we can differentiate anything!

[@retired5548]

I anticipate that suggestions of nondifferentiable functions will be quite popular. Perhaps exploring the properties of such a function, such as Weierstrass' function or Bolzano's function, and showing where the reasoning for differentiability breaks down in such cases might help increase the understanding of why continuity does not necessarily imply differentiability.

[@SaurabhKumar-jo6dp]

Right my opinion was also same🤞

[@volodymyrgandzhuk361]

When you say that Weierstrass's function has no derivative, that's an answer to the question "What is the derivative of Weierstrass's function?" On the other hand, it's impossible, or at least too hard, to compute the integrals of certain functions.

[@PinusTF2Spy]

True. I want bprp to show why the subdifferential (generalized derivative) of the norm function doesn't exist at 0 and also differentiate a "quadratic" function, except a quadratic that doesn't look like x^2, but rather x^T*A*x.

[@josefwakeling7103]

*laughs in the weierstrass function*

[@pardeepgarg2640]

NO!!!!

[@soumyojitpal3399]

i am new to calculus, can you explain what is the weierstrass function ?

[@jaxoncr]

@@soumyojitpal3399 a function that is continuous everywhere, but differentiable no where. If a function is differentiable, it implies it is also continuous. However, the converse is not true. A classic example is the function abs(x) about x = 0, the function is continuous but not differentiable at that specific point. This is similar to the weierstrass function except we cannot differentiate it at any point.

[@Taigokumaru]

@@jaxoncr Omg. I had to Google it to confirm the graph. The Weierstrass function is basically the aura around the characters powering up in the Dragonball Z anime!

[@liamwelsh5565]

6:45 I do this so much. I spend like a minute or two trying to figure out what's the best way to write the final answer even though it doesn't matter at all. lol

[@karannair3035]

You should do a legitimate series on derivatives of big expressions like this. Really cool to watch and solve along.

[@bprpcalculusbasics]

I plan to!

[@coderanger7708]

Differentiate Weierstrass function next. By differentiation just find the slope at each interval and show us.

[@dafureveerbhadra2772]

Not st each just find slope at any point on the xy plane

[@zhiyuanliu9533]

Can you please find the second derivative of that equation?

[@dinosaric4862]

You wish..

[@adrishbora]

This series is going to be absolutely fun!!! Love your content man!!!

[@mathboy8188]

One gotcha with implicit differentiation: you've got stop and ask if there's even a function there at all. For instance here, when x = 0, ANY y such that sin(y) > 0 is a solution, so you're either locally looking at a vertical line (so no derivative), and/or, as I'd guess, a lot of "branches" converging onto intervals of the y-axis. Even if the formula spits out a value for dy/dx at some point where x=0, you don't have "the derivative there", but rather only the beginning of an interpretation of what's going on with that relationship there. I checked graphically (wolfram alpha), and there's a unique x value corresponding to y = pi/6 (x ~ 0.2), so you likely have a clean local branch of a function near there, but in general, implicitly defined "functions" can be a real mess. (There's an advanced calculus theorem, The Inverse Function Theorem, which will give you some local guarantees, but of course that demands carefully checking its assumptions hold.) Easy Example of the kinds of situations to look out for: Use implicit differentiation on x^2 + y^2 = 0. You get 2x + 2y y' = 0, so y' = - x / y. Thus you get a formula for the derivative everywhere where y isn't 0. Problem is, y = 0 is the only y value for that relationship. The implicit function there is the most trivial one imaginable: { 0 } --> { 0 }, having domain and range both a single element. No function with a discrete domain is going to have any derivatives anywhere... the concept doesn't even make sense. So even though you get a nice simple relationship, x^2 + y^2 = 0, and a nice clean formula, y' = - x / y, that derivative formula means absolutely nothing. Here's an even more dramatic example: x^2 + y^2 = -1. Again, if you do implicit differentiation, you'll get y' = - x / y... but there's no points satisfying that example, so certainly no local branch function there, so certainly no derivative.

