You know its about to get real when he starts using the blue pen (-:

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.

8:11 Talking to your GF

8:25 my brain to me after a test

10:43 I really love the satisfaction I get when my mind snaps and know how the demostration continues before the video. Great video!

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.

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

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

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.

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.

Elegant proofs for the derivative of ln(x). I like the intelligent and creative ways you used to develop and establish your proofs. Thanks.

Well done guy! You sort it out! Keep it up! Go always deep n in every detail to enlightening. Again you've done it!

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)

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

I loved calculus in college and now i am 71 and watch these videos to fill the gaps in my understanding and keep dementia away

this math professor dripping out with tha preme jacket

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

Thank YOU so much for sharing your beautiful smile and passion!! It makes me so much more excited to learn and genuinely happy :))

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!.

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))

dayuumm now that's impressive, finding the derivative of ln(x) using the standard definition of a derivative

Just found your channel. Thanks for creating this content and keep up the good work.

Dear friend, you are not only genius but you a great guru (teacher). My regards - Sudarshan🙏

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.

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.

Absolutely beautiful. Great explanations! Thank you.

How easily he changes markers is amazing to watch

Finally, since the basketball secret has been revealed I can find some sleep!

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.

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

It's so beautiful ♥ Great explanation!

This was so intertwining I was guessing what to do and when he showed what to do it made sense feels amazing

This mad lad really just used the limit definition. Can we get this guy a medal?

YOU ARE PHENOMENON!!!!!

جميل ورائع ومميز ما يقوم به هذا الشاب،،، فعلا عقليه فذه،، 🌹🌹🌹

Dang, when he finally pulled out the e term, I got super excited. Nice job!

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 love you, plain and simple.

I love u-sub when doing algebra and calculus. SO useful.

I love these type of people on KZfaq

Very elegant description of this important derivative

The second lim going into the continues function was so eye opening and satisfying

Smart moves and thank you. To avoid confusion in approach 1 instead of twice using u I will use U and then w.

Thank you, you are the best explaining ♥️

Such a great video on this!

Excellent, as usual!

I wonder if there is a numerical analysis class he teaches. This guy is a good teacher.

Wonderful work!

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!!! All other videos I found only explained how you used the derivative not actually showing proof on why it’s 1/x

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

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.

I watch your videos for inspiration and help as I just started year 7

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

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.

That was pretty cool!

thanks brother for clear information ❤🤗🤗😎😎

Awesome explanation

Thanks bro. Awesomeness

This was so cool!

That was beautiful!

you’re so awesome!!

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!!!

Thank You so much sir!😇

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).

The limit definition of the derivative of ln(x) is a nice one!

Amazing job man

I like all the math problem and solutions 👍👍

Which was proven first, the derivative of e^x or that of ln 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 is so awesome.

youre my inspiration

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

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.

Thank you so much sir.

so beautiful !

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.)

Love the second proof of lnx's derivative

well done professor

Well done!

Wait, doesn't the power 1/h belong inside the brackets?

wow! thank's a lot for the explanation, now i know why it's 1/x :)

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?

How do we relate X and Y for any given value of X and randomly changing Y?

Awesome man

Nice piece of Mathematics.

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

in 6:40 when he says that when t approaches infinity doesnt u approach 0 only from the positive side, so it approaches 0+?

I literally love this channel holy shit

This is 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.

the first definition are amazing

great teacher

Very satisfying and pretty!!!

Very helpful

That moment when a channel about two colors of pen PULLS OUT A THIRD COLOR

Proving this was actually a question on one of our calc exams

Great! Thanks!