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@kennethgee2004

5 ай бұрын

answer is still no. You are taking the derivative of the gamma not factorial. I also do not think that the gamma function is valid as it breaks the rules of factorial and ends up with imaginary numbers in the negative, which actually do exist for all negative values in factorial. It is just the negative side does not go towards anything nice. Derivatives by definition must be on continuous functions. The gamma function is not continuous.

@seroujghazarian6343

4 ай бұрын

​@@kennethgee2004nah

"That's a good place to start" I undesrtood the reference hahahaha

@banana6108

3 жыл бұрын

Me too 😂

@SergioLopez-yu4cu

3 жыл бұрын

I know you...

@jimjim3979

3 жыл бұрын

Michael Penn?

@emilioparedesbarba4032

3 жыл бұрын

Qué haces aquí :v

@MatesMike

3 жыл бұрын

@@jimjim3979 yes!

Take the derivative of floor function. Take derivative of ceiling function. Take derivative of modulo function. Take integral of error function. Take Fourier Transform of Step function (Heaviside). Take integral of e^e^x.

@alexterrieur6858

3 жыл бұрын

For the floor and ceil fonction, derivitives areF:R\Z -> R x |---> 0

@pedrosso0

2 жыл бұрын

1. 0 for x is not an integer 2. 0 for x isn't an integer 3. Modulo has two variables, you need to specify which one is to be derived 4. x*erf(x)+e^(-x^2)+C by integration by parts It's even in the wiki en.wikipedia.org/wiki/Error_function

@pedrosso0

2 жыл бұрын

7. is defined with help of the exponential integral. Ei(e^x)

@hassanshaikh3451

2 жыл бұрын

The first 2 are impossible, they are both defined by not being continuous. If you took the derivative of small sections like between 0 and 1 or something, the derivative would be 0 as they have no "velocity"

@spacebusdriver

2 жыл бұрын

@@hassanshaikh3451 it is possible if you use the distributional derivative, but strictly speaking you wouldn't get a function in the normal sense but a distribution which is basically a linear functional on the hilbert space of functions if i remember correctly

Derivative turned into integration!!! Unexpectedly awesome.

@BriTheMathGuy

3 жыл бұрын

Right?!

@Cjnw

6 ай бұрын

​@@BriTheMathGuythis should be the case, because each component of e^x is based on a reciprocal of a factorial.

The Gamma function is simply amazing.

@BriTheMathGuy

3 жыл бұрын

I know right?!

@David-km2ie

3 жыл бұрын

No

@failurehaus5979

3 жыл бұрын

@@David-km2ie based

@ssaamil

Жыл бұрын

I love it too much.

@KingGreenscreenKid420

Жыл бұрын

θ

I have uncontrollable urge to click on maths videos that intrigue me, and this definitely one of them, very great!

@BriTheMathGuy

3 жыл бұрын

Wow, thanks!

What else can you "not" take the derivative of?

@mathevengers1131

3 жыл бұрын

How about differentiating super cube root of x?

@mathevengers1131

3 жыл бұрын

Can this be integrated? (Φ^(x))-(-Φ)^(-x))/√5 Where Φ is the golden number.

@ydg_me

3 жыл бұрын

Me

@mathevengers1131

3 жыл бұрын

x/0

@aashsyed1277

3 жыл бұрын

HMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

What's the inverse of the factorial? Is there a function: f(120) = 5 --> Or generally: f(x) = y where y! = x And how do you invert the gamma-function (in positive 'R')?

@ezequielangelucci1263

3 жыл бұрын

gamma doesnt have inverse but the gamma function at 1 to infinity?

@angelmendez-rivera351

3 жыл бұрын

You cannot invert the factorial function, since it is not injective, nor is it surjective.

@andrasfogarasi5014

2 жыл бұрын

@@angelmendez-rivera351 Well, it does have a right inverse (on the domain of non-zero reals because fuck zero). But not a left inverse, which is the one you're likely referring to.

