how do we know the derivative of ln(x) is 1/x (the definition & implicit differentiation)
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- Опубликовано: 4 июл 2024
- We will show that the derivative of ln(x), namely the natural logarithmic function, is 1/x. We will use the definition of the derivative and also implicit differentiation.
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You know its about to get real when he starts using the blue pen (-:
yup, that's right!
Somewhat relevant xkcd: xkcd.com/849/
He has also used a purple pen a few times.
😂😂😂😂😂😂😂😂
Multiple Colour Pen
8:11
Talking to your GF
under rated comment
Perfect
Lol
ayyy
Very good lmao
I have graduated 3 months ago, at the start of the calculus class 2 years ago i hated calculus but here i am, loving calculus and enjoying every second of your awesome videos.
Alex Ramyeon thank you!!!!!!
Honestly speaking calculus is more fun than GTA and all other video games.
@@blackpenredpen my way docs.google.com/document/d/e/2PACX-1vQ0SB1cs5gR0S17zmIhfFuQqmhGsw8_jn_QoL1n6AjI26wsu2bOPIxzCrw1D0SK-fCca1FUR-xAQ-gI/pub
@@tintinfan007 you're probably speaking about some random mobile gta spinoff
I love the way he teaches
8:25 my brain to me after a test
Underrated
Lol
Lol
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lol
10:43 I really love the satisfaction I get when my mind snaps and know how the demostration continues before the video. Great video!
Clicked at around 7:50 for me, so satisfying
was about to cmment that same thing! It's such a great feeling
How do we know ln(x) is a logarithm? I once had a professor “define” ln(x) as a function whose derivative is 1/x. He then proceeded to show the ln(x) is indeed a logarithm, and it has the base e. I’d like to see this again. It was very inspiring, but I have forgotten how it was done.
Are you asking how to prove the properties based on that definition? If so, I have a video here ruclips.net/video/4D-M5qB_l6k/видео.html
@hobo doc id be happy to receive those pages on scrubster@gmx.de
I was taught that ln(x) is by definition, log base e of x. The term ln itself means the natural logarithm.
@@lewisbotterill4948 look
@@cellcomsanggau424 ?
Another proof using parametric equation:
x = e^t
y = t
dx/dt = e^t
dy/dt = 1
(dy/dt)/(dx/dt) = dy/dx = 1/e^t = 1/x
Wonder proof buddy! Three different proofs: Limits, implicit differentiation and parametric equations.
Wonderful proof buddy! Three different proofs: Using parametric equations, limits and implicit differentiation.
Just brilliant. Congratulations, I’m going to teach this one tomorrow
nicely done
For any log , 1/x .ln a, a its the base of the log , If a = e, the derivative is 1/x
I did a general proof
Just watched it again as there were a few things I wasn’t sure of. I really liked it when he explained one trick to use was because the natural log is a continuous function, and the limit of a continuous function is a continuous function of the limit, you can move the limit inside the parentheses to simplify things. Cool stuff.
I really enjoyed your last few videos, and I am glad you're back to uploading more videos containing your explanations
Its a shame we dont get teached this stuff in school but are just supposed to remember f'(x)=1/x of F(x)=ln(x)
I remember my school teaching us a variation of the 2nd method, namely
y = ln x
=> e^y = x
Therefore dx/dy = e^y
dy/dx = 1/(dx/dy) = 1/e^y = 1/x
FF same situation(((
Of f(x)=ln(x). Capital f often implies integration. Especially because integrated function is defined by it F'(x)=f(x) then you are ok.
they told us:
e^ln(x)=x
diferentiating gives:
e^ln(x)*d/dx(ln(x))=1
d/dx(ln(x))=1/e^ln(x)=1/x
This is just the application of the first principle definition of the derivative. You know how to do limits and should be well versed in algebra manipulation. It's not a big leap to do this problem. This is the sort of exercise a student should do away from school.
I have always wanted a more detailed explanation of this result. This is the best I’ve seen on the subject. Considering things like Euler’s identity and the quantum wave equation and other uses of the exponential function, it seems to me it’s the most useful of all the special functions.
Another way :
exp(ln(x)) = x
Derivative of both sides :
ln(x)' * exp(ln(x)) = 1
Replace exp(ln(x)) by x and divide the whole equation by it :
ln(x)' = 1/x
wow!
it's literally in the video
Elegant proofs for the derivative of ln(x). I like the intelligent and creative ways you used to develop and establish your proofs. Thanks.
Wonderful videos. It is a long time ago that I studied complex variables, differential and integral calculus and algebra. So it is great fun watching this guy do with ease what most of us struggled with when learning the basic elements of these important mathematical techniques. I can generally follow him right to the end once I see where he headed. The mathematical manipulations seem to be firmly rooted in my brain. The algorithms he applies for problem solving are much less so.
You are my new favorite high school math teacher. In my AP calculus class, we were never taught how to derive this. Only taught to memorize that d/dx ln(x) = 1/x
Great video man! I feel like you've made me so much smarter; this time I was actually able to see ahead a little bit, that the argument of ln would end up being e^1/x (this was around when you brought the derivative into the u world)
this math professor dripping out with tha preme jacket
Gavin Burns lmaooo
Well done guy!
You sort it out!
Keep it up!
Go always deep n in every detail to enlightening.
Again you've done it!
dayuumm now that's impressive, finding the derivative of ln(x) using the standard definition of a derivative
Oh my god you are incredible! I learned a thing or two because of you! Loved it ❤️
So clear explanation, Greatest Math Teacher in the WORLD, Thank You Sir!.
Thank YOU so much for sharing your beautiful smile and passion!! It makes me so much more excited to learn and genuinely happy :))
blackpenredpen could you solve the non elementary integral of x^x. You did the (easier) derivative so please do the difficult integral or let Payem do it
Ahsoka Tano How is he suppose to solve it if it is non-elementary? Do you understand what solving an integral is? And do you understand what non-elementary is?
Angel Mendez-Rivera ima be real with you that made no sense
Zach Cate if an integral is non-elementary, by definition, that means you cannot solve it. It will be defined by a special function. For example, the fresnel integral
Okay for anyone that is confused this is a matter of pedanticism. "Solving an integral" technically refers to definite integrals. The original comment probably just wants the indefinite integral and is using the word "solve" to mean "to do" as in ordinary english.
Again, all a matter of mathematical vs normal language.
I've always been told that the derivative rule for f'(x) of ln(x) has always been 1/x but I've never understood how that was proven. Thank you for the explanation.
There's usually one of these proofs for it somewhere in the textbook. Since the teacher probably sees proving them as reinventing the wheel, and not necessary to understand the subject, they probably just skip showing why these derivative rules work.
This reminds me of when I was a COBOL programmer, we would have discussions about whether you could have positive zero and negative zero. This was because the sign of a number was contained in the units digit. So, when comparing numbers it was important to take this into account. But I would say to my colleagues that zero was neither positive nor negative, it was separate from other numbers.
Just found your channel. Thanks for creating this content and keep up the good work.
This mad lad really just used the limit definition. Can we get this guy a medal?
Dear friend, you are not only genius but you a great guru (teacher). My regards - Sudarshan🙏
How easily he changes markers is amazing to watch
Finally, since the basketball secret has been revealed I can find some sleep!
Marian P. Gajda in fact, in was in that previous video as well, just allllll the way at the end.
I live close to where you recorded that basketball video! I was pleasantly surprised when I saw that
Derek Anderson
Are you serious???
How did u even recognize that place!!!
@@blackpenredpen COINCIDENCE?
@@blackpenredpen it is possible, but very unlikely
You can do this in two ways. You can use the integral definition of log(c) and use the fundamental theorem of calculus or you can note that log(x) is the inverse function of exp(x), and just use the expression for differentiating the inverse function.
This was so intertwining I was guessing what to do and when he showed what to do it made sense feels amazing
The second lim going into the continues function was so eye opening and satisfying
Absolutely beautiful. Great explanations! Thank you.
I wonder if there is a numerical analysis class he teaches. This guy is a good teacher.
Dang, when he finally pulled out the e term, I got super excited. Nice job!
Very elegant description of this important derivative
I love u-sub when doing algebra and calculus. SO useful.
It's so beautiful ♥
Great explanation!
جميل ورائع ومميز ما يقوم به هذا الشاب،،، فعلا عقليه فذه،، 🌹🌹🌹
YOU ARE PHENOMENON!!!!!
