22:19,520 --> 00:22:21,420 Sorry, 7 over x. 183 00:22:33,760 --> 00:22:40,918 We could have done this one other way. How could I have changed this function? An 184 00:22:40,918 --> 00:22:42,260 exponent inside a 185 00:22:42,260 --> 00:22:50,407 logarithm becomes a coefficient. Now we don't need a chain rule, the constant 186 00:22:50,407 --> 00:22:51,660 multiple rule. 187 00:22:54,960 --> 00:23:03,360 It's 7 times the derivative of ln of x which is 1 over x and you're done. 188 00:23:13,360 --> 00:23:18,611 Most of the time in this class we're really just going to deal with the natural log 189 00:23:18,611 --> 00:23:18,920 to 190 00:23:18,920 --> 00:23:22,980 log base 4 or some other base. But not always. 191 00:23:24,620 --> 00:23:27,000 And you can always change bases too before you 192 00:23:27,000 --> 00:23:30,920 want it. So for like this, we could have 193 00:23:30,920 --> 00:23:32,660 used the change of base formula. 194 00:23:34,580 --> 00:23:37,840 What would have been the change of base formula? If we want to use the natural log, 195 00:23:39,320 --> 00:23:40,640 it's ln of the inside 196 00:23:44,980 --> 00:23:46,020 divided by 197 00:23:47,380 --> 00:23:48,680 ln of the base. 198 00:23:51,740 --> 00:23:52,980 And then this is just the coefficient. 199 00:23:53,360 --> 00:23:58,900 one over ln 4 times ln of sine x plus x cubed. 200 00:23:59,440 --> 00:24:01,200 Anyways, and then if you differentiate this, 201 00:24:02,140 --> 00:24:05,940 you end up with this times... 202 00:24:05,940 --> 00:24:08,040 Oops, that's a 4, right? 203 00:24:09,640 --> 00:24:12,520 Times 1 over ln of 4, and you get that same thing. 204 00:24:16,900 --> 00:24:19,520 Okay, let's try another one. 205 00:24:30,700 --> 00:24:33,780 Okay, you definitely have to use the chain rule. 206 00:24:36,180 --> 00:24:49,560 Derivative of ln u is 1 over u times the derivative of 5x plus 6. 207 00:24:49,560 --> 00:24:51,620 Five. 208 00:25:15,920 --> 00:25:18,380 What kind of function is this? 209 00:25:20,100 --> 00:25:21,240 It's a piecewise. 210 00:25:24,300 --> 00:25:28,940 Uh oh. Anything involving absolute value is a piecewise function. 211 00:25:37,160 --> 00:25:39,740 It's going to be ln of x, if what? 212 00:25:42,860 --> 00:25:53,900 if X is greater than zero and it's going to be if X is less than zero, Ln of 213 00:25:53,900 --> 00:26:02,460 negative X, oops, get it right this inside is the definition of the absolute value. 214 00:26:03,820 --> 00:26:06,820 What is the domain here? 215 00:26:11,600 --> 00:26:12,840 All real numbers 216 00:26:12,840 --> 00:26:13,740 except for 217 00:26:14,480 --> 00:26:17,720 zero. Unlike 218 00:26:21,020 --> 00:26:22,220 this function 219 00:26:22,220 --> 00:26:24,820 just ln of x, what is the domain here? 220 00:26:29,640 --> 00:26:30,540 Uh oh. 221 00:26:30,800 --> 00:26:32,160 Only positive numbers. 222 00:26:34,760 --> 00:26:39,240 Okay, right, we talked about the graph of this already. Ln of x looks like that. 223 00:26:50,580 --> 00:26:52,760 What does the graph of this look like? 224 00:26:55,780 --> 00:26:58,060 To the right of zero, it is ln of x. 225 00:27:00,420 --> 00:27:03,260 What does this look like? 226 00:27:05,140 --> 00:27:07,940 This is like that graph like reflected on the 227 00:27:09,680 --> 00:27:12,000 y axis like in the classes. That's correct. 228 00:27:13,140 --> 00:27:14,040 And how'd you know that? 229 00:27:15,760 --> 00:27:17,400 Okay, that's a good guess. 230 00:27:19,680 --> 00:27:24,260 A negative in front of an x inside a function reflects about the y axis. 231 00:27:26,240 --> 00:27:27,320 For example, 232 00:27:27,320 --> 00:27:38,302 If you plug in negative 1 for x, you get ln of negative negative 1, which is ln of 233 00:27:38,302 --> 00:27:38,880 1. 