@angelmendez-rivera351

2 жыл бұрын

@@andrasfogarasi5014 The word "inverse" in mathematics typically refers to the two-sided inverse, unless otherwise specified. If you ignore 0, which is arbitrary, then it is surjective, but not injective, so it would have a right-inverse, but not a left-inverse, and therefore, no inverse. For it to have an inverse, or alternatively, to be invertible, it must have both a left-inverse and a right-inverse.

@dafureveerbhadra2772

2 жыл бұрын

@@angelmendez-rivera351 it is bijective in its own domain and range, we want to define it only where x is actually a factorial of something

I am happy to see that your channel is getting bigger. Soon gonna be 100k, thanks for all videos, greetings.

@BriTheMathGuy

3 жыл бұрын

Thank you very much!! Really appreciate the support!

I really approciate you explaining everything and not just saying _'trust me bro!'_ . Helps a lot understanding the video at the first watch. Subscribed!

As a followup it might be nice to introduce the digamma function (or polygamma functions in general). One thing I particularly like is the derivative of the gamma function at integer points which involves the ubiquitous Euler-Mascheroni constant.

I actually don’t care about derivating the factorial function, but I knew this was going to be about the gamma function and the demonstration of why it’s a good candidate to extending the factorial to |R really was impressive

I still didn't take calculus or higher maths until now, but I still kinda understood. Your explanation is great! I like the idea.

Very great explanation!!

Little precisation that none will care about: swapping the integral with the derivative requires some hyphothesis. For example the integrand must be continous with continous derivative(and here this is easily verified). Otherwise the step may be incorrect. Good video!

@clementboutaric3952

3 жыл бұрын

you must find a function g such that for all x and t, |f(x,t)| < g(t) with g being integrable over the same domain as f with respect to t.

@BriTheMathGuy

3 жыл бұрын

Thanks for pointing this out! I tend not to be totally rigorous on KZread :)

@clementboutaric3952

3 жыл бұрын

@@BriTheMathGuy As someone else pointed out in another comment, the purpose of your video is not to focus on such tedious details.

@francesco1777

3 жыл бұрын

@@clementboutaric3952 I know that it's not the purpouse of the video, I just thought it would be interesting to provide insights on why this actually works. I think that tediousness is a relative concept, for me they are not tedious details.

@clementboutaric3952

3 жыл бұрын

@@francesco1777 Well once you know that one can not swap however one wants, you can't go back. These details become everything but details.

Great approach to an interesting problem. The question is how to address the integral? Is there a follow up video? Thanks

I will advertise this channel as much as possible. It helped motivate me to get through precalc and now discrete because I get to see the cool stuff I get to do later.

@BriTheMathGuy

3 жыл бұрын

Wow thanks so much!

@randallmcgrath9345

3 жыл бұрын

@@BriTheMathGuy so question, if you have time. Have you taken any interest in cryptology?

Great video. Do you have a video that determines all relative extrema of the gamma function?

The derivatives of the gamma function become extremely self-referential and nested within each other, it's actually really cool! (Look up the digamma and polygamma functions)

@ydg_me

3 жыл бұрын

Not that I understand it but sounds cool 😎

Very interesting (something like that) : With sum of inverse factorial you get e With half factorial squared you can get pi Thats very very interesting for me!!

@paolomorseletto3030

3 жыл бұрын

How u get pi from half factorial squared?

@capjus

3 жыл бұрын

@@paolomorseletto3030 pi=4*(1/2)!^2

Muy buen video Genio!

I was waiting for this.

@BriTheMathGuy

3 жыл бұрын

Hope it was worth the wait!

@mathevengers1131

3 жыл бұрын

Yes it was.

This was well explained. Thank you.

@BriTheMathGuy

3 жыл бұрын

Glad you enjoyed it!

Thanks!

You are thinking out of the box❤️

Wow, that was pretty easy), thanks

It's terrific, thanx, I'll show it to my sonny for the scientific high school math exam preparation

Nice video! Now we need to integrate the factorial (if possible)

@Noam_.Menashe

3 жыл бұрын

On what bounds? It would be a double integral for X and t.

When can we switch the derivatives and integral? Like there's dominated convergence theorem for Summation and Integration?

Your animations are so neat,,is it manim or which software??

how did you turn a derivative of a factorial to an integral?! This is amazing!

