we can differentiate anything!
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- Опубликовано: 30 сен 2024
- We are going to differentiate anything that we want! For this video, we will use implicit differentiation to find dy/dx for x^sin(y)+y^cos(x)=1. This is a great practice problem for calculus 1 students! What's next?
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*laughs in the weierstrass function*
NO!!!!
i am new to calculus, can you explain what is the weierstrass function ?
@@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.
@@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!
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.
Right my opinion was also same🤞
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.
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.
Differentiate Weierstrass function next. By differentiation just find the slope at each interval and show us.
Not st each just find slope at any point on the xy plane
Math majors are triggered. Not all functions are differentiable everywhere :D
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).
You should do a legitimate series on derivatives of big expressions like this. Really cool to watch and solve along.
I plan to!
Can you please find the second derivative of that equation?
You wish..
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
Why does bprp say chain rule as chandu?
You could differentiate the Lambert W function. It really suits your channel, and it’s quite nice using implicit differentiation.
Just noticed you already did that!
Yea. But I can make a remastered version or a harder one.
@@bprpcalculusbasics sounds good
@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
@ニコラ-NR You can integrate any continuous function. It may not be possible to express the integral with elementary functions, though.
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.
We can differentiate anything! what about functions like Weierstrass
That was a pretty wild ride! Very interesting problem, and a neat solution.
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)
Wow
This series is going to be absolutely fun!!! Love your content man!!!
I ask you to solve the resulting exact differential equation from this implicit differentiation
Me as calc3 student : why i am doing this ? 😂😅😂
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
Could've used apostrophes for less clutter: (g^h)' = (g^h) * [h * g' / g + h' * ln(g)]
@@JansthcirlU ah, thx
Try differentiating x^(tan(y)) = y^(tan(x))
Can you try differentiate x^y^x^y ?
We need an equation. So what would you like to be =?
@@bprpcalculusbasics equal to sinx for example
@@bprpcalculusbasics set it equal to an arbitrary constant ="🙂"
@@bprpcalculusbasics
"the fish"
Entered the original function into Wolfram Alpha. Well...
😆
🤭
I think it is easier to use the y’ notation rather than dy/dx in this situation.
now do the second derivative.
Differentiate x!*(x^2)
The symmetry with this derivative is angelic, i’m not gonna lie. Math is so satisfying
...and now solve the differential equation you just created. 🤓😁
I. Solved this very quickly
Haters will say that means you can't differentiate what you can differentiate vs what you can't differentiate.
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...
The graph of that function is just a bunch of squared with rounded corners haha
weierstrass function? ok.
and dirichlet function next
Is the factorial function differentiable? How would one tackle that problem? Do we use the gamma or pi function?
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
Sounds fun
Do some radical function derivatives
you look different. what could it be hmm lol
Him: Chain Rule
Subtitle: Chandu
Differentiate the gamma function
5:21 this and this and this and that sounds understandable
differentiate the infinite tetration of x
differentiate x^y^x^y..... = πy
Second derivative of this one?
bring back the longer hair!!
This is in class,in the test - find y'' 🤣
Now integrate the result
Now integrate it to get rid of the dy/dx 🤡
Felt amazing on solving it within 3 minutes....🙂😃
You're a very cool guy. :)
Dirichlet Function
Nice, now find y 🤣🤣
Differentiate x! :)
u gonna do calc 3 content?
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))
Case one:
D/DX of D/DX = D^2/DX^2
Case two:
Cancel d and d out
D/DX of X = 1
and yes, i do like it!
now solve for y
Try x! maybe?
That was great
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. :"(
solve for y
d/dx sin(x)^cos(x)^tan(x)
More !!!
How about differentiate matrix function? Or differentiate vector function wrt verctor?
differentiate y=icosxsiny/(2sinxtanx+1),w.r.t. x
Next time i suggest you to find the derivative of (tanx)^(e^x)
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…
could you do some episodes on split-complex and dual numbers?
I am come on this channel not immediately but definately
No, check by integration. 😅
Do half derivative of it.
🤑🤑🤑
now take the second derivative >:)
🤞 yeah
Differentiate I_Q(x) R->R, where I_Q(x) is 1 if x is rational and 0 otherwise...
Johann Peter Gustav Lejeune Dirichlet likes your comment
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.
thanks for saying “he or she” instead of the “they” bullshit
yoooooo i just graphed this on desmos and it looks so cool!