Here’s a link to the sequel: Fractional derivatives of exponential and trigonometric functions kzread.info/dash/bejne/nWaIkruinLPYh9o.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.

@fimblewinter7806

6 жыл бұрын

so what would that mean you can have a ith derivative

@flatfingertuning727

6 жыл бұрын

In electronic filtering applications, an integrator may be used as a low-pass filter that attenuates signals by ~6dB/octave (a factor of two in amplitude for a factor of two in frequency) and a differentiator may be be used as a 6dB/octave high-pass filter. Integration and differentiation thus behaves as filters whose amplitude/frequency function, plotted on a log/log scale, would have slopes of -1 and +1, respectively. I would expect that non-generalized derivatives and integrals would behave as filters whose slope on a log/log scale is the order of the derivative.

2=1+1

@OonHan

6 жыл бұрын

7=4+3

@GetSmart008

6 жыл бұрын

Prove 1+1 is 2 or site Cantor`s proof. Doc P can frCTIONAL DERIVATIVES BE DEFINEd using Fourier pseudodiff operators? If so do a video. TIA

@Gameboygenius

6 жыл бұрын

*gives blackpenredpen a cookie* (Oreo brand, of course.)

@hassanalihusseini1717

6 жыл бұрын

2+2=5

@chasemarangu

6 жыл бұрын

and 1+1=2 communicative property of addition don't forget that either

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

@neck2b

6 жыл бұрын

PCreeper394 god damnit

@srimayikorrapati9423

6 жыл бұрын

OH MAN HAD TO READ THAT THRICE BECAUSE I KNEW I WAS MISSING SOMETHING

@OfficialMrQ

6 жыл бұрын

let's try complex derivatives

@rajdeepyadav4590

5 жыл бұрын

@@OfficialMrQ That could be complex

@aliciatamayo6680

4 жыл бұрын

a+b

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

@drpeyam

6 жыл бұрын

Hahaha, they say that sleeping is addi(c)tive! :P

@brian554xx

6 жыл бұрын

Never realized how mathematical a bed can be. Multiplication _and_ addi(c)tion?

@drpeyam

6 жыл бұрын

Hahaha, a bed is an algebra then 😂

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

@JaydentheMathGuy

5 жыл бұрын

I'm not ready for Ph.D. level math yet! I'm still in high school!

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

@blackpenredpen

6 жыл бұрын

I did too!!

@mihaiciorobitca5287

6 жыл бұрын

i learned today that 2=bprp+dr. peyam i love '2' !!!

@tricky778

2 жыл бұрын

but also that 2≠½+1+½

@k_wl

Жыл бұрын

and 7 = 4 + 3

Now do the half-integral !

@drpeyam

6 жыл бұрын

Poetu Hahaha, great idea!!! :D I'm guessing it should be constant times x^3/2 :)

@drpeyam

6 жыл бұрын

OMG, guess what!!! If you assume that the half-derivative of the half-integral of a function is just the function itself, then: Claim: The half-integral of a function is just the half derivative of the ordinary integral! Here's why: By definition, the half integral int^(1/2) should satisfy: int^(1/2) (int^(1/2) f) = int f (the integral of f) Now take half derivatives on both sides: D^(1/2) int^(1/2) (int^(1/2) f) = D^(1/2) int f Now assuming that the half derivative of the half integral of a function is just the function itself, we then get int^(1/2) f = D^(1/2) (int f) Ta-daa!!!! :D

@OonHan

6 жыл бұрын

Dr. Peyam's Show amazing!

@MagicGonads

6 жыл бұрын

I totally read that in your voice

@Gameboygenius

6 жыл бұрын

Remember, if you do the half integral, make sure you only add C/2 at the end!

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

@VikeingBlade

4 жыл бұрын

Lmao

@alexanderm5728

4 жыл бұрын

As soon as I saw this video I looked for this comment.

@gdash6925

4 жыл бұрын

Well Dr """"""""""Peyam"""""""" yoshi.

@epalegmail

4 жыл бұрын

God tier coment

@deedewald1707

3 жыл бұрын

Is this a half video TOO !

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!

@firebrain2991

6 жыл бұрын

Hell, I did the same thing, but I looked up the gamma function, tried to read the Wikipedia article, and gave up.

@afadeevz

6 жыл бұрын

I was thinking about negative-order derivative

@gnikola2013

6 жыл бұрын

Alexander Fadeev there would be integrals. I think Peyam shows this in other video or in this one. The point is that if the derivative of f is D^1(f), then considering that the grade of the derivative is equal to the sum of the "exponents", and that any function is its own 0th derivative, then D^-1(D^1(f)) = f. Considering the fundamental theorem of calculus, of derive a function and then integrate it you get the original function. Also, considering that integration also satisfies the linear transformation properties, we can assume that D^-1(f) is the integral of f. (Technically you also have a constant of integration, but I've neglected it for demonstration purposes)

@bcthoburn

5 жыл бұрын

SAME. Me too, I considered it as he did but wasn’t thinking about the linearity thing and never tried one.

