Here’s a link to the sequel: Fractional derivatives of exponential and trigonometric functions kzfaq.info/get/bejne/oZiEY7ya0bbWh6s.html Some applications of fractional derivatives: There are surprisingly many applications of this, because it turns out that some differential equations in physics are written in terms of fractional derivatives, see en.wikipedia.org/wiki/Fractional_calculus#Applications There are three other ones I can think of: 1) In functional analysis, it's an important problem to find a square root of an operator (I don't really know why, maybe to decompose that operator?), and what we really did is to find a square root of the derivative operator, because if you apply D1/2 twice, you get D, so (D1/2)^2 = D, so D1/2 = sqrt(D) in some sense. 2) There is the nice formula in Fourier analysis that says that the Fourier transform of f' is integral of x e^(i something), and we have a similar formula for the fractional derivative, (I think, don't quote me on that) that the Fourier transform of D^(1/2) f is integral of x^(1/2) e^(i something). 3) Fractional derivatives allow us to define nice spaces of functions (for example, those whose fractional derivatives exist and are square integrable), and sometimes in differential equations you have a solution that is not defined in the classical sense (i.e. continuously differentiable), but might belong to this nice space, which allows us to study those equations.

2=1+1

It now seems obvious that not all derivatives should be positive integers. In fact, when you think about it, negative derivatives are integral to math and science

1:25 Maan, you're gonna kill me with high-level mathematics.

If I watch this video while half asleep, and watch it again while half asleep, will I have watched it in my sleep?

Now do the half-integral !

What I learned today: 2=1+1. Thanks Dr. Peyam!

"A derivative is a derivative, you can't say it's only a half" Joking aside, really great video

I am super excited to watch this video, because actually I have thought about this concept of non-interger order derivatives some years ago, when I was like 16 or so. However I obviously didn't have the tools nor the knowledge to actually develop the idea. But know I'm watching someone who thought about this like I did! Amazing!

Your enthusiasm about this beautiful art is contagious, sir. You are amazing.

This is the quality content I came here for. Please explore the properties of non-integer derivatives of some non-polynomials!

"A derivative is a derivative. You can't say it's only a half." - TJ Henry Yoshi

A few years back, I thought of the idea of a half-derivative. I realized that sinx and e^x work quite well, being sin(x + pi/4) and e^x. The only thing was, I almost felt like people would laugh at me for proposing something so ridiculous. Of course now I'm actually quite relieved to see that at least one other person was just as crazy as me, and I kinda wish I wouldn't have convinced myself that it having no apparent application it was useless. I mean like a lot of math was discovered before it was needed, so... I think I should probably finally figure out the half-derivative chain rule.

Both you and blackpenredpen rock! Keep those awesome math videos coming!

KZfaq started this rabbit hole for me, and I’m glad it finally brought me to the video that has the best explanation I’ve seen so far!

This is great stuff! Never thought of the concept of fractional order derivatives, but it comes so naturally. Thanks Dr P!

This was a fascinating video, it had everything: Derivatives, Gamma Function and the Gaussian Integral. Thank you Dr. Peyam!

I just learnt derivatives in School and watched this entirely video without getting a single thing and enjoyed the hell out of it only from watching the hype and excitement. Wished i had you as a teacher xD. Keep it up ^^ :3

this dude is just filled to the brim with positive energy

Your excitement is contagious. Love your personality.

What a fun video. Interesting topic, and it was great to see you guys having so much fun with the maths. Subbed :)

This is only the first video of yours I've seen, but it's so dang good I had to subscribe!

I nenver, ever imagine that this could be posible. Your channel is amazing.

Wow I really wonderred if there's any way to define derivative 'continuously' when I first knew the second and third derivative and this video answers my question in 20 mins! I really enjoyed this video and now I wonder how to define half derivative in analytic way. Thank you Dr. Peyam!!