234 00:27:38,880 --> 00:27:39,900 a1, ln of 1. 235 00:27:39,940 --> 00:27:41,940 So it just basically reflects all these values. 236 00:27:42,640 --> 00:27:48,200 Anyways, so to find the derivative of a piecewise, what do we have to do? 237 00:27:52,040 --> 00:27:53,640 Take two separate derivatives. 238 00:27:53,640 --> 00:28:03,240 So let's start with this one. If x is greater than 0, then f of x which equals 239 00:28:03,240 --> 00:28:14,380 ln of x equals ln of x. And the derivative would be, derivative ln of x is 1 over x. 240 00:28:25,580 --> 00:28:35,280 Right now we have f prime of x is 1 over x if x is greater than 0. 241 00:28:37,420 --> 00:28:48,640 And then if x is less than 0, f of x equals ln the absolute value of x. 242 00:28:49,600 --> 00:28:55,640 Since x is less than zero, our function is this. 243 00:29:07,640 --> 00:29:10,420 Now when we take the derivative, we've got to use the chain rule. 244 00:29:10,420 --> 00:29:23,100 So, the derivative of ln of u is 1 over u, 1 over negative x times the derivative of 245 00:29:23,100 --> 00:29:30,280 the inside derivative of negative x, negative 1, just 1 over x. 246 00:29:40,820 --> 00:29:46,480 Since it's the same, we just write it as one thing. 247 00:29:47,160 --> 00:29:52,840 F' of x is simply 1 over x. 248 00:29:58,240 --> 00:30:04,484 And we already know that domain, all numbers except for zero, because you can't plug 249 00:30:04,484 --> 00:30:04,900 in 250 00:30:04,900 --> 00:30:05,800 zero there. 251 00:30:11,260 --> 00:30:17,680 So, the derivative of ln x is 1 over x. 252 00:30:20,520 --> 00:30:28,280 The derivative of ln of the absolute value of x is also equal to 1 over x. 253 00:30:33,700 --> 00:30:38,680 So what's the only, actually let me write it differently. 254 00:30:46,100 --> 00:30:47,420 Same story thing. 255 00:30:59,920 --> 00:31:02,640 So what is the difference between these two then? 256 00:31:07,600 --> 00:31:09,220 The domain is the difference. 257 00:31:14,020 --> 00:31:16,760 What is the domain of this? 258 00:31:19,960 --> 00:31:21,120 From 0 to infinity. 259 00:31:21,380 --> 00:31:22,500 The only positive numbers. 260 00:31:25,220 --> 00:31:31,215 That tells us the domain of this. There are no slopes of tangent lines to the left 261 00:31:31,215 --> 00:31:31,965 of zero. 262 00:31:34,440 --> 00:31:38,500 The domain here is also zero to infinity. 263 00:31:41,020 --> 00:32:01,160 This is the graph of ln x and this is the graph of its derivative, 1 over x. 264 00:32:01,160 --> 00:32:11,009 Here the domain is all numbers except for zero: negative infinity to zero, union 265 00:32:11,009 --> 00:32:13,120 zero to infinity, 266 00:32:15,530 --> 00:32:29,710 which is also the domain of this. 267 00:32:30,150 --> 00:32:31,270 Okay. 268 00:32:35,270 --> 00:32:36,390 Okay. 269 00:32:36,950 --> 00:32:39,330 Ln of the absolute value of x. 270 00:32:41,230 --> 00:32:42,310 Looks like this. 271 00:32:45,970 --> 00:32:53,414 and this is this which exactly the same on the right hand side over here the only 272 00:32:53,414 --> 00:32:54,290 difference is 273 00:32:58,030 --> 00:32:59,710 We have a different domain. 274 00:33:00,710 --> 00:33:12,713 Okay. So when we're finding derivatives, even though the derivative is 1 over X 275 00:33:12,713 --> 00:33:13,570 here, 276 00:33:14,990 --> 00:33:16,370 the domain is restricted. 277 00:33:16,690 --> 00:33:17,990 It's not the domain of this. 278 00:33:18,330 --> 00:33:20,650 The domain is only positive numbers because the domain 279 00:33:20,650 --> 00:33:22,210 of the function is only positive numbers. 280 00:33:23,490 --> 00:33:29,065 So that is very important. If the domain is not all real numbers for the function, 281 00:33:29,065 --> 00:33:32,550 the domain is not all real numbers for the derivative. 