@sowmyag5142

5 жыл бұрын

R u Indian?

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

@dominickrobinson332

4 жыл бұрын

Computer pannenkoek2012 heck yes

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

@drpeyam

6 жыл бұрын

Thanks so much!!! bprp and I really appreciate it!!!

@blackpenredpen

6 жыл бұрын

Thanks!!!

@roygalaasen

6 жыл бұрын

Is it blackpenredpen that I can hear in the background? Edit: err probably since he commented on this comment lol...

@blackpenredpen

6 жыл бұрын

roygalaasen loll yes.

KZread 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!

Your excitement is contagious. Love your personality.

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

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.

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

this dude is just filled to the brim with positive energy

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

@drpeyam

6 жыл бұрын

Thanks so much!!!! :D

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

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?

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

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!!

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

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

Incredibly amazed. Great video!!!

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.

@davidwright8432

6 жыл бұрын

Thanks for the kudos, Dr Peyam! The Cheshire Cat now awards himself a generous saucer of double cream.

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

This was soooooooo fucking legit

@drpeyam

4 жыл бұрын

Thank you!!!

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

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

"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.

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.

@drpeyam

6 жыл бұрын

Ahsoka Tano Indeed the -1th derivative of f is just the integral of f, because by definition we should have D(D-1 f) = f, and similarly the -alpha derivative of f is the integral of the alpha derivative of f. Not sure about complex numbers, but since Gamma is defined for complex numbers this might actually work! I'll check it out and see what happens, but I'm guessing it's just a constant times x^(1-i)

@drpeyam

6 жыл бұрын

Ahsoka Tano Oh, and there are indeed differential equations with fractional derivatives! Check out one of the comments below where I put some applications!

@GermanSnipe14

6 жыл бұрын

Wait but wouldn't the kth derivative (where k is a negative natural number) not exist for x^n since that would yield a negative natural number in the gamma function, which isn't defined?

@OonHan

6 жыл бұрын

Dr Peyam yay

@MagicGonads

6 жыл бұрын

And yet, GermanSnipe14, we can very easily integrate x to obtain it's kth integrals (where k is positive natural number), so I think this generic definition of the derivative is incomplete, do we need to make a special gamma function so that it has satisfactory values?

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

@turbopotato4575

6 жыл бұрын

Why not? He claimed it is a linear operator so polynomials and power series are half-differentiable and thus so are all holomorphic functions which even though still doesnt cover all continuously differentiable functions it includes most elementary functions

@sea34101

6 жыл бұрын

turbo potato As an exercise try to calculate the half derivative of x->1 (this only requires basic linear algebra), this will lead to a contradiction.

@maxvilla5005

6 жыл бұрын

Isn't "f(x)=x" a continuous, differentiable function? So the last result shown in the video is not valid?

@mpalssonur

6 жыл бұрын

Wait... what do you mean?

@douggwyn9656

6 жыл бұрын

Thanks for supplying a little bit of actual math here.

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

Wonderful, I really enjoyed it Dr Payam Show.

I have never thought about this before. Amazing!

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

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

@drpeyam

6 жыл бұрын

See the pinned comment for applications :)

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)?

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

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

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

@drpeyam

6 жыл бұрын

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.

@drpeyam

6 жыл бұрын

Ahsoka Tano This is the comment I was referring to!

@markzero8291

6 жыл бұрын

Thanks Dr. Peyam!

Excellent video, thank you very much sir!

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.

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

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).

@drpeyam

4 жыл бұрын

Check out my playlist, I do precisely that

This stuff is amazing! Good work guys!

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

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

Fantastic work, thanks for sharing this wonderful work

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?

He is the happiest person I've ever seen.

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

is this thing even consistent with power series expression of e^kx and another video on this topic.?

You just rekindled my inner math fire! ❤️😍

Sir you are a legend. Mind blown!!!

as always :) just beautiful !! thanks Dr Peyman

How amazingly u explained , thanku so much 😃😃

@drpeyam

5 ай бұрын

My pleasure 😊

Wonderful! Your best video yet.

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

Can this be extended to power series? And, if so, does it satisfy that the half derivative of the exponential is itself?

@drpeyam

2 жыл бұрын

Check out my fractional derivative playlist :)

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

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

Is showing that the two half derivatives equal to the first degree derivative sufficient condition to suggest the existence of half derivatives? Indeed, can we show that these half derivatives can compute all degree n derivatives?

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

@drpeyam

4 жыл бұрын

❤️

does it gives the same definition as the half derivative you get by using fourrier transform ?

