Derivative of The Factorial Function

In this video, I showed how to differentiate the factorial function obtained from the shifted Gamma function, the pi function.

[@toastdog214]

I love how you always find an intersting topic and go down a deep rabbit hole of making maybe 10 videos about that topic. Truly shows your passion for mathematics and the true desire to learn more. Never stop learning

[@darickmendes969]

I honestly enjoy seeing your enthusiasm for mathematics , you have way more passion and better teacher then all the math profs I had in my university haha

[@PrimeNewtons]

Thank you

[@darrynreid4500]

It's a great choice of a problem for students to build an understanding of what's going on. I can see how you put a lot of thought into example selection, and your subsequent delivery for an audience is something to be admired.

[@roberthowes6614]

You, Sir, are the epitome of what teaching with passion is all about.

[@jethrobo3581]

Wow! You're one of the most fantastic instructors I have ever seen! Great video!

[@devcoolkol]

I was just wondering about this a few days ago, can't stop living!

[@kianushmaleki]

I like it when you smile. Love the videos ❤️

[@anthonydevellis6708]

these are the most wholesome advanced calculus videos ive ever seen in my life. i say advanced calculus only because my high school calculus teacher was a devoutly religious, elderly vietnamese woman who stood 4'11"

[@umbertocostabitencourt8417]

This is a really nice video. Your enthusiasm is so natural! Thanks for bringing this to us

[@rav3nx33]

They are some clean as hell blackboards you got there. 😜 You do good work man, love the pace and energy

[@journeymantraveller3338]

Great delivery and informative.

[@jeanagulay3479]

Sir your videos helps me a lot.. From Iloilo Philippines ❤❤❤

[@josephwellinghoff1259]

Very clearly explained...thanks

[@WhiteGandalfs]

Well, it's useful to have a sufficiently appropriate "coarse feeling" of the value. The integral at the end is not straightaway self-explanatory, so lets make sense of it! Maybe for the "coarse feeling" of the derivative, we don't need to take the exact value of the gamma function. By taking the difference over a full 1 in x, then taking the "appropriate" average... difference one up: (x+1)! - x! == x!*(x+1) - x! == x! * ((x+1)-1) == x! * x difference one down: x! - (x-1)! = (x-1)! * (x-1) Since the series is growing by multiplication (by a rather constant factor, since the difference between x and x+1 for the growths is the smaller the bigger x becomes), it is appropriate to take the geometric average from the difference up und down to get a pretty good fitting approximation of the value for the difference at spot x: average (one up, one down) = sqrt( x! * x * (x-1)! * (x-1) ) == sqrt( x!^2 * (x-1) ) == x! * sqrt(x-1) The "-1" in the sqrt we can qietly ignore since the whole thing goes about a "coarse feeling" anyways, thus we land at: derivative (x!) ≈ x! * sqrt(x) That's a very easy to remember (but very coarse) approximation for practical usage. Check with Wolfram Alpha yields that this is actually better approximated by: derivative ((x-1)!) ≈ x! / (sqrt(x) * ln(sqrt(x))) The "-1" on the LHS because the Gamma function is one of against the factorial function. To rectify that for easier use: derivative (x!) ≈ x! * sqrt(x) / ln(sqrt(x)) That is sufficiently easy to remember and to calculate and in the range of a few percentage off the exact value. And it gives a good "feeling" for the look of that derivative function.

[@xenmaifirebringer552]

Thanks for the extra insight and explanation! I think for a coarse approximation you could also differentiate Stirling's factorial formula. I'm curious if that'd look anything similar to the approximation you explained.

[@AaryanK-wp6vi]

I think you are very very ... passionate about mathematics. The 10s of videos you make about the same topic in different ways show this. And I like your way of explanation that is different from other YT people. I hope you do more videos like this

[@Supercatzs]

Great videos! Love the scripture at the end.

[@kragiharp]

Thank you, Sir! ❤️🙏

[@douglasstrother6584]

You, Michael Penn & Papa Flammy all make me miss *real* chalkboards.

[@KaushikAdhikari]

6:42 John 1:4? Amen Thanks for the tutorial ❤

Good job. You can actually represent the derivative of the gamma function using the definition of the digamma function and its series representation. Keep up the good work!