282 00:33:32,830 --> 00:33:36,350 It's maximum of whatever the domain of the function is. 283 00:33:38,990 --> 00:33:47,773 The distinction between these two is going to be very important in several months 284 00:33:47,773 --> 00:33:50,910 when we're talking about antiderivatives. 285 00:33:52,450 --> 00:33:59,410 What are antiderivatives? Antiderivatives are you're given a derivative and we want 286 00:33:59,410 --> 00:34:00,570 to find 287 00:34:00,570 --> 00:34:08,391 a function whose derivative is that. Okay, and this is gonna be very important 288 00:34:08,391 --> 00:34:09,141 later. 289 00:34:09,250 --> 00:34:18,732 But the main point of this is the domain of the function is also the most possible 290 00:34:18,732 --> 00:34:19,290 domain 291 00:34:19,290 --> 00:34:28,110 of this right when I say most possible like the square root of X the derivative 292 00:34:28,110 --> 00:34:43,930 is 1 over 2 square root x. The domain here is zero to infinity. What is the domain 293 00:34:43,930 --> 00:34:54,330 here? Zero to infinity, not including zero. Okay, so sometimes the numbers in the 294 00:34:54,330 --> 00:35:07,910 domain go down, right? Now we're missing one number. But anyways, my point was the 295 00:35:07,910 --> 00:35:12,910 domain of the function is at most the domain of the derivative. I don't know if 296 00:35:12,910 --> 00:35:19,650 I'm saying that quite right but it can be more restricted like it is here. 297 00:35:26,010 --> 00:35:28,770 Okay one more thing. 298 00:35:39,190 --> 00:35:42,590 I told you your function. 299 00:35:53,350 --> 00:35:58,530 Is this a power function? 300 00:36:00,470 --> 00:36:03,890 Anyway, if that's the case, we're going to use the power rule. 301 00:36:06,890 --> 00:36:09,730 What is the definition of a power function? 302 00:36:12,590 --> 00:36:22,318 X to any power. And the power has to be any number, any real number. So is this a 303 00:36:22,318 --> 00:36:22,830 power 304 00:36:22,830 --> 00:36:34,492 function? Here is a variable, it's not a real number. So it's not a power function. 305 00:36:34,492 --> 00:36:35,950 So we 306 00:36:35,950 --> 00:36:36,990 We can't use power rule. 307 00:36:42,670 --> 00:36:43,850 Is it an exponential function? 308 00:36:50,130 --> 00:36:53,270 So for an exponential function, what does the base have to be? 309 00:36:56,410 --> 00:37:00,570 Greater than zero and not equal to one. 310 00:37:03,410 --> 00:37:05,050 Is this an exponential function? 311 00:37:05,050 --> 00:37:07,370 No. No, why? 312 00:37:09,890 --> 00:37:12,890 Because the base is not a number. 313 00:37:14,290 --> 00:37:16,590 So this is neither a 314 00:37:16,590 --> 00:37:21,090 power function or an exponential function. So we can't use 315 00:37:21,090 --> 00:37:25,030 the power rule or we can't use the rule for derivative of exponential function. 316 00:37:26,790 --> 00:37:28,390 So what in the world are we going to do? 317 00:37:33,990 --> 00:37:35,030 How do you 318 00:37:35,030 --> 00:37:38,870 Okay, so we can't use any of the rules that we know 319 00:37:40,610 --> 00:37:42,970 because it's neither of those type of functions. 320 00:37:43,570 --> 00:37:45,910 So we're gonna first manipulate the function 321 00:37:45,910 --> 00:37:49,250 or the equation and then take derivatives. 322 00:37:50,670 --> 00:37:52,990 And as we said, we're gonna start 323 00:37:52,990 --> 00:37:56,390 by taking the natural log of both sides or any logarithm. 324 00:37:59,010 --> 00:38:00,870 We always choose the natural log 325 00:38:00,870 --> 00:38:02,610 because that's the simplest derivative. 326 00:38:03,830 --> 00:38:07,990 So we take the natural log of both sides. 327 00:38:17,670 --> 00:38:19,630 And then what can I do with this? 328 00:38:22,350 --> 00:38:23,390 It becomes a coefficient. 