@drpeyam

5 жыл бұрын

I believe so!

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

Brilliant video. You just earned a sub

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

what would the the ( 1 / 3 ) derivative of ln^3 ( x ) be ?

Doesn’t that kth derivarive formula only work for k>0?

Wish my calc 2 course covered this, neat stuff

Great video Dr., thanks

What is the function that meets the following condition int(ln(f(t))dt)=×^2 Thank you

@MarianoArias1

5 жыл бұрын

maybe e^2x ?

can you make any sense of the half derivative spacialy? if i take the half derivative of displacement vs. time, what does that give me?

@drpeyam

5 жыл бұрын

Think like it as a half velocity, like stepping on your gas pedal. If you do it twice it gives you the velocity

@ianmcstruthers9937

5 жыл бұрын

@@drpeyam That's not helpful to me - I still can't wrap my mind around what that concept would be.. velocity, acceleration, jerk - are all concepts I can grapple with, but "half velocity" .. what is that? Thanks for the reply

Does this have a graphical significance as well?

Interesting ! I learned something today.

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

Can you solve Cauchy-Euler differential equations that contain fractional derivatives?

great stuff and camerawork!

Is this the only solution? What about other alternatives for factorials, like not logarithmically convex functions?

I'm wondering if it's possible to show an example for the derivative of transcendental order

@tmogoreanu

6 жыл бұрын

Not in terms of gamma function

I know im late as hell but isn't the gaussian integral equal to sqrt(π) instead of half of that ? @18:57

yes, but how do I visualize it?

Would there also be a the quotient and product rule for fractional derivatives?

@drpeyam

6 жыл бұрын

Great question! Apparently no! It might not even be true that D^alpha D^beta f = D^beta D^alpha f. Pretty crazy, no? :O

@MrRyanroberson1

6 жыл бұрын

well, if you can break a derivative into halves, and you can break it into whatever uniform & identical fraction as you want (like 5 fifth derivatives) then you get the same identities associated with exponentials, that 5^2 *5^4=(5 *5) *(5 *5 *5 *5)=5 *5 *5 *5 *5 *5= 5^6, as even without a transitive identity you can still squish them back from the right upwards. as for product, d^(ab) would naturally be simultaneously d^a applied b times and d^b applied a times

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.

wow awesome video, interesting content presented in an interesting way

Dr. Peyam what might be the application of this. I know that to reach this general formula it was very entertaining but I am just wondering if there is a connection to the real world with fractional derivative.

@drpeyam

6 жыл бұрын

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.

@kamineko4565

6 жыл бұрын

Dr. Peyam's Show I want to go to your class now D:

If I take f(x)= x^5 say, and differentiate multiple times I will get eventually get to zero. I can graph each derivative quite easily. Now how could I graph fractional derivatives between each integer value of derivative? Would I finish up with a set of curves that progressively transition between each integer value curve?

I know I'm a year late but up seeing the setup, I solved this using the Laplace Transform. Is that transformation valid for all of this? I took the laplace of x (lets say L(x)=1/s^2) and multiplied it by SQRT(s) because L(x')=L(1)=1/s=s*L(x). From my Laplace table, L^(-1) (s^(1/2)/s^2 )= L^(-1) (1/s^(3/2)) = 2/SQRT(pi) L^(-1) (SQRT(pi)/2/s^(3/2) = 2SQRT(x/pi). I want to try to use this for the ith derivative of x but that will require me to go well beyond my handy dandy Laplace table. Edit: Yeah. This method worked to calculate the imaginary derivative of x too. It took a while to convert it to your answer. I got that the (ith derivative of x^n) = x^(n-i)/Gamma(n+1-i) which can be confusing to simplify but does equal what your video said for n=1.

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

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.

@sherllymentalism4756

5 жыл бұрын

What's it called?

@RAJAT6555

5 жыл бұрын

Here it is: b-ok.cc/book/2315244/757a6e

@baskara3668

4 жыл бұрын

@@RAJAT6555 Thank you so much...

How does this apply to functions that aren't polynoms, e.g. ln(x), sin(x), arctan(x) etc. ? Are you going to approach by taking the half-derivative of their taylor-series approximations?

@drpeyam

6 жыл бұрын

Yes, you could find fractional derivatives of e^x and sin(x), and I’ok do that in another video! I’m not sure about ln or arctan, and the Taylor series approach might work we’ll see

So what about transcendental functions? I figure √d/√dx (e^x) remains e^x, and log x becomes more complex, but what about sine & cosine?

@drpeyam

6 жыл бұрын

Firebrain I’ll do a video on that; it’s super interesting!

I wonder what kind of phisical phenomenon may we associate to a certain half or 1/n th derivative. In any case Gamma function here seems to assume new significance.

@drpeyam

4 жыл бұрын

Usually fractal things or Brownian motion