I have shown this in 2017 to my theoretical physics prof and her assistent. They once asked if anyone could do the derivative of x!. I said yes and 1 thing is if x is discreet, we can do the log (f(x)) = log(x!) derivative , but in this case there are only integers llowed, and the other way for all x is working with the derivative! Im Glad that someone else did the exact same thing like me! Keep it up friend ! You are awesome!

for differentiate an integrating with different variable don't you need to use Leibniz integral rule ?

Hi, I've been working on a proof myself, and stumbled on to x!, having to differenciate it. I knew about the gamma function and it's derivatives, but I was wondering If whe restricted the x value to be an integer, would the derivative have a non-integral form? I couldn't found anything about it, so I thought maybe with complex analysys use the fact that Gamma is meromorphyc and then have a power series. But then again i trouble myself with calculating the n'th derivative in a certain point (integer) because i couldn't find a non-integer form of an n'th derivative of the gamma function on an integer

Brilliant presentation !! vow !!

@BriTheMathGuy

2 жыл бұрын

Thank you! Cheers!

Derivative of a fractal curve. Something like the Weierstrass function

Can we somehow return to the integer input values?

Can it be written in the form of digamma function?

i was thinking about this thing since i was 13 years old. THANKS

And someone who’s currently taking ap calc ab right now. I can’t comprehend any of this

Nicely done.

@BriTheMathGuy

3 жыл бұрын

Thank you! Cheers!

Fantastic can you please explain. CArdano Method of solving Cubic equation

03:50 Polynomial (or power function) ("t^x"), natural logarithm ("ln(t)") and expotential function ("e^(-t)") in one place (and all three multiuplied) - amazing! Btw It is interesting what are the results of these questions: 1) d/dx( x^2 * ln(x) * e^x ) = ? 2) ʃ ( x^2 * ln(x) * e^x ) dx = ? 1') d/dx( x^2 * log.2(x) * 2^x ) = ? 2') ʃ ( x^2 * log.2(x) * 2^x ) dx = ?

Please explain why did you apply by parts integration method to the gamma function? Like why integrate it?

@coursmaths138

2 жыл бұрын

No. The goal is to prove that the gamma function satisfies the same relation as factorial

How do you solve that integral that comes from this derivative. I know I can change the variable t into W(u) but I don’t know how to integrate W. Where does the series for W lambert come from?

Derivative of the W-Lambert funktion W(x)

And we can also write this as a product of the digamma function and the gamma function, since Ψ(z+1)=Γ'(z+1)/Γ(z+1). Thus, d/dx(x!)=x!•Ψ⁰(x+1)

It was easier, you're a great teacher

Super interesting, a little too fast for me, I'll be watching over before I try it myself. Thanks for sharing!

My brain at 3a.m. : let's differentiate factorials XD

I'm fascinated by what this is don't understand any of it how it works or what it actually solves. But I am watching and again find it very interesting. Great video.

how about using the pi function?

U could have used the expression of 1 over gamma function where the euler-mascheroni constant shows up and derive from it the expression of the digamma function . Hence derivkng the formula of the derivative of x factorial ( WHAT THE FACTORIAL)

I swear two days ago i looked for video about derived of factorial and found nothing literally nothing so thank u very much for reading my mind

@BriTheMathGuy

3 жыл бұрын

🤯 You're very welcome!

@griffisme4833

3 жыл бұрын

kzread.info/dash/bejne/nZ5_m7mfna-rnNI.html

@griffisme4833

3 жыл бұрын

kzread.info/dash/bejne/dZeG3KyJgd3gn6w.html

@griffisme4833

3 жыл бұрын

kzread.info/dash/bejne/m6CD1MhxfaWbj9o.html

@griffisme4833

3 жыл бұрын

You didn't look very well...

It was exactly what I thought it would be, that math degree came in handy

Is there a way to then simplify that integral at all?