Your video is cool! The half derivative of a function is really a great thing; I want to learn more about it. But, I want to know what is the geometrical interpretation of half derivative?

That got a lot more tricky than I anticipated. Well played!

I love how he always sneaks in pop culture references into his videos

Good Gamma, what a mind blower! I need a straight jacket!

Such a happy presentation. I get to go to sleep thinking about fractional calculus, thanks. : )

Incredibly amazed. Great video!!!

I have never thought about this before. Amazing!

Mind blowing half derivative.... of x Simply amazing. Dr. Peyam always master of mathematics

Thanks, Dr. Peyem! Very clear; each step made sense Now I need to think, re-view and internalize the whole thing! Then I'll be able to grin a Cheshire-cat-like grin, and know what I'm talking about.

"When Dr. Peyam teaches It's a show" believe me It is absolutely true.💜💜💜 Dr. Peyam show is very addictive.if someone enter this show,he/she never go out.

Thank you so much! I always asked myself if this could be possible. Very nice job

I have done so many proofs in Undergraduate Maths but I never enjoyed it. finding you I am enjoying learning mathematics. Thank you Sir.

I really love your videos, much more than before!!

This stuff is amazing! Good work guys!

Fantastic work, thanks for sharing this wonderful work

2 =1+1 BPRP : LAUGHS LIKE CRAZY Fast forward to 2019: BPRP: 2=1+1

Wonderful! Your best video yet.

Dr Payem! What about negative derivatives, are they possible? Or is that an integral? What about complex derivatives? (The i-th derivative of x) Maybe this can be used to create some very hard differential equations. This is an eye opener thanks for the amazing video.

as always :) just beautiful !! thanks Dr Peyman

I didn't know Kakà is now a calculus teacher... great explanation by the way!

Fun fact: It is not possible to extend that result to any continously differentiable function.

I'm just waiting for a practical use of this incredible and ridiculously complicated piece of art.

Excellent video, thank you very much sir!

Interesting ! I learned something today.

Hi ,10 ans passés que j'attends cette démonstration , en fait j'ai lu dans livre de distributions mathématiques mais sans aucune indication ,c'est magique ,les mathématiques avancent plus vite que la physique ,certainement il y aura l application de cette formule ,MERI Dr, c'est génial. write in French.

Sir you are a legend. Mind blown!!!

This is amazing! 6 min in and im already impressed!

Wonderful, I really enjoyed it Dr Payam Show.

Great video Dr., thanks

Brilliant video. You just earned a sub

Wish my calc 2 course covered this, neat stuff

I can't say nothing just wooooowww! I haven't seen something like this. Everything makes sense. You're awesome 😮

He is the happiest person I've ever seen.

Dr Peyam! WHERE HAVE YOU BEEN ALL MY LIFE???!!!

For those wanting to read up on this subject, there is a book published by Dover Publications and authored by KB Oldham and J Spanier. It is a good introduction to the subject.

You all are enjoying yourselves. Thx. For the video.

This was soooooooo fucking legit

This was a really awesome video, but I had a couple questions. Is it mathematically rigorous to treat the derivative operators as variables when you kind of add the "exponents"/n-th order on the derivative operator? If not, what is the mathematically rigorous way to defined fractional derivatives? Also, what do fractional derivatives conceptually mean (I know what a regular derivative conceptually mean, but it's hard for me to visualize a fractional derivative)?

Awesome lesson!

what happens when you take the derivative of the value you get when you change the order of the derivative of a function?

Thanks for the great video! Do you know of any applications of fractional derivatives? Why might someone want to calculate fractional derivatives?

I love their energy !!!

Great video! I was wondering if the linearity of the half derivative is ever proved, or if we just assume it?

great stuff and camerawork!