[@YoavZilka]

You could differentiate the Lambert W function. It really suits your channel, and it’s quite nice using implicit differentiation.

[@YoavZilka]

Just noticed you already did that!

[@bprpcalculusbasics]

Yea. But I can make a remastered version or a harder one.

[@YoavZilka]

@@bprpcalculusbasics sounds good

[@YoavZilka]

@ニコラ-NR you can, but it’s kinda complicated. You need to take a u sub where u=W(x), and then find dx in terms of du by differentiating u (which is just W(x)). From there it’s pretty straight forward using integration by parts

[@cH3rtzb3rg]

@ニコラ-NR You can integrate any continuous function. It may not be possible to express the integral with elementary functions, though.

[@fanamatakecick97]

The symmetry with this derivative is angelic, i’m not gonna lie. Math is so satisfying

[@michaeledwardharris]

That was a pretty wild ride! Very interesting problem, and a neat solution.

[@kepler4192]

You can also memorize the formula for differentiating g(x)^h(x): f prime= (g(x)^h(x))[(h*dg/dx)/g + dh/dx*ln(g)] God it’s so hard writing math in comments

[@JansthcirlU]

Could've used apostrophes for less clutter: (g^h)' = (g^h) * [h * g' / g + h' * ln(g)]

[@kepler4192]

@@JansthcirlU ah, thx

[@SeeTv.]

Math majors are triggered. Not all functions are differentiable everywhere :D

[@cH3rtzb3rg]

This totally triggered me as well! In fact, barely any functions are even continuous everywhere. And every function which is differentiable is also integrateble (because every continuous function is).

[@cobovega27]

Is the factorial function differentiable? How would one tackle that problem? Do we use the gamma or pi function?

[@soupisfornoobs4081]

It's not differentiable in the standard way because it is a discreet function, but you can use the gamma function and the result is pretty cool

[@cobovega27]

Sounds fun

[@YossiSirote]

I think it is easier to use the y’ notation rather than dy/dx in this situation.

[@Firefly256]

Why does bprp say chain rule as chandu?

[@conanedojawa4538]

I ask you to solve the resulting exact differential equation from this implicit differentiation

[@nalingoel6971]

Try differentiating x^(tan(y)) = y^(tan(x))

[@acuriousmind6217]

5:21 this and this and this and that sounds understandable

[@raifegeozay687]

find the second devirate next: y=((x^x^x+tan(tan(tan(x!))) -cos(sin(x))+arctan(arctan(x)))!)^((x^x^x+tan(tan(tan(x!))) -cos(sin(x))+arctan(arctan(x)))!) (yes there are some factorials) (use gamma function to differentiate factorial)

[@yk4r599]

Wow

[@nekothecat]

Can you try differentiate x^y^x^y ?

[@bprpcalculusbasics]

We need an equation. So what would you like to be =?

[@nirmankhan2134]

@@bprpcalculusbasics equal to sinx for example

[@BiscuitZombies]

@@bprpcalculusbasics set it equal to an arbitrary constant ="🙂"

[@ДенисКосько-н9и]

@@bprpcalculusbasics "the fish"

[@henkhu100]

The principle is only allowed if the relation defines y as a differentiable function of x, so there are certain conditions for a relation before we can perform implicit differentiation. For instance (in real number system): sqrt(-(x-3)^2) + y^(lnx) + y=2 is a relation that has a meaning for only x=3 So if we see y as a function of x the domain is just {3} and there is no derivative of y. And that means that the dy/dx you find using the method in the video does not have any meaning. What I miss in the video is: does the relation fit the conditions for implicit differentiation? So what about the title of the video? Anything? How do you differentiate the function y=sqrt(-(x-3)^2).

[@AriosJentu]

How about differentiate matrix function? Or differentiate vector function wrt verctor?

[@teelo12000]

That sounds like a challenge. Differentiate: d/dx(my completely made up function where I'm not going to tell how you it works of (x))

[@m3nny125]

Case one: D/DX of D/DX = D^2/DX^2 Case two: Cancel d and d out D/DX of X = 1

[@quadruplay9788]

could you do some episodes on split-complex and dual numbers?