@angelmendez-rivera351

3 жыл бұрын

There is not, not really

@BriTheMathGuy

3 жыл бұрын

There are probably different representations but as far as I know that's as simply as it gets 😬

He asks at the end was that what u were expecting like yeah sure cuz i worked it out with my best friend and photomath lol

By taking logs of both sides of y = x! and differentiating we find y’ = x! (1/x + 1/(x-1) + 1/(x-2) + … ) = x! ψ(x) where ψ(x) is the digamma function, the differential of ln(Γ(x+1))

Could you differentiate the Fibonacci sequence?

Here o was thinking he’d use the Stirling Approximation for factorials and then differentiate that

How did you learn all of this?

thanks u brother

@BriTheMathGuy

3 жыл бұрын

You're welcome!

3:19 we can slip the derivative inside in this specific case using a theorem but it's not always true unfortunately

good solution and i learned gamma function

Pls can you take indefinite integral of x! I need it

There are lots of continuous functions that match the factorial function on the integers. The (shifted) gamma function is certainly the most popular one, and indeed in some circles people do use the ! notation to refer to it, but calling this the derivative of the factorial function seems like a stretch.

@radadadadee

2 жыл бұрын

pure clickbait

Brilliant thinking🤔

@BriTheMathGuy

3 жыл бұрын

Thanks! Have a great day!

What values of x could you find a value for the indefinite integral at the end? The weird power log exponential product looks like finding an exact answer would be annoying

What does t represent in the equation

Is there an inverse function to gamma(x)? i know that there technically infinite inverses but i only care about roots for gamma(x) where x>1 so there is only 1 root in this function.

Just as an experiment, i tried doing integral of x! and apparently it's almost the exact same thing but the ln(t) appears in the denominator

1:46 where does the x come from

Is the Gamma function the only function that satisfies Gamma(x+1) = x*Gamma(x) and Gamma(1) = 1? Or is there a whole group of functions that satisfy this?

0:44 should 0! be one? Why is there a vertical asymptote?

I wonder if there are other functions f with this "factorial feature", that f(x) = x*f(x-1), but with other variations. Like you have to shift the Gamma function by -1, that it matches the original factorial values.

@andrewgrebenisan6141

3 жыл бұрын

There are

@handschich7736

3 жыл бұрын

@@andrewgrebenisan6141 Do they have a certain name?

@angelmendez-rivera351

3 жыл бұрын

Look at the Pi function. It is defined so that it matches the factorial exactly. In other words, Π(x) = Γ(x + 1).

Why not use the pi function?

For a little intuition for how fast the factorial function is growing, it's Stirling's approximation to the rescue: ln(x!) ~= x ln(x) - x d(ln(x!))/dx ~= ln(x) dx!/dx ~= x! ln(x)

I like the graphics but I realized watching this versus the vids where the formulas are written on the board by hand that this one seems like it’s slightly too fast paced, there’s not enough time to read and really think about the equations as they appear on screen. When they were written by hand it created a naturally slower pace to the script where something would be explained since it took a little more time to write things out. I think the best of both worlds is keep the new graphics, they’re great, but intentionally speak a little more slowly and include some slight pauses to give the viewer time to digest what just appeared.

@BriTheMathGuy

3 жыл бұрын

I totally agree. This was my first time trying a totally animated video, so I'm still getting the pacing down. I really appreciate the detailed and thoughtful critique! Thank you for watching!

I understand why the Gamma function is x! (For discrete values) and as a result can be generalised to non-discrete values but how does one go about discovering the Gamma function?

For us statisticians this function is very important. We use this expression thousands of times throughout our careers, but I never thought of the derivative of this function. Very ilustrative, very impressive. Thank you.

@BriTheMathGuy

3 жыл бұрын

Thanks for sharing and thanks for watching!

That was cool

Is gamma the only continuous function that matches the factorial? Or can there be others?

@Blaqjaqshellaq

2 жыл бұрын

@@mastery4667 Or gamma(x) + n*sin(m*x*pi), where n and m are any integers!

I was expecting some pi(-ish) thing from this derivative but it isn't. But, as always nice video

@BriTheMathGuy

3 жыл бұрын

Glad you liked it!

why is the limit for x=r and r going to infinity from r^x * e^-r /x =0 ? for x=0 i understand but that i dont understand

Muito bom!

good one “Factorials _!_ “

so much so fast! I cant learn math like this. I had to pause so often, and even then... sheesh. Looks like I need to brush up on partial derivatives.

