The Lambert W Function's Derivative
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- เผยแพร่เมื่อ 4 ต.ค. 2024
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Let us calculate this bad boi's derivative using implicit differentiation. Also providing you with a graphical representation to show you, in which situation said derivative does not exist! =D Have fun c:
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![Lambert W Taylor Series Expansion [ Lagrange Inversion Theorem ]](http://i.ytimg.com/vi/CX-5orQxNAg/mqdefault.jpg)
![Lambert W Taylor Series Expansion [ Lagrange Inversion Theorem ]](/img/tr.png)
If you have 1+W(z)=0 then W(z)=-1. Then z=(-1)e^(-1) because of the definition of W(z). Way easier!
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you are the papa now
0:00 German general realizes the drawn path leads to an allied ambush 1945 colourized
Andrew W. Dotson. The W stands for Lambert
There is a small error in the video. The derivative is defined at 0 because the substitution e^W(z)=z/W(z) does not apply for z=0 because you are dividing both sides of W(z)e^W(z)=z by zero (because W(0)=0). The expression 1/[e^W(0)*(1+W(0))] is defined and equal to 1.
You can also see from the graph that it really does have a derivative at 0.
"Right boii..."
That escalated quickly
The singularity at z = 0 is removable; lim W'(z) -> 1 as z -> 0 (which shows in the graph).
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0:25 *sighs* The Allies have won :'(
The first step is recognizing you have an addiction: hi, i'm papa flammy, and i'm addicted to greek letters
He's also addicted to blackboards😂 he owns 5 if i'm not mistaken
i have started a new approach to math and science to help me stay motivated, i look at it as if its some sort of magic and that makes it a lot more interesting for me! :3
I actually managed to solve it on my own!
I'm proud of myself
Papa is back with a bang
He brought papa Lambert with him
5:48 : I don't understand why we can say it's the representation of a function : it can have multiple solutions to a single number plugged into it, so it doesn't respect the definition of a function
Flammable Maths
Oh okay so we consider each interval separately ? :)
Flammable Maths
Okay I understand now :) Thank you !
Welcome back to naa video
2:15 chill man!
2:20 OMFG I LOST IT Lmfao
I do not think that the point z=0 is a point of singularity. Note that at z=0, W(z)=0.
Hence, the term W(z)/z --> 0/0 and the derivative is not actually going towards infinity (as clear from the graph of W(z)).
In fact W'(z) @ z=1 is 1.
Yo, i know i'm late but we could also do it like this, arguably easier:
1. y = xe^x => x = ln(y/x) = W(y)
2. dW(y)/dy = 1/dy/dx = 1/(xe^x + e^x) = 1/e^x(x+1) = 1/e^W(y)(W(y)+1) = 1/(y/W(y))(W(y)+1) = W(y)/y(W(y)+1)
variable is a dummy one so it doesn't really matter
lmao the "just channel has evolved into an English channel" bit killed me
2:54 the "Chen Lu" blackpenredpen is shaking 😔😔
Loved it boi!
Why bother switching e^w(z) with z/w(z)? It seems that if you keep it in the original form, then the derivative is only undefined for z = -1/e
I mean, to be fair, English does produce the vast majority of good memes, papa.
And this is half way to a shit posting channel at this point, so your English preference makes a lot of sense.
Man, you just sparked my curiosity!
Well then, can you find the limit as n approaches zero of [W(n)/(n*(W(n)+1))] ?
(Knowing the derivative might be useful now!)
Time to watch this from many different ip addresses to give papa them views
Helo!,you must know that araund the world some peopple ;like me,dont underestand in english;because of this is importan to speak as easy as you can;and write more big on the blackboard ;let me say you that im from ARGENTINA,and im 65 years old;so i say THANKS YOU VERY MUCH;I see you must be very clever because all of these maths are not easy,and you handle it so easily way.
Regards from Argentia.
I missed your beautiful voice, papa
Most powerful explosive in the universe: 1/0
It can expand universe to infinity.
For the principle branch, it looks like the function should have a positive slope at (0,0). But you said that z cannot be zero. What’s wrong with this?
Can you solve this for me the integral of sqrt of sin (x) from 0 to pi
Papa flammy! Can you solve the spicy integral of e^-x^2 from -infinity to infinity ?
Whoops sorry
The song that comes at the start of the video sounds good. From where you got it?
Interesting! You had a episode on the "right" pronunciation of German mathematician ... but Lambert is maybe not German enough ... or you changed your mind !