THANK YOU!!! All other videos I found only explained how you used the derivative not actually showing proof on why it’s 1/x
Thank you for this awesome video!
Also, I could make the proof shorter by using equivalence "ln(1 + h/x) ~ h/x" on the 2nd step in your proof
this approximation is based on derivative of ln(x)
6:30 Hum. Ah. But when we say limit(h->0) that implies in any direction right ?
As we work with real numbers we can have h < 0 or h > 0 and both directional limits (or whatever the proper name for that is) must give the same result for the derivative to exist.
But when we substitute for h->infinity, we only check the side h>0, right ?
So shouldn't we also substitute u=-1/t and verify that we have the same result ?
Or else prove that the derivative must exists in which case only one side is enough to get the value.
Hey , I recently started reading Thomas calculus and found that lnx was actually defined as definite integral of 1/t from 1 to x. So i think a proof is not needed stating the definition is enough. Anyway hats off to the great content
How it is defined, really depends on who you ask. Historically, natural log was discovered before the number e, and it was defined as this integral. But in modern times, we usually define it as the inverse of e^x, and define e^x as the special case of the exponential where it is its own derivative. The modern definition is much more useful, to learn what logs are for the first time.
These two definitions are internally consistent, but you need to start with one to prove the other.
Smart moves and thank you. To avoid confusion in approach 1 instead of twice using u I will use U and then w.
I love you, plain and simple.
I love these type of people on RUclips
I just wanted to say, that for some reason, LOGb(X)=ln(X)/ln(b) has always been my favorite relationship in "Logarithmic Functions" and THANKS for the bonus at the end!!!
X = b ^ logb(x), then take logd of both sides and bring the exponent down. Then solve for logb(x).
12:05 when I saw this, I was like... OMG I just realized what the hell I've been watching for the past 12 minutes... I was more intrigued by what he was able to do in terms of modifying the formulae, but then noticed he brought it down to 1/x, I love this guy.
Thank you, you are the best explaining ♥️
I watch your videos for inspiration and help as I just started year 7
The limit definition of the derivative of ln(x) is a nice one!
Thanks i like you so much, maths is magic ♥️. I try to find this by focus on the definition of a function wich is derivating if this limite was not infinity and i encounter a lot of problème by not knowing this definition of e and also "the limit of a continuous function is the function of the limit. Thanks a lot ♥️
Sorry i dont speak english very well but i learned more and more each days
You can also use the formula for inverse derivatives. This is how I did it:
Let g(x) = the inverse of f(x)
g’(x) = 1/(f’(g(x))
Let f(x) = e^x
Therefore f’(x) = e^x & g(x) = lnx
g’(x) = 1/(e^lnx)
g’(x) = 1/x
Therefore the derivative of lnx is 1/x.
To prove the formula I used, you can let
g(x) = inverse of f(x)
So,
x = f(g(x))
Differentiating both sides, you get:
1 = f’(g(x))*g’(x)
g’(x) = 1/(f’(g(x))
It basically what he does from 13:00, without explicitly using the formula for the derivative of a reciprocal function.
Which was proven first, the derivative of e^x or that of ln x?
The most impressive thing about these videos is not the math, it's his ability to write with 2 or 3 markers in the same hand while holding them all at the same time. And that his writing is still legible while he does it. I can barely read my own handwriting when i write with just 1 pencil
And hold a microphone in the other hand. Might as well start juggling at that point.
Such a great video on this!
Love the second proof of lnx's derivative
Back when I learned this we defined the logarithm function in terms of the integral 1/x dx, then proved that this function had the properties expected of a logarithm.
CIERTO BRO YO TAMBIEN LO APRENDI AL REVES,QUE EL LOGARITMO SE DEFINE JUSTO POR LA INTEGRAL DE 1/X
I have a fourth proof: If we differentiate e^ln x, instead of resulting in x, we use the chen lu, where u = ln(x). That results in e^(ln x) * du/dx. However, if we use the power rule, it results in 1. Therefore, x * du/dx = 1. We solve for du/dx = 1/x.
Awesome explanation
Hi, great video and work as usual :) thanks.
Could you elaborate on 12:52 from d( ln(X))/dx to y=ln(x).
I struggle a bit (lot) in seeing this by intuition.
In my mind I would have begun with dy/dx=ln(x).
The frace of the change in ln(X) in relation to X does not make sense to me.
