1 00:00:00,000 --> 00:00:00,370 2 00:00:00,370 --> 00:00:03,450 In the first version of the video of the proof of the 3 00:00:03,450 --> 00:00:10,859 derivative of the natural log of x, where the first 4 00:00:10,859 --> 00:00:13,179 time I proved this is a couple of years ago. 5 00:00:13,179 --> 00:00:16,960 And the very next video I proved that the derivative 6 00:00:16,960 --> 00:00:19,640 of e to the x is equal to e to the x. 7 00:00:19,640 --> 00:00:23,410 I've been charged with some of making a circular proof, and 8 00:00:23,410 --> 00:00:26,160 I'm pretty convinced that my proof wasn't circular. 9 00:00:26,160 --> 00:00:28,660 So what I want to do in this video, now that I have a little 10 00:00:28,660 --> 00:00:31,480 bit more space to work with, a little bit more sophisticated 11 00:00:31,480 --> 00:00:34,280 tools, I'm going to redo the proof and I'm going to do these 12 00:00:34,280 --> 00:00:38,910 in the same video to show you at no point do I assume this 13 00:00:38,909 --> 00:00:40,919 before I actually show it. 14 00:00:40,920 --> 00:00:42,329 So let's start with the proof. 15 00:00:42,329 --> 00:00:45,409 So the first thing I need to do is prove this thing up here. 16 00:00:45,409 --> 00:00:46,319 I want to keep track of this. 17 00:00:46,320 --> 00:00:49,730 I don't assume this until I actually show it. 18 00:00:49,729 --> 00:00:52,369 So let's start with the proof, the derivative of 19 00:00:52,369 --> 00:00:54,309 the natural log of x. 20 00:00:54,310 --> 00:00:57,730 So the derivative of the natural log of x, we can 21 00:00:57,729 --> 00:01:01,579 just to go to the basic definition of a derivative. 22 00:01:01,579 --> 00:01:07,939 It's equal to the limit as delta x approaches 0 of the 23 00:01:07,939 --> 00:01:15,170 natural log of x plus delta x minus the natural log of x. 24 00:01:15,170 --> 00:01:19,879 All of that over delta x. 25 00:01:19,879 --> 00:01:22,149 Now we can just use the property of logarithms. 26 00:01:22,150 --> 00:01:25,100 If I have the log of a minus the log of b, that's the same 27 00:01:25,099 --> 00:01:27,899 thing as a log of a over b. 28 00:01:27,900 --> 00:01:29,800 So let me re-write it that way. 29 00:01:29,799 --> 00:01:36,489 So this is going to be equal to the limit as delta 30 00:01:36,489 --> 00:01:38,899 x approaches 0. 31 00:01:38,900 --> 00:01:40,900 I could take this 1 over delta x right here. 32 00:01:40,900 --> 00:01:48,370 1 over delta x times the natural log of x plus delta 33 00:01:48,370 --> 00:01:50,590 x divided by this x. 34 00:01:50,590 --> 00:01:53,810 Just doing the logarithm properties right there. 35 00:01:53,810 --> 00:01:55,829 Then I can re-write this -- first of all, when I have this 36 00:01:55,829 --> 00:01:58,159 coefficient in front of a logarithm I can make 37 00:01:58,159 --> 00:01:59,479 this the exponent. 38 00:01:59,480 --> 00:02:01,189 And then I can simplify this in here. 39 00:02:01,189 --> 00:02:06,939 So this is going to be equal to the limit as delta x approaches 40 00:02:06,939 --> 00:02:11,870 0 of the natural log -- let me do this in a new color. 41 00:02:11,870 --> 00:02:14,379 Let me do it in a completely new color. 42 00:02:14,379 --> 00:02:18,659 The natural log of -- the inside here I'll just 43 00:02:18,659 --> 00:02:20,020 divide everything by x. 44 00:02:20,020 --> 00:02:22,390 So x divided by x is 1. 45 00:02:22,389 --> 00:02:26,599 Then plus delta x over x. 46 00:02:26,599 --> 00:02:30,310 Then I had this 1 over delta x sitting out here, and I can 47 00:02:30,310 --> 00:02:31,490 make that the exponent. 48 00:02:31,490 --> 00:02:34,030 That's just an exponent rule right there, or 49 00:02:34,030 --> 00:02:35,439 a logarithm property. 50 00:02:35,439 --> 00:02:38,810 1 over delta x. 51 00:02:38,810 --> 00:02:40,099 Now I'm going to make a substitution. 52 00:02:40,099 --> 00:02:43,909 Remember, all of this, this was all just from my definition 53 00:02:43,909 --> 00:02:44,500 of a derivative. 54 00:02:44,500 --> 00:02:46,129 This was all equal to the derivative of the 55 00:02:46,129 --> 00:02:47,740 natural log of x. 56 00:02:47,740 --> 00:02:51,250 I have still yet to in any way use this. 57 00:02:51,250 --> 00:02:54,550 And I won't use that until I actually show it to you. 58 00:02:54,550 --> 00:02:57,600 I've become very defensive about these claims 59 00:02:57,599 --> 00:03:00,180 of circularity. 60 00:03:00,180 --> 00:03:03,040 They're my fault because that shows that I wasn't clear 61 00:03:03,039 --> 00:03:05,949 enough in my earlier versions of these proofs, so I'll try 62 00:03:05,949 --> 00:03:07,139 to be more clear this time. 63 00:03:07,139 --> 00:03:09,119 So let's see if we can simplify this into terms 64 00:03:09,120 --> 00:03:10,840 that we recognize. 65 00:03:10,840 --> 00:03:15,860 Let's make the substitution so that we can get e in maybe 66 00:03:15,860 --> 00:03:17,790 terms that we recognize. 67 00:03:17,789 --> 00:03:26,829 Let's make the substitute delta x over x is equal to 1 over n. 68 00:03:26,830 --> 00:03:28,490 If we multiply -- this is the same thing. 69 00:03:28,490 --> 00:03:29,890 This is the equivalent to substitution. 70 00:03:29,889 --> 00:03:32,909 If we multiply both sides of this by x, as saying that 71 00:03:32,909 --> 00:03:36,159 delta x is equal to x over n. 72 00:03:36,159 --> 00:03:37,199 These are equivalent statements. 73 00:03:37,199 --> 00:03:40,489 I just multiplied both sides by x here. 74 00:03:40,490 --> 00:03:46,480 Now if we take the limit as n approaches infinity of this 75 00:03:46,479 --> 00:03:49,780 term right here, that's equivalent -- that's completely 76 00:03:49,780 --> 00:03:53,669 equivalent to taking the limit as delta x approaches 0. 77 00:03:53,669 --> 00:03:56,679 78 00:03:56,680 --> 00:03:59,890 If we're defining delta x to be this thing, and we take the 79 00:03:59,889 --> 00:04:02,589 limit as its denominator approaches 0, we're going 80 00:04:02,590 --> 00:04:05,120 to make delta x go to 0. 81 00:04:05,120 --> 00:04:06,520 So let's make that substitution. 82 00:04:06,520 --> 00:04:12,810 So all of this is going to be equal to the limit as -- now 83 00:04:12,810 --> 00:04:14,275 we've gotten rid of our delta x. 84 00:04:14,275 --> 00:04:19,000 We're going to say the limit as an approaches infinity of the 85 00:04:19,000 --> 00:04:23,379 natural log -- I'll go back to that mauve color -- the natural 86 00:04:23,379 --> 00:04:28,899 log of 1 plus -- now, I said that instead of delta x over x, 87 00:04:28,899 --> 00:04:31,579 I made the substitution that that is equal to 1 over n. 88 00:04:31,579 --> 00:04:35,009 So that's 1 plus 1 over n. 89 00:04:35,009 --> 00:04:37,860 And then what's 1 over delta x? 90 00:04:37,860 --> 00:04:41,689 Well delta x is equal to x over n, so 1 over delta x is going 91 00:04:41,689 --> 00:04:42,875 to be the inverse of this. 92 00:04:42,875 --> 00:04:47,139 It's going to be n over x. 93 00:04:47,139 --> 00:04:50,699 And then we can re-write this expression right here -- 94 00:04:50,699 --> 00:04:52,670 let me re-write it again. 95 00:04:52,670 --> 00:04:58,600 This is equal to the limit as n approaches infinity of the 96 00:04:58,600 --> 00:05:02,730 natural log of 1 plus 1 over n. 97 00:05:02,730 --> 00:05:05,610 What I can do is I can separate out this n from the 1 over x. 98 00:05:05,610 --> 00:05:08,129 I could say this is to the n, and then all of 99 00:05:08,129 --> 00:05:11,199 this to the 1 over x. 100 00:05:11,199 --> 00:05:13,729 Once again, this is just an exponent property. 101 00:05:13,730 --> 00:05:16,290 If I raise something to the n and then to the 1 over x, I 102 00:05:16,290 --> 00:05:18,110 could just multiply the exponent to get 103 00:05:18,110 --> 00:05:19,660 to the n over x. 104 00:05:19,660 --> 00:05:21,660 So these two statements are equivalent. 105 00:05:21,660 --> 00:05:24,720 But now we can use logarithm properties to say hey, if this 106 00:05:24,720 --> 00:05:26,730 is the exponent, I can just stick it out in front of the 107 00:05:26,730 --> 00:05:29,100 coefficient right here. 108 00:05:29,100 --> 00:05:32,300 I could put it out right there. 109 00:05:32,300 --> 00:05:34,829 And just remember, this was all of the derivative with respect 110 00:05:34,829 --> 00:05:37,709 to x of the natural log of x. 111 00:05:37,709 --> 00:05:38,680 So what is that equal to? 112 00:05:38,680 --> 00:05:41,340 We could put this 1 out of x in the front here. 113 00:05:41,339 --> 00:05:44,039 In fact, that 1 out of x term, it has nothing to do with n. 114 00:05:44,040 --> 00:05:45,550 It's kind of a constant term when you think 115 00:05:45,550 --> 00:05:46,290 of it in terms of n. 116 00:05:46,290 --> 00:05:49,000 So we can actually put it all the way out here. 117 00:05:49,000 --> 00:05:50,060 We could put it either place. 118 00:05:50,060 --> 00:05:55,490 So we could say 1 over x times all that stuff in mauve. 119 00:05:55,490 --> 00:06:03,400 The limit as n approaches infinity of the natural log 120 00:06:03,399 --> 00:06:08,679 of 1 plus 1 over n to the n. 121 00:06:08,680 --> 00:06:10,560 The natural log of all of that stuff. 122 00:06:10,560 --> 00:06:14,670 Or, just to make the point clear, we can re-write this 123 00:06:14,670 --> 00:06:19,550 part -- let me make a salmon color -- equal to 1 over x 124 00:06:19,550 --> 00:06:27,420 times the natural log of the limit as n approaches infinity. 125 00:06:27,420 --> 00:06:30,015 I'm just switching places here, because obviously what we care 126 00:06:30,014 --> 00:06:33,829 is what happens to this term as it approaches infinity, of 127 00:06:33,829 --> 00:06:37,550 1 plus 1 over n to the n. 128 00:06:37,550 --> 00:06:39,910 Well what is -- this should look a little familiar to you 129 00:06:39,910 --> 00:06:42,760 on some of the first videos where we talked about e -- this 130 00:06:42,759 --> 00:06:45,019 is one of the definitions of e. 131 00:06:45,019 --> 00:06:46,029 e is defined. 132 00:06:46,029 --> 00:06:46,879 I'm just being clear here. 133 00:06:46,879 --> 00:06:50,649 I'm still not using this at all. 134 00:06:50,649 --> 00:06:55,019 I'm just stating that the definition of e, e is equal 135 00:06:55,019 --> 00:07:00,759 to the limit as n approaches infinity of 1 plus 136 00:07:00,759 --> 00:07:02,750 1 over n to the n. 137 00:07:02,750 --> 00:07:04,490 This is just the definition of e. 138 00:07:04,490 --> 00:07:07,329 And natural log is defined to be the logarithm 139 00:07:07,329 --> 00:07:09,669 of base, this thing. 140 00:07:09,670 --> 00:07:12,890 So this thing is e. 141 00:07:12,889 --> 00:07:14,509 So I'm saying that the derivative of the natural log 142 00:07:14,509 --> 00:07:18,680 of x is equal to 1 over x times the natural log. 143 00:07:18,680 --> 00:07:19,860 This thing right here is e. 144 00:07:19,860 --> 00:07:22,050 That's what the definition of e is. 145 00:07:22,050 --> 00:07:24,800 I'm not using the definition of the derivative e, or 146 00:07:24,800 --> 00:07:28,360 the definition of the derivative of e to the x. 147 00:07:28,360 --> 00:07:30,449 I'm just using the definition of e. 148 00:07:30,449 --> 00:07:35,120 And the definition of natural log is log base e. 149 00:07:35,120 --> 00:07:38,370 This says the power that you have to raise e to to get to e, 150 00:07:38,370 --> 00:07:40,870 well this is just equal to 1. 151 00:07:40,870 --> 00:07:43,620 There we get that the derivative of the natural log 152 00:07:43,620 --> 00:07:47,399 of x is equal to 1 over x. 153 00:07:47,399 --> 00:07:52,109 So, so far I think you'll be satisfied that we've proven 154 00:07:52,110 --> 00:07:56,449 this first statement up here, and in no way did we use 155 00:07:56,449 --> 00:07:58,019 this statement right here. 156 00:07:58,019 --> 00:08:01,539 I just used the definition of e, but that's fine. 157 00:08:01,540 --> 00:08:04,420 I mean we assumed we know the definition of e, even when we 158 00:08:04,420 --> 00:08:07,069 just talk about natural log, we assume that it's base e. 159 00:08:07,069 --> 00:08:10,819 In no way did I assume this to begin with. 160 00:08:10,819 --> 00:08:13,759 Now, given that we've shown this and we didn't assume 161 00:08:13,759 --> 00:08:16,750 this at all, let's see if we can show this. 162 00:08:16,750 --> 00:08:22,269 So the derivative -- let's do a little bit of an exercise here. 163 00:08:22,269 --> 00:08:26,609 164 00:08:26,610 --> 00:08:30,170 Actually, I could probably do it in the margins. 165 00:08:30,170 --> 00:08:34,220 Let's take the derivative of this function. 166 00:08:34,220 --> 00:08:37,245 The natural log of e to the x. 167 00:08:37,245 --> 00:08:40,250 168 00:08:40,250 --> 00:08:42,139 So there's two ways we can approach this. 169 00:08:42,139 --> 00:08:45,210 The first way we could simplify this and we could say this is 170 00:08:45,210 --> 00:08:47,519 the exact same thing as the derivative. 171 00:08:47,519 --> 00:08:51,740 We could put this x out front of x times the 172 00:08:51,740 --> 00:08:53,820 natural log of e. 173 00:08:53,820 --> 00:08:55,910 And what's the natural log of e? 174 00:08:55,909 --> 00:09:01,459 The natural log of e we already know is equal to 1. 175 00:09:01,460 --> 00:09:03,930 So this is just the derivative of x. 176 00:09:03,929 --> 00:09:08,029 And the derivative of x is equal to 1. 177 00:09:08,029 --> 00:09:09,189 So that's pretty straightforward. 178 00:09:09,190 --> 00:09:13,540 The derivative, in no way did we assume this to begin with. 179 00:09:13,539 --> 00:09:16,139 We just simplified this expression to just this is the 180 00:09:16,139 --> 00:09:18,259 same thing as the derivative of x, because this 181 00:09:18,259 --> 00:09:19,659 term cancels out. 182 00:09:19,659 --> 00:09:21,600 And the derivative of x is just 1. 183 00:09:21,600 --> 00:09:23,210 Or we could do it the other way. 184 00:09:23,210 --> 00:09:25,379 We could do the chain rule. 185 00:09:25,379 --> 00:09:29,639 We could say that this could be viewed as the derivative of 186 00:09:29,639 --> 00:09:33,960 this inner function, of this inner expression, so the 187 00:09:33,960 --> 00:09:35,180 derivative of the inner expression, I don't 188 00:09:35,179 --> 00:09:35,969 know what that is. 189 00:09:35,970 --> 00:09:37,660 I'm not assuming anything about it. 190 00:09:37,659 --> 00:09:38,719 I just don't know what it is. 191 00:09:38,720 --> 00:09:40,410 So I'll write it in yellow right here. 192 00:09:40,409 --> 00:09:42,649 So it's equal to the derivative with respect 193 00:09:42,649 --> 00:09:45,079 to x of e to the x. 194 00:09:45,080 --> 00:09:46,080 I don't know what this is. 195 00:09:46,080 --> 00:09:48,350 I have no clue what this is, and I haven't assumed 196 00:09:48,350 --> 00:09:49,430 anything about what it is. 197 00:09:49,429 --> 00:09:51,149 I'm just using the chain rule. 198 00:09:51,149 --> 00:09:54,069 If the derivative of this inside function with respect to 199 00:09:54,070 --> 00:09:57,920 x, which is this right here, times the derivative of this 200 00:09:57,919 --> 00:10:01,409 outside function with respect to the inside function. 201 00:10:01,409 --> 00:10:04,079 So the derivative of natural log of x with respect 202 00:10:04,080 --> 00:10:05,730 to x is 1 over x. 203 00:10:05,730 --> 00:10:08,159 So the derivative of natural log of anything with 204 00:10:08,159 --> 00:10:10,309 respect to anything is 1 over that anything. 