Excellent explanation and presentation of the non-integer derivation of the monoms. I would have two other candidates for similar cases to examine: 1) You could take the taylor development of the square root and apply it to the derivative operator. 2) You could take the Fourier analysis of the function you want to derivate and then the n-th derivative of sin(omega x) is (omega^n)*sin(omega x + n*pi/2) and the n-th derivative of cos(omega x) is (omega^n)*cos(omega x + n*pi/2).

You just rekindled my inner math fire! ❤️😍

I would love to see an animation of a plot as you let the derivative vary. I wonder if it would be a smooth animation

How beautiful this prove steps, thanks

Man, you're so cool!! I wish i was as good as you , in maths

wow awesome video, interesting content presented in an interesting way

Wow.... That's why I love math. And that's why I love you,Sir....

I was solving integrals the other day and writing up the n-th derivative of certain functions, and I was wondering, what if we plug in a fraction into n? Looks like it wasn't that crazy of a thought after all :D

How amazingly u explained , thanku so much 😃😃

Dr. Peyam is a great man.

Really good work 👍.

You kinda look like Ramanujan xD

Do you extend this to non-polynomial functions by converting them into a Taylor series and applying the fractional derivative term wise?

The donkey kong joke caught me off guard hahaha so good

I was just wondering, does this have any application or is it just a fun thing?

Very interesting video! Can you do another one about non-polynomial functions?

Can you do video explaining what are cosets and how do they partition the group and why is that equivalence relation defined on group in that way? I just cannot get that no matter how much i read

This is a great and natural extension of derivatives. Just like we define the 3rd operation (exponentiation) first as repeatedly multiplying ^n , then finding an inverse ^1/n and therefore having rational exponents, you apply this to the "multiplication" of differential operators. On another note, I am always annoyed by the shift in the operand of the gamma function. I prefer having Π(n) = n!, which is much more intuitive and removes some unnecessary thinking efforts, just like with π=τ/2 (why alway 2 π, if you mean 1 period ? why always Γ(x) = x-1!).

Is there a better way to calculate the half derivative of trigonometric, exponential or logarithmic functions than using the method shown above to their respective Taylor's series?

Thanks for another great video, Dr. Peyam !! How can we generalize this for any arbitrary function of x? Perhaps via the following argument? Any "reasonable" function of x can be expressed as a power series in x, within an "applicable neighborhood". Half derivative can be applied to each individual term of the expansion, and then combined back by virtue of linearity?

wow!! this was awesome!

You are every bit as good as the 3b1b and bprp's of this world

I had the same idea as well. But instead of x^n, I chose xe^x. The nth derivative of xe^x is (x+n)e^x which is very simple and doesn't require any analytically continued functions.

hey dr peyam i have never seen such thing in my maths class,although i studied calculus upto class 12. so could you please suggest me some resourses or some good books to study these stuffs of calculus

I'm rewatching this just to tell you you're awesome

How about Prdouct and Chain rule for fractional derivative?. Do more on Fractional Calculus..

I would kill for one calc class with Dr. Peyam

Cool stuff. Does is also work for an one half integral ?

I would like to see this concept described using vectors with a polynomial basis and the derivative transformation matrix. given that the square of the half derivative would give you the derivative, you should be able to describe half derivatives using the derivative transformation matrix and finding its square root. you should be able to easily generalize this to nth fractional derivatives using the eigenvalues of the matrix and then computing the fractional powers. does that seem right or am i completely off? Immediately i am becoming skeptical of imaginary derivatives, negative derivatives, irrational or transcendental derivatives etc. because of the branch nature of the nth root mapping would you find extraneous fractional derivatives? or that the n solutions would be on footing somehow, implying that there are n 1/nth fractional derivatives? so many questions i can hardly formulate them properly.

Amazing!

Amazing video, as usual. :D Is there a way of defining fractional derivatives in terms of limits and best linear approximations?? Secondly, would half derivatives in higher dimensions have some connections with square roots of positive semi-definite linear operators?

Can you make a video explaining physically or geometrically the meaning of non-integer order derivatives, even complex number order derivatives?