How Leibniz Invented The Product Rule

[@developsome209]

The proper approach. Not just serving things as set and done, to just 'shut up and calculate' as the saying goes, but showing how things came about, the stories behind them. There is interesting stuff to see in a nutshell.

[@tuitia]

Thanks, this is what we strive to teach with Tuitia.

[@bjornfeuerbacher5514]

You are right - that would be the proper approach. But at least with this video here, they failed completely, they did not show at all how Leibniz actually derived this, they simply reproduced the standard calculation which one can find in essentially all textbooks.

[@allenanderson5529]

While inventing/discovering the Product Rule, he's more famous for inventing/discovering the Chain Rule!

[@markbracegirdle7110]

Leibniz never a post at a university, which made his achievements all the more remarkable.

[@Censorededs]

Considering how limit notation wasn't invented until well after Leibniz passed away, this is certainly not how he derived the product rule.

[@borincod]

Leibniz just used dx and dy, where d is exactly delta approaching zero. Prone to inaccuracies sometimes, yes, but the derivation will be the same - just without 'lim' and aasuming dx=lim(h) and df(x)=lim(f(x+h)-f(x)). Whats the problem?

[@bjornfeuerbacher5514]

@@borincod So how do you do the step of "inserting" -u(x+h)v(x)+u(x+h)v(x) when using the notation with df and dx?

[@borincod]

@@bjornfeuerbacher5514 I guess you didn't account this: d(uv) = u(x+dx)v(x+dx) - u(x)v(x) Then you do the mentioned "inserting" with dx instead of h.

[@bjornfeuerbacher5514]

@@borincod But that is _not_ how Leibniz actually did do that. Leibniz did not even use the concepts of functions, he did not write u(x) or v(x), he simply used the product xy. What Leibniz actually did do was writing d(xy) = (x+dx)(y+dy) - xy, the multiplying the brackets and in the end arguing that one can ