The Weak Derivative

Have you ever wondered how to differentiate a function that is not differentiable? In this video, I will show you how! It all relies on a simple integration by parts and a tiny bit of linear algebra. I will start by calculating the derivative of x^2, then I will calculate the derivative of |x|, and, more surprisingly, the second derivative of |x|. Be prepared to be amazed! :D

[Rundas]

When you use integrals to calculate derivatives you have achieved true transcendence.

[taubone]

Any day that Dr. Peyam uploads a video is a great day.

[cipriannegoescu]

one of the coolest dudes on youtube! i've become addicted to your videos

[ozzyfromspace]

When you said Heaviside function in example 2, I think you meant sgn() function. A Heaviside function is basically a unit step. That said — phenomenal lecture! Just like this, now I understand the big idea of generalized derivatives of distributions. Thanks, Dr. Peyam! 🙏🏽😊🎊

[BBB]

I remember you as a graduate student at Berkeley, and I was also jealous when people would walk out of your section with cake before my class.

[Stelios.Posantzis]

Woah... does this bring back a ton of memories... Having to deal with the derivative of |x| (the absolute value) tormented me daily... Generalized derivatives.. a great topic (although I didn't end up using them). Heaviside functions is one of my favourite math inventions - I just find this theory very elegant.
A great little video but I really can't read anything off the blackboard.

[AhnafAbdullah]

Please differentiate something like the Weierstrass function: continuous but non-differentiable everywhere :D

[PrincessEev]

*sees video title*
*mind explodes*
*clicks immediately*

[MrRyanroberson]

"how to get an integer quotient in n/m for integers 0<n<m"

[mattetor]

Thank you for keeping inspiring me! Do you have any recomended book for learning distribution theory on a graduate level? I have some books, but im always looking for more ways to learn this :)

[tonypalmeri]

I'm not sure I completely understood the technique. I'll watch it again later when I have time, but I'd like to see the technique applied to some more interesting non-differentiable (or even, merely "hard-to-differentiate") function. Nothing too wild and crazy, but something more interesting. Also, it would be interesting if there was some discussion of graphical interpretations of this technique (if there exists some sensible way to visualize it).

[MrQwefty]

he differentiatt
he integratt
but most importantly
he infinitt

[OonHan]

you looked like you dressed up for a lecture
thanks, Dr. Peyam. great lecture

[michaelgolub]

If one studies generalized functions and integral transforms (Laplace's or Fourier's) and convolutions, then the similar thing is done with Heaviside and Dirac functions. Does the `distribution theory' claim that we can choose any compact support infinitely differentiable kernel with zero boundaries we can have the similar effect, moreover, it has relation to intergral transforms?

[ItsLukkio]

Dear Dr. Peyman, you're the only one on youtube I've seen giving "tutorials" on functional analysis. Is there any chance you would consider something related to Banach Algebra and Spectral analysis for bounded/unbounded operators?

[MichaelMiller-rgor]

This one was a but beyond my level of mathematical education

[leonardromano]

7:30 The GEN DER of f, I see what you did there :)

[GreenMeansGOF]

Dr. Peyam, could you do a video about the derivative of x! ? I have been curious about this. I’m sure that using the gamma function would help but I don’t know how to do it.

[mudkip_btw]

Do I see a Dirac delta distribution coming? Great video!

[thomasjefferson]

So, im pretty sure im off into the woods when peyam is the only video I can find on this.

im gonna try to understand this first without the video. As a hobbiest I need to learn how to not be reliant on others and use my head. But for sure you gave me the first idea of how to do this by the description.

Now my question is what test function should i use? Do test functions like from the Finite Element Method do well in these kinds of situations? or do I have to create unique test functions?