Please help
I agree
I think it could have been made a bit more clear at 3:29 that the 1/h exponent is supposed to be evaluated for (1+h/x) before the log is taken. (But I still got the point.)
2nd part blew my mind, I am not familiar of the note which is equivalent to e, what particular topic is that note? thanks
Wonderful work!
To differentiate ln(x) I use this trick:
1 = 1
1 = d/dx x
1 = d/dx [e^(ln(x))]
1 = e^(ln(x)) * d/dx(ln(x))
d/dx(ln(x)) = 1 / [e^(ln(x))]
d/dx(ln(x)) = 1/x
This also works for all inverse functions like arcsin(x), arcos(x) & arctan(x).
I realized that its actually an additional set of functions (the inverse) that can be differentiated, apart from constant, sum, product, difference, quotient and subfunctions. But I still struggle on the mapping of it. the why f((-1)(f(x))) =x. I understand, also f((-1)f((-1)(f(x)))) =f(x). But I dont see the image clearly. Could you elaborate please. Thanks in advance
thanks brother for clear information ❤🤗🤗😎😎
it's always cool seeing derivations for things you learned without the reasoning behind them
Excellent, as usual!
such a long proof but very well thought out. I was definitely doing a shorter proof for my test (luckily, not sure if I could survive writing this for my test.. lol).
Dloga(x)=1/x*ln(a)
D(log(e^x))=1/xlne=1/xloge(e)=1/x*1=1/x
but of course mine is already making assumptions (that derivative of loga(x)=1/x*ln(a)) instead of figuring it out with definition of e. Great work, definitely I learned something.
How do we relate X and Y for any given value of X and randomly changing Y?
A most elegant solution to d/dx Ln(x)...I didn't imagine it would take 3 substitutions.
11:49 how can we say that the formula in the ln function and the definition of e he derived in the corner are the same since they define u differently?
U is a variable in both expressions. U is not a constant.
I like all the math problem and solutions 👍👍
Thanks bro. Awesomeness
16:43 Yeah! It even works for e: ln(e)=1 so you get back the 1/x :)
That was pretty cool!
Thank You so much sir!😇
Proving this was actually a question on one of our calc exams
Nice piece of Mathematics.
well done professor
That was beautiful!
I like your channel, its content and overall disposition (subscribed long back).
I would like to mentiom that a lot of times you seem to show some very simple or basic algebraic manupilation in great detail as if you are showing that to some beginner who is not very bright. I request you to be consistent in knowledge density (idk how to describe thos) throughout the vdo and spread your time evenly on the topics and ponder over stuff which really calls for it.
In this vdo, time spent on definition of e with t to u is probably 6 times than necessary, my personal feeling.
Lastly, as I like this channel, I complained. If I was indifferent, I wouldn't have cared.
you’re so awesome!!
Amazing job man
Thank you so much sir.
That moment when a channel about two colors of pen PULLS OUT A THIRD COLOR
Hi, could you make a video about de Fourier Transform?
youre my inspiration
This was so cool!
the first definition are amazing
in 6:40 when he says that when t approaches infinity doesnt u approach 0 only from the positive side, so it approaches 0+?
wow! thank's a lot for the explanation, now i know why it's 1/x :)
The closer he gets to 1/x, the bigger his smile becomes. :D
Hello, thanks for the vid!. I have the question, lets say y = x^2, if i want to differentiate y with respect to x^2, (not x) what would be the result in this case then?
Given y=x^2
Let w = x^2
Thus: y=w
We'd like to know what dy/d(x^2) is, or in other words, what is dy/dw?
Well, since we have y in terms of w, we don't even need to think about x. Simply take dy/dw of y=w, and get 1.
So there is your answer
For y=x^2, the solution for dy/d(x^2) = 1
y=lnx
e^y=x
(e^y)’=x’
(e^y)*y’=1
y’=1/(e^y)
(lnx)’=1/x (substitution of the given parts)
The proof of the derivative of e^x uses the exact same ln lim n->0 so it’s better to derive with limits and not other operations that rely on said limits because it can end up begging the question
@@AhmedKhan-qk3xi idk what u are saying but ig ur right
@@AhmedKhan-qk3xi I think it is just way shorter
how do i determine the n-th derivative of e^(3x).(sin x)^2
I literally love this channel holy shit