I Found Out How to Differentiate Factorials!

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Have you ever wondered how to find the derivative of a factorial? In this video I'll show you how to differentiate factorial functions! It's time to find out how to differentiate the factorial of x. \
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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

[@BriTheMathGuy]

🎓Become a Math Master With My Intro To Proofs Course! www.udemy.com/course/prove-it-like-a-mathematician/?referralCode=D4A14680C629BCC9D84C

[@kennethgee2004]

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]

​@@kennethgee2004nah

[@benyseus6325]

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]

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

[@pedrosso0]

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]

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

[@hassanshaikh3451]

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]

@@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

[@MatesMike]

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

[@banana6108]

Me too 😂

[@SergioLopez-yu4cu]

I know you...

[@jimjim3979]

Michael Penn?

[@emilioparedesbarba4032]

Qué haces aquí :v

[@MatesMike]

@@jimjim3979 yes!

[@hasanjakir360]

Derivative turned into integration!!! Unexpectedly awesome.

[@BriTheMathGuy]

Right?!

[@Cjnw]

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

[@tasneemmidhat6741]

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

[@BriTheMathGuy]

Wow, thanks!

[@micheltenvoorde]

The Gamma function is simply amazing.

[@BriTheMathGuy]

I know right?!

[@David-km2ie]

No

[@failurehaus5979]

@@David-km2ie based

[@ssaamil]

I love it too much.

[@KingGreenscreenKid420]

θ

[@mehmet2247]

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

[@BriTheMathGuy]

Thank you very much!! Really appreciate the support!

[@Handelsbilanzdefizit]

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]

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

[@angelmendez-rivera351]

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

[@andrasfogarasi5014]

@@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]

@@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]

@@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

[@rosskrt]

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

[@subthermosphere]

Very great explanation!!

[@davidgillies620]

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.

[@lemonteurdesanuseur9686]

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

[@pleasegivemeaciggy]

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

[@randallmcgrath9345]

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]

Wow thanks so much!

[@randallmcgrath9345]

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

[@BriTheMathGuy]

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

[@mathevengers1131]

How about differentiating super cube root of x?

[@mathevengers1131]

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

[@ydg_me]

Me

[@mathevengers1131]

x/0

[@aashsyed1277]

HMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

[@francesco1777]

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]

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]

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

[@clementboutaric3952]

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

[@francesco1777]

@@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]

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

[@mathevengers1131]

I was waiting for this.

[@BriTheMathGuy]

Hope it was worth the wait!

[@mathevengers1131]

Yes it was.

Muy buen video Genio!

[@capjus]

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]

How u get pi from half factorial squared?

[@capjus]

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

[@muratkaradag3703]

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!

[@manucitomx]

This was well explained. Thank you.

[@BriTheMathGuy]

Glad you enjoyed it!

[@guyedwards22]

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]

Not that I understand it but sounds cool 😎

[@saimsiddiqui1544]

You are thinking out of the box❤️

[@andreaspapageorgiou2445]

Thanks!

[@nuranichandra2177]

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

[@dr.rahulgupta7573]

Brilliant presentation !! vow !!

[@BriTheMathGuy]

Thank you! Cheers!

[@user-mn3nc6xt8t]

Wow, that was pretty easy), thanks

[@EddieVBlueIsland]

Nicely done.

[@BriTheMathGuy]

Thank you! Cheers!

[@namangoyal1297]

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)

[@stevenzimmerman3945]

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

[@stinkybohoon71]

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

[@nicolasbustos6234]

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

[@damianbla4469]

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 = ?

[@adilcrafts1460]

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

[@zaidsalameh1]

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

[@Noam_.Menashe]

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

[@chessmaster2041]

It was easier, you're a great teacher

[@umami0247]

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.

[@bapanbiswas4678]

Brilliant thinking🤔

[@BriTheMathGuy]

Thanks! Have a great day!

[@gardenmenuuu]

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

[@hafizajiaziz8773]

Derivative of a fractal curve. Something like the Weierstrass function

[@jurimonikalita6068]

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

[@richard_larrain]

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

[@ianrobinson8518]

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))

[@dominicellis1867]

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?

[@eriktempelman2097]

Anorher quality video... Nice!

[@BriTheMathGuy]

Glad you think so!

[@Bernhardseckm]

Derivative of the W-Lambert funktion W(x)

[@rennoc6478]

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

[@mdrifaturrahman2403]

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

[@redasatisfaction9638]

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)

[@oklongdy8692]

thanks u brother

[@BriTheMathGuy]

You're welcome!

[@HoangTran-rg7nb]

good solution and i learned gamma function

[@KaiCrafted]

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

[@user-xy9ip4my3k]

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

[@zygoloid]

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]

pure clickbait

[@tarunpurohit6522]

Brilliant

[@alexterrieur6858]

That was cool

[@yoavshamariz7362]

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]

🤯 You're very welcome!

[@griffisme4833]

kzbin.info/www/bejne/oZuuaoqcorF4otE

[@griffisme4833]

kzbin.info/www/bejne/eZS1q32Ght-tpas

[@griffisme4833]

kzbin.info/www/bejne/n52yo5lugqdoldk

[@griffisme4833]

You didn't look very well...

[@handschich7736]

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]

There are

[@handschich7736]

@@andrewgrebenisan6141 Do they have a certain name?

[@angelmendez-rivera351]

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

[@vinayakchawla2510]

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

[@coursmaths138]

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

[@KurdaHussein]

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

[@zetacrucis681]

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)

[@joshuanugentfitnessjourney3342]

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

[@rikschaaf]

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?

[@SoufianeIdrissi123]

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

[@sans1331]

good one “Factorials _!_ “

[@romajimamulo]

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

[@angelmendez-rivera351]

There is not, not really

[@BriTheMathGuy]

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

[@piyushpathak1186]

Nice video

[@jaikrishnanp2006]

U can substitute y=x! y=x(x-1)(x-2)(x-3).... Taking natural logarithm on both sides lny=lnx(x-1)(x-2)(x-3).... ln(ab)=lna+lnb lny=lnx+ln(x-1)+ln(x-2)+ln(x-3)...... Differentiating on both sides (1/y)dy/dx=1/x +1/x-1 +1/x-2 +1/x-3....... dy/dx=y(1/x +1/x-1 +1/x-2 +1/x-3...) Substituting y we get dy/dx=x!(1/x +1/x-1 +1/x-2 +1/x-3...)

[@khanhtran-gw3pm]

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

[@lukostello]

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.

[@baldski8866]

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

[@Bodyknock]

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]

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!

[@mathsandsciencechannel]

Well done sir. Great contents.

[@BriTheMathGuy]

Thanks very much! Glad you enjoyed it.

[@thesecondderivative8967]

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?

[@p_square]

Can it be written in the form of digamma function?

[@Catman_321]

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.

[@TheDigiWorld]

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

[@casa1420]

Muito bom!

[@cheems1337]

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)

[@o5-1-formerlycalvinlucien60]

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

[@ameerunbegum7525]

This is so cool.

[@BriTheMathGuy]

Glad you think so!

[@linguafranca7834]

Easier than the expectations

[@xanderlastname3281]

Easy, x! Is also (x^1) * ! So using just your standard power rule, bring the power down, subract one, and then you get 1*(x^0)*! Therefore, d/dx(x!) = !

[@gamedever]

One can use Stirling's Approximation as well. X! ≈ [√(2πX)] • (X/e)^X The answer is way too long and complicated to write here. You might try it once. My answer is (it is gigantic and it definitely could be wrong) : (-√(2π) xˣ√x)/eˣ + (√(2π)xˣ√x(2xlnx + 2x + 1))/2xeˣ

[@johndoe7017]

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

[@charbeleid193]

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

[@chumblewumble2422]

Are you sure

[@kapsel-yg2sk]

Can we somehow return to the integer input values?

[@p_square]

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

[@BriTheMathGuy]

Glad you liked it!

[@armisol00]

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

[@philippemts88]

How did you learn all of this?

[@Cjnw]

e^x is always a sum of factorial reciprocals

[@hetsmiecht1029]

Could you differentiate the Fibonacci sequence?

[@MrCigarro50]

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]

Thanks for sharing and thanks for watching!

[@cosmicvoidtree]

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

[@mastershooter64]

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

[@herbcruz4697]

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]

@@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]

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

[@angelmendez-rivera351]

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

[@purim_sakamoto]

Γ functionはわかりやすくて楽しいですね🙂

[@ssaamil]

よくできましたね!

[@jandely8038]

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

[@gregorymorse8423]

Numeric differentiation was not considered so this is kind of half an answer if we want to consider factorials over integers and not a continuous function. Yes it uses numeric approximation methods, but it would be interesting if taking the approximation as the number of discrete terms considered goes to infinity... how far off would this method be from the continuous version? Honestly a very interesting idea IMO

[@damianbla4469]

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

[@BriTheMathGuy]

‼️

[@clementboutaric3952]

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]

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]

@@angelmendez-rivera351 well you're right.

[@karelspinka3031]

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.