205 00:10:10,309 --> 00:10:13,709 So it's going to be equal to -- so the derivative of natural 206 00:10:13,710 --> 00:10:15,850 log of x with respect to e to the x is equal to 1 207 00:10:15,850 --> 00:10:19,190 over e to the x. 208 00:10:19,190 --> 00:10:23,860 Once again, I in no way assumed this right here. 209 00:10:23,860 --> 00:10:26,919 So far in anything we've done, we haven't assumed that. 210 00:10:26,919 --> 00:10:30,579 But clearly, my derivatives, either way I solve it -- one 211 00:10:30,580 --> 00:10:32,210 way I solve it I got 1. 212 00:10:32,210 --> 00:10:34,650 The other way, I kind of didn't solve it. 213 00:10:34,649 --> 00:10:37,029 I got this expression right here. 214 00:10:37,029 --> 00:10:39,225 They must be equal to each other. 215 00:10:39,225 --> 00:10:40,759 So let me write that down. 216 00:10:40,759 --> 00:10:42,330 This must be equal to that. 217 00:10:42,330 --> 00:10:44,259 It's just we just looked at it two different ways and got 218 00:10:44,259 --> 00:10:45,620 two different results. 219 00:10:45,620 --> 00:10:47,340 But I still don't know what this thing is. 220 00:10:47,340 --> 00:10:49,399 I just left it kind of open. 221 00:10:49,399 --> 00:10:51,009 I just said whatever the derivative of e to 222 00:10:51,009 --> 00:10:52,700 the x happens to be. 223 00:10:52,700 --> 00:10:56,480 But we know, since these two expressions are equal, we know 224 00:10:56,480 --> 00:11:01,149 that the derivative with respect to x of whatever e to 225 00:11:01,149 --> 00:11:03,634 the x -- so whatever the derivative with respect to x of 226 00:11:03,634 --> 00:11:07,139 e to the x happens to be, we know that when we multiply that 227 00:11:07,139 --> 00:11:11,340 times 1 over e to the x -- that's when we just did the 228 00:11:11,340 --> 00:11:13,960 chain rule -- that we should get the same result as when we 229 00:11:13,960 --> 00:11:15,930 approached the problem the other way. 230 00:11:15,929 --> 00:11:18,839 That should be equal to this approach because they're both 231 00:11:18,840 --> 00:11:21,070 different ways of looking at the derivative of the 232 00:11:21,070 --> 00:11:22,530 natural log of e to the x. 233 00:11:22,529 --> 00:11:24,990 So that should be equal to 1. 234 00:11:24,990 --> 00:11:27,090 Well, we're almost there. 235 00:11:27,090 --> 00:11:29,649 We could just simplify this and solve for our mystery 236 00:11:29,649 --> 00:11:31,250 derivative of e to the x. 237 00:11:31,250 --> 00:11:34,519 Multiply both sides of this equation by e to the x, and you 238 00:11:34,519 --> 00:11:38,889 get the derivative with respect to x of e to the x is 239 00:11:38,889 --> 00:11:41,629 equal to e to the x. 240 00:11:41,629 --> 00:11:43,789 And I want to clarify this. 241 00:11:43,789 --> 00:11:47,649 At no point in this entire proof, at no point 242 00:11:47,649 --> 00:11:50,299 did I assume this. 243 00:11:50,299 --> 00:11:52,089 In fact, this is the first time that I'm even 244 00:11:52,090 --> 00:11:54,879 making the statement. 245 00:11:54,879 --> 00:11:58,080 I didn't have to assume this when I showed you that the 246 00:11:58,080 --> 00:12:00,850 derivative of the natural log of x is 1 over x. 247 00:12:00,850 --> 00:12:04,580 And I didn't have to assume this to kind of get to it. 248 00:12:04,580 --> 00:12:07,320 So in no way is this proof circular. 249 00:12:07,320 --> 00:12:10,520 So anyway, I didn't want to appear defensive, but I 250 00:12:10,519 --> 00:12:12,079 wanted to clarify this up. 251 00:12:12,080 --> 00:12:17,080 Because I don't want to in any way blame those who think that 252 00:12:17,080 --> 00:12:18,910 my original proof was circular. 253 00:12:18,909 --> 00:12:21,370 It's my fault because I didn't explain it properly. 254 00:12:21,370 --> 00:12:23,580 So hopefully this should provide a little bit of 255 00:12:23,580 --> 00:12:25,910 clarity on the issue. 256 00:12:25,909 --> 00:12:26,076