[@gamingmusicandjokesandabit1240]

Haters will say that means you can't differentiate what you can differentiate vs what you can't differentiate.

[@Bilbobaggiins.0]

I am come on this channel not immediately but definately

[@sriprasadjoshi3036]

Him: Chain Rule Subtitle: Chandu

[@genius5625]

Me as calc3 student : why i am doing this ? 😂😅😂

[@Ashirene22]

u gonna do calc 3 content?

[@dinosaric4862]

The graph of that function is just a bunch of squared with rounded corners haha

[@zhiyuanliu9533]

Differentiate x!*(x^2)

[@Ou_dembele]

Do some radical function derivatives

[@yoav613]

This is in class,in the test - find y'' 🤣

[@samwitkowski3459]

now do the second derivative.

[@deep24543542]

d/dx sin(x)^cos(x)^tan(x)

[@rafaelgcpp]

Differentiate I_Q(x) R->R, where I_Q(x) is 1 if x is rational and 0 otherwise...

[@ДенисКосько-н9и]

Johann Peter Gustav Lejeune Dirichlet likes your comment

[@mathboy8188]

Differentiate f(x) = x^2 if x is rational, and 0 if x is irrational. Differentiate f(x) = 1 + sin(x) if x is rational, and 0 if x is irrational.

[@SlemTurnThePowerOn]

Second derivative of this one?

[@shadowgamerA]

Felt amazing on solving it within 3 minutes....🙂😃

[@pierrotinturquoise]

What about a video about Weierstrass Function next? :> I couldn't find a single good video on this topic in youtube. Please make one on this. :"(

[@henningnagel1977]

Entered the original function into Wolfram Alpha. Well...

[@bprpcalculusbasics]

😆

[@SaurabhKumar-jo6dp]

🤭

[@holyshit922]

We can differentiate anything! what about functions like Weierstrass

[@twelfthdoc]

So this is where we paramaterize x and y in terms of t and figure out what dy/dx is terms of t?... Maybe another day...

[@plutothetutor1660]

differentiate the infinite tetration of x

[@NC28-En]

Yay new series :)

[@GreenMeansGOF]

Differentiate the gamma function

[@AndDiracisHisProphet]

Dirichlet Function

[@yoav613]

Next time i suggest you to find the derivative of (tanx)^(e^x)

[@ДенисКосько-н9и]

and yes, i do like it!

[@AnakinSkywalker-zq6lm]

It would have taken me forever and idk why… but he did it in under ten min so it must be true that studying solutions makes you good at math…

[@ДенисКосько-н9и]

weierstrass function? ok. and dirichlet function next

[@jamessprenkle6355]

More !!!

[@dinosaric4862]

Now integrate the result

[@yk4r599]

Nice, now find y 🤣🤣

[@joakimharbak7485]

Try x! maybe?

[@distinctperception2776]

Differentiate x! :)

[@frentz7]

You're a very cool guy. :)

[@MohitKumar-eu4pz]

I. Solved this very quickly

[@hoen3561]

differentiate x^y^x^y..... = πy

[@sharele98]

solve for y

[@agabe_8989]

now take the second derivative >:)

[@adamforte9530]

That was great

[@mathevengers1131]

Do half derivative of it.

[@quadruplay9788]

now solve for y

[@sanjayavamachaari623]

differentiate y=icosxsiny/(2sinxtanx+1),w.r.t. x

[@shaftous]

bring back the longer hair!!

[@IoT_]

No, check by integration. 😅

[@SaurabhKumar-jo6dp]

🤞 yeah

[@novidsonmychanneljustcomme5753]

...and now solve the differential equation you just created. 🤓😁

[@Thealiszt]

Now integrate it to get rid of the dy/dx 🤡

[@Theo-ki3lu]

thanks for saying “he or she” instead of the “they” bullshit

[@GaryFerrao]

you look different. what could it be hmm lol

[@nirmankhan2134]

🤑🤑🤑