I understand algebraically why the derivative doesn’t exist at z=0, and looking at the lower branch of the graph of W(z) I can see that the slope approaches -inf as z approaches zero. But the upper branch has no discontinuity at zero whatsoever. If I just wanted to consider the upper branch of W(z) as its own function, would the singularity of the derivative at z=0 disappear?
Hans, bring mir den Flammenwerfer, wir müssen die Lambert W-funktion differenzieren!
Isn't W(z) differentiable at z=0, at least for the main branch?
I guess that the substitution e^W(z)=z/W(z) implicitly added it in the last step.
The graph shows that is valid for the second branch, that is not defined for z=0
Why not use the inverse function theorem? d(f⁻¹)/dx = (df/dx)⁻¹ = 1/(df/dx)
This boi is on fire
You are awesome 🙌🏻
Hmmm.. alright. Let w(x)=y, therefore ye^y=x, 1/(y+1)e^y = w'(x). Didn't I also comment this on another video with integral(w)? Lol. I guess we get 1/(x+x/w(x)) = (w/(w+1))/x = (1-1/(w+1))/x, to minimize how many w functions while simplifying.
the Peyam shoutout
Can you do some video on Riemannian geometry or measure theory or Hodge theory or E8 geometry? Thanks
thats a good video
Can make video on the proof Rolle theorem
Derivative at 0 is 1! Although W(z)/(z(1 + W(z)) is undefined at z = 0 the limit is perfectly well defined, and derivatives are defined as limits.
We can show that the limit is 1 like this:
Let x = u*e^u
W'(x) = lim_(x -> 0) W(x)/x = lim_(u -> 0) W(u*e^u)/(u*e^u)
= lim_(u -> 0) u/(u*e^u) = lim_(u -> 0) 1/(e^u) = 1
It's clear from the graph that z can be zero though. W(0) = 0, so you can just L'Hopital that shit.
Are you in a hurry... did you need to catch the train 🙃
But you did derivative of Lambert - W function previously ?
Something feels weird... at x=0 that graph seems to have a tangent line that could fit. Am I missing something?
3:04 LMAO
Very nice! But what about W'(0)? The graph doesn't look like it doesn't exist.
Wild guess using grandpa l'Hôpital's rule: lim z->0 W(z)/z(1+W(z)) = lim z->0 W'(z) / W'(z)(1+W(z)) = 1 as W(0)=0.
Never mind I made an error in the denominator's derivative, so the W'(z) doesnt cancel...
W(0)=0 so the derivative becomes a 0/0 situation. Pretty weird! The derivative isn't defined but the coordinate is. Only the limit as z goes to 0 of W(z) is defined.
We have to use our 1/e^W(z)(1+W(z)) which will become 1 when plugging in z=0. So W'(0)=1
Ah wait! So the W(z)/(z(1+W(z)) is a wrong simplification, it doesn't have the right domain
How's the music that plays in your intros called, papa flammy?
0:06 no, Brooklyn 9 9 ... Or Hawaii 5 0
Wait, if we're looking at the principal branch, then it seems like the derivative should be well-behaved at 0 (and I think just equal to 1). So what's going on?
Yes, it does, you are correct.
He lost the derivative at 0 by dividing by z in order to make the formula look better. Actually just plug in 0 in the first formula (with exponentials) and you will get 1.
Hey Papaflammy, I have not been able to find a definition of the principle branch of a function. I feel like there is no definition, do you know of one?
Ist Analysis in Deutschland das gleiche wie Real Analysis in Amerika?
How did you get that smart in math?
Show us the Taylor boi
@@PapaFlammy69 Oooof missed that one.
The drawing of Lamber W function indicates that first of all it is not a function :)
Papa
2:17 lmao
Wait a minute... Is lambert w function actually a function? If yes, it should be defined just for z>0
Lambert W function is a complex-valued function, therefore it's not "only defined for z >0"
@@PackSciences oh! Thanks
LamBOIIIIrt W function
papa "flambert"....😉😉😉
Nah, you can't just say Nazi, you have to emphasize an Amercan accent and say "Natsee"
i get W(x)/(xW(x) + x) using implicit diff
I ate beans to this
You could just have shown how to differentiate every inverse function. The proof is pretttttyy close to your approach.
chen lu is best lu
Lambert W 😖😥
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Fuck! Neeeeiinn! There are the German roots ;)
Please don't do those laud sounds amidst your videos. They really hurt one's ears over the headphones.
100th comment :) yay
Next, prove whether NEIN NEIN is rational or irrational.
fak 9 9
Chen Lu !!! Yeaah papa !!! The best rule...
Is there anyway I can contact you papa if I have any questions ?
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