329 00:38:39,270 --> 00:38:44,276 Now here we just got some function. We don't have to take the derivative of ln now. 330 00:38:44,276 --> 00:38:47,810 Here we got a product. We're going to use the product rule. 331 00:38:49,090 --> 00:38:52,712 And we know the derivative of ln of x. So since there's functions on both sides, 332 00:38:52,712 --> 00:38:54,070 what are we going to use? 333 00:38:56,990 --> 00:38:58,750 implicit differentiation. 334 00:39:06,570 --> 00:39:09,050 The derivative of ln y is 335 00:39:10,970 --> 00:39:17,490 1 over y times the derivative of y, which is 336 00:39:18,990 --> 00:39:20,270 dy dx. 337 00:39:22,850 --> 00:39:25,170 Okay, that's just really the left side. 338 00:39:27,750 --> 00:39:31,910 Now, the derivative of the right side, we need the product rule. 339 00:39:37,050 --> 00:39:39,830 So, derivative of the 3x is 3. 340 00:39:45,110 --> 00:39:45,670 The 341 00:39:45,670 --> 00:39:48,830 derivative of 3x is 3, and the derivative of ln x is 1 over x. 342 00:40:03,110 --> 00:40:05,590 These happen to simplify, right? Plus 3. 343 00:40:07,490 --> 00:40:11,390 And we're trying to find dy dx, the derivative. 344 00:40:12,290 --> 00:40:13,470 So we solve for this. 345 00:40:23,210 --> 00:40:24,670 Times y. 346 00:40:27,790 --> 00:40:31,310 Except now we don't want x's and y's. 347 00:40:33,230 --> 00:40:38,790 What was y equal to? x to the 3x. 348 00:40:40,370 --> 00:40:41,770 So. 349 00:40:53,030 --> 00:40:56,430 And there it is. 350 00:41:08,550 --> 00:41:17,610 Okay, so when are we going to do this? 351 00:41:17,890 --> 00:41:19,010 First take the natural log. 352 00:41:25,590 --> 00:41:30,690 Whenever you have a variable both in the base and in the exponent. 353 00:41:32,090 --> 00:41:34,770 Because it's not a power function, it's not an exponential function. 354 00:41:35,390 --> 00:41:38,990 So this is called logarithmic 355 00:41:40,910 --> 00:41:45,470 I spell logarithmic. 356 00:41:49,970 --> 00:41:52,530 I'm blanking. 357 00:41:53,150 --> 00:41:54,330 Logarithm. 358 00:41:57,230 --> 00:41:58,130 There we go. 359 00:42:00,730 --> 00:42:01,630 Differentiation. 360 00:42:12,610 --> 00:42:14,630 Use whenever 361 00:42:17,950 --> 00:42:21,390 there is a variable 362 00:42:27,530 --> 00:42:36,670 And both the base and the exponent. 363 00:42:41,270 --> 00:42:49,210 And all you do is start by taking logs of both sides 364 00:42:57,590 --> 00:43:01,130 and then use implicit differentiation. 365 00:43:21,690 --> 00:43:26,210 Let's try another one. 366 00:43:40,270 --> 00:43:42,030 Let's see what this function looks like. 367 00:43:58,020 --> 00:43:59,180 It is kind of cool. 368 00:44:02,700 --> 00:44:05,220 Why are we missing a bunch of gaps in it? 369 00:44:10,380 --> 00:44:16,092 So, there's definitely some asymptote right here, but there is no, the function is 370 00:44:16,092 --> 00:44:16,500 not 371 00:44:16,500 --> 00:44:17,800 defined right here. 372 00:44:18,960 --> 00:44:27,240 It is not defined from pi to 2 pi. 373 00:44:27,760 --> 00:44:28,660 Why is that? 374 00:44:34,380 --> 00:44:35,900 What's that? 375 00:44:38,480 --> 00:44:39,540 It has to do with the 376 00:44:42,840 --> 00:44:46,300 We cannot have a negative base. 377 00:44:51,600 --> 00:44:52,500 Oh. 378 00:44:53,800 --> 00:44:56,080 We can't have negative numbers in a base. 379 00:44:57,620 --> 00:44:58,280 Like, negative numbers in a base. 380 00:44:58,280 --> 00:45:02,420 4 to the power of x is not an exponential function. The base has to be positive. 381 00:45:03,560 --> 00:45:04,580 So long story short, 382 00:45:07,520 --> 00:45:10,900 whenever sine is a negative number, the function doesn't exist. 