Anorher quality video... Nice!

@BriTheMathGuy

3 жыл бұрын

Glad you think so!

Brilliant

Well done sir. Great contents.

@BriTheMathGuy

3 жыл бұрын

Thanks very much! Glad you enjoyed it.

This kind of video always overlook the most difficult and tedious part : proving that the function is defined on R+*, proving that you can make the integration by part, proving that you can derivate inside the integral.

@angelmendez-rivera351

3 жыл бұрын

Yes, and that is intentional. These videos are not meant to be rigorous. If you want rigor, then you should just watch university channel videos instead.

@clementboutaric3952

3 жыл бұрын

@@angelmendez-rivera351 well you're right.

@karelspinka3031

2 жыл бұрын

1) The integrand is non-negative so the integral exists, and is finite because "the exponential function with minus in the exponent decreases much faster than any polynomial" (some criterion about convergence of integrals around plus infinity) 2) Do people still check assumptions for per-partes? I've done it maybe once in my lifetime. 3) A bit tricky, but the integrand is non-negative, so the absolute value of the integrand is the integrand itself, and the integral itself is finite (see point 1) so you can interchange, basically. There are a few more assumptions (like the derivative of the integrand exists almost surely) but these seem trivial since we deal with polynomials and an exponential.

Easier than the expectations

You can actually find the factorial derivative without defining gamma function, just use the limit definition of derivative and you'll find that if f(x) = x! then f'(x) = f(x)*(f'(0) + H_x)

Why is the gamma function offset from the factorial function by 1 unit in the x direction?

@hetsmiecht1029

2 жыл бұрын

I don't know, but there's a thing called the pi function with the only difference being that the shift isn't present.

@hetsmiecht1029

2 жыл бұрын

I don't know, but there's a thing called the pi function with the only difference being that the shift isn't present.

now time to figure out how to take the partial derivative of single variable function

@herbcruz4697

3 жыл бұрын

Well, if you had something like f(x)=2x-5 in the single-variable case, we could make this a multivariable equation, instead, if we wanted to (i.e., f(x,y)=2x-5) (This is now a function in terms of both x and y). Then, taking the first-order partial derivatives of this function, we get that ∂f/∂x=2 (The derivative of 2x (with respect to x) is just 2, and the -5 is a constant, so its derivative with respect to anything is just equal to zero (0)) and ∂f/∂y=0 (Here, since we are differentiating with respect to y, we treat x like a constant, but since there are no y's in this function, the whole 2x-5 is treated like a constant, so the partial derivative of this function (2x-5) with respect to y is just equal to zero (0)). We could even go further and make this a function of 3 (i.e., f(x,y,z)=2x-5, in the case of the function being a function of 3 variables) or more variables, if we wanted to, and the first-order partial derivatives with respect to those other variables would also be equal to zero (0), for the exact same reasoning as when we took the (first-order) partial derivative of the above function with respect to y (So, if we had f(x,y,z)=2x-5, then ∂f/∂z=0, etc.).

@mastershooter64

3 жыл бұрын

@@herbcruz4697 f(x, y) = 2x - 5 f(x) = 2x - 5 f(x, y) = f(x) wait what? did I do something wrong here? how can a multivariable func be single variable at the same time?

@herbcruz4697

3 жыл бұрын

@@mastershooter64 It just reduces down to the single-variable case.

@angelmendez-rivera351

3 жыл бұрын

The partial derivative of a single variable function is well-defined. We just call it the ordinary derivative, though.

I wish people would use the Pi function instead of the Gamma function for factorial since it needlessly over complicates things and the Pi function is a little tidier than the Gamma function

e^x is always a sum of factorial reciprocals

Try differentiating e^(phi*x) with respect to x over and over again. You'll get a pretty cool series

@chumblewumble2422

3 жыл бұрын

Are you sure

And that exclamation mark in the title of this video ;)

@BriTheMathGuy

3 жыл бұрын

‼️

How about the derivative of x^(1/x^(1/x^(1/x ... up to infinity?