[@andy_lamax]

you are so close to discovering the di-gamma function

[@DEYGAMEDU]

I was waiting for this

[@ttaylor3rd]

nicely done!

[@garrytalaroc]

Cool but how are you gonna evaluate more complicated integral

[@maths-pro-by-khan-sir]

YOU ARE AMAZING

[@ricardopaula4082]

calculus is my favorite maths topic, its so fun

[@pk2712]

There is another maybe shorter way to show that the partial derivative with respect to x of t^x is ln(x)t^x . We know that t is considered as a constant . The derivative with respect to x of y=e^(ax) is ae^(ax) . Start with t = e^(lnt) ( where t and also lnt are constants ) and substitute this into t^x = (e^(lnt))^x = e^[(lnt)x}] . Now the derivative with respect to x of this last expression is lnxe^[(lnt)x} . But , in this last equation we know that e^[(lnt)x} = t^x ; therefore , the partial derivative with respect to x of t^x is (lnt)t^x .

[@iithomepatnamanojsir]

Very nice lecture

[@rajesh29rangan]

Thank you.

[@cbbohn8107]

He is awesome

[@Ghostwriter_zone]

Now it's time for integral x factorial

[@ingiford175]

Saw an interesting definition of the gamma function: lim (n goes to infinity) n! * u^n / Product (other Pi function) ( v as v goes from 0 to n) of (u+v) u > 0 In an old 1960's Finite Differences textbook.

[@Harrykesh630]

I would like to enroll in your class this year!!

[@polzinger]

Very nice writing.

[@PrimeNewtons]

Thanks a lot 😊

[@lornacy]

All I could think of is that the derivative would be huge, quickly. Factorials grow fast 😅 I am going to have to rewatch this to really get my head around it.

[@mickodillon1480]

Interesting one there. Good video.

[@alejandropulidorodriguez9723]

splendid

[@MrMusicM67]

Love the shirt! Where did you get it?

[@Misteribel]

You can simplify using the digamma function, though (if you can really call that a simplification).

[@rknowling]

Thankyou for a fun and useful result! 😄 In the early pages of Bleistein & Handelsman "Asymptotic Expansions of Integrals", they talk about: \limits_{N \to \infty } \left[ {{{\left( { - 1} ight)}^N}N!x{e^x}\int\limits_x^\infty {\frac{{{e^{ - t}}}}{{{t^{N + 1}}}}dt} } I have been wrestling with this for some time; thanks to your videos combining the Leibnitz rule, l'Hopital, second FTC etc with limits, I am (slowly! haha) gaining some traction. Much appreciated!

[@beapaul4453]

Can you upload videos about complex geometrical problems(drawing graphs), like polygons? That would be great to see.

[@PrimeNewtons]

Sounds like something I don't know yet

[@paraskumar9850]

@@PrimeNewtons never stop learning, those who stop learning ! stops living

[@lornacy]

​@@paraskumar9850 He never said he wasn't willing to figure it out ... Looks to me like a way for him to sustain life!

[@miguelmarcoscatalina3872]

Hace mucho que no practico matemáticas, pero me parece, solo me parece, que hay un grave error en cambiar una función que solo es continua en puntos concretos y aislados en una función continua en todo el intervalo. Lo considero un error, aunque puedo estar equivocado

[@kotylka90]

Mister I think leibniz rule hold for proper integrals. How would you justify using it for the improper integral here?

[@kianushmaleki]

❤️❤️

[@lemon.linguist]

i love your videos! i have a question that's unrelated to the video but still mathematical i can put it in the replies of this question if you'd like

[@PrimeNewtons]

An email with be better. [e-mail]

[@herlandarmantotampubolon8135]

Sir, it seems to me that you could use Lambert Function to continue the last result.

[@makramaarid6598]

This is the gamma function

[@user-ky5dy5hl4d]

I did not understand much of it without delving into it more. But the beginning is interesting by making the x factorial as pi of x. I think you can do that with any irrational number, so why not chose square root of 2? Or another irrational number.

[@mrngochoi89]

But i dont know the define of x! if x in R

[@szymonharpula1217]

Wouldnt it be easier to use stirlings aproximation

[@jacobgoldman5780]

the bounds are in terms of t or x?

[@anuragguptamr.i.i.t.2329]

t

[@raghuvanshiedit]

Hey sir, a doubt is can't we write ln(t) t^x as ln(t)^(t^x) which would give x?

[@sammtanX]

sir, for the power of t, shouldn't it be x-1? Because the y = x!, not y = (x-1)! hence it should be gamma of x, so t's power has to be (x-1)

[@PrimeNewtons]

I used the π function

[@salahouldaya4958]

why don t you ask if this fuction is derivable before anything

[@MathSync]

i ❤ Mathematics

[@awrRoman25]

You could just differentiate Stirling formula.

[@nyksik001]

Is this channel for postgraduates?

[@Ahmad-yi6d]

Oops derivative of a factorial function 🥶

[@MATHS_FOR_FUN]

Dy/Dx = X! [ Sum from {i = 0 to x-1} (1/(X-i))] Isn't it ?

[@hydraim9833]

Hi! I am curious, why is there no way? At the end of the video you had the intention to replace t^x e^-t with x! ? You didnt do it because it would be abusive notation or im missing the smth?

[@awrRoman25]

You can not replace t**x*exp(-t) with x! because integral(t**x*exp(-t)) from 0 to inf equals x!, not function inside.

[@petr_duduck]

Учитывая, что Гамма функция- это интеграл, найти от неё производную не так уж сложно

[@user-ld7cm5jj6h]

Please try to solve this equation (X+1/x)^x=2

[@user-ld7cm5jj6h]

Please

[@user-ld7cm5jj6h]

Please

[@bridgeon7502]

x = 1 (I just guessed)

[@allozovsky]

x = 1 is a trivial solution

[@IoT_]

Since the function on the left always increasing , there can be maximum one solution. One may guess that it's x=1 , but I am afraid , you have to use numerical ways to solve it, like Newton's method.

[@allozovsky]

I guess the next derivative would square the logarithm.

[@eliaskhanmeh7399]

Not zero x/ y if y=0 that mean not knowing

[@danielaromero8474]

Γ(x) = (x-1)! → x! = Γ(x+1) (Gamma function) Γ'(x) = Γ(x)(ψ(x)) → Γ'(x+1) = Γ(x+1)(ψ(x+1)) d(x!)/dx = x!(ψ(x+1)) → ψ(x+1) = ψ(x) + 1/x (Digamma function) ψ(x) = Hₓ₋₁ - γ (Harmonic number & Euler's constant) d(x!)/dx = x!(Hₓ₋₁ - γ + 1/x)

[@himadrikhanra7463]

( x-1)!...?

[@harris5140]

Factorial is part of N, not R.

[@allozovsky]

Abuse of notation is pretty common in math (as long as it is clear from the context what a given notation mean). After all, there are not so many math symbols to denote the variety of similar concepts.

[@fabiopilnik827]

Well in that case the derivative of x! is (x+1)! - x! = x!(x+1 - 1) = x!x. But technically that’s a difference not a derivative.

[@anigami01]

anyone from India ( JEE aspirant) here

[@cparks1000000]

Taking the derivative under an integral requires some justification.

[@hussainrashed4453]

The FTOC.

[@eliaskhanmeh7399]

X Munier multiply by zero the result zero

[@eliaskhanmeh7399]

Thé chaîne de zéro is unknowing

[@Berin.Jervin]

X! is not continuous, so has no derivative.

[@thedudethatneveruploads2617]

Correct; however, he differentiated the Pi function, which is a popular extension of the factorial function to all reals except negative integers, essentially making a continuous factorial function

[@scheillaraffaelli]

A non differentiable function may still be partially differentiable. He's correct here.

[@porcospino289]

Ugh.

[@salahouldaya4958]

This fuction is not continu how could it be derivable ???

[@v8torque932]

Derivatives an anti derivative

[@Leoscacchi08]

Hey bro here y=x ok? Emh no sorry i didn't understand I said that y=x Emh i'm sorry, can you repeat, i can't hear you Oh are you deaf? I said y=x!

[@tonyscott1658]

You can go further. That derivative you speak can be obtained in terms of what is called the digamma function (Psi) . en.wikipedia.org/wiki/Digamma_function i.e. Int(t^x*ln(t)*exp(-t), t = 0 .. infinity) = Psi(x+1)*GAMMA(x+1)

[@RottenWoodInPower]

Me totally forgot gamma function

[@kk12181]

May I know the name of this educator! Respect 🫡