383 00:45:12,160 --> 00:45:13,320 That's why there's a bunch of gaps. 384 00:45:15,420 --> 00:45:18,240 So, let's find dy by dx. 385 00:45:24,440 --> 00:45:29,937 The derivative. And we're going to have to use logarithmic differentiation. Why is 386 00:45:29,937 --> 00:45:30,687 that? 387 00:45:33,560 --> 00:45:41,897 There's a variable in the base and there's a variable in the exponents. Okay. Can't 388 00:45:41,897 --> 00:45:44,120 use the power rule. 389 00:45:44,120 --> 00:46:00,480 can't use the exponential rule. So first step is take the log of both sides and 390 00:46:00,480 --> 00:46:06,580 what happens it will be do this exponent can become the coefficient 391 00:46:12,900 --> 00:46:16,640 Okay, so it's the same function as written differently. 392 00:46:20,860 --> 00:46:25,560 And now we differentiate using implicit. 393 00:46:29,980 --> 00:46:36,460 The derivative of ln of y is 1 over y times the derivative of the inside, which is 394 00:46:36,460 --> 00:46:36,820 dy 395 00:46:36,820 --> 00:46:37,720 dx. 396 00:46:41,020 --> 00:46:43,280 Then we've got a product rule. 397 00:46:49,600 --> 00:47:04,120 The derivative of cosine is negative sine, times ln of sine x, plus cosine x times 398 00:47:07,780 --> 00:47:34,100 the derivative of ln of sine x, which is one over sine x times cosine x. 399 00:47:50,300 --> 00:47:53,760 And then again, we're solving for dy dx. 400 00:47:53,760 --> 00:47:55,820 We multiply both sides by y. 401 00:48:18,560 --> 00:48:21,560 And then we replace Y with where it was, which was what? 402 00:48:21,560 --> 00:48:36,800 sine x raised to cosine x. There it is. 403 00:48:36,800 --> 00:48:47,646 functions. We've never even thought about before. And when do we use logarithmic 404 00:48:47,646 --> 00:48:48,480 differentiation? 405 00:48:51,720 --> 00:48:57,291 Variable in the base and a variable in the exponent. You have to use it or it's the 406 00:48:57,291 --> 00:48:57,600 only 407 00:48:57,600 --> 00:49:06,069 option. Okay you could use logarithmic differentiation for other functions. Okay for 408 00:49:06,069 --> 00:49:07,480 example, and 409 00:49:07,490 --> 00:49:19,710 wouldn't do this but let's say we have this sine of x to the fourth power we 410 00:49:19,710 --> 00:49:24,050 would normally just use the chain rule. 411 00:49:27,190 --> 00:49:34,750 You get 4 sine cubed x times cosine x. 412 00:49:34,750 --> 00:49:41,368 It's easier. You could use logarithmic differentiation by first manipulating the 413 00:49:41,368 --> 00:49:42,118 equation. 414 00:50:04,750 --> 00:50:14,010 You have to use implicit differentiation, so 1 over y times dy dx is simply 4 times 415 00:50:14,910 --> 00:50:16,170 derivative of ln assigned. 416 00:50:17,470 --> 00:50:23,090 For ln of u would be 1 over u times the derivative of sine and cosine. 417 00:50:31,590 --> 00:50:39,550 over sine of x times y. What was y equal to? 418 00:50:40,950 --> 00:50:42,850 sine of x to the fourth. 419 00:50:55,670 --> 00:50:58,030 And then we can simplify this. 420 00:51:00,670 --> 00:51:01,450 sine of x 421 00:51:01,450 --> 00:51:04,550 divided by, or sine of x to the fourth, this becomes 422 00:51:04,550 --> 00:51:06,690 sine of x cubed. 423 00:51:08,030 --> 00:51:09,070 We have four, 424 00:51:10,070 --> 00:51:19,533 cosine of x times sine of x cubed, which is this right here. It's a long story 425 00:51:19,533 --> 00:51:28,997 short. You could use it for other problems we rarely ever do. Once or while it makes 426 00:51:28,997 --> 00:51:30,110 things easier. 427 00:51:31,270 --> 00:51:33,930 It definitely did not make it easier here, right? 428 00:51:36,430 --> 00:51:38,250 But you could use logarithmic 429 00:51:39,810 --> 00:51:41,870 differentiation. You have to use it 430 00:51:41,870 --> 00:51:45,690 whenever you've got a variable both in the base and in the exponent. 431 00:51:47,630 --> 00:51:49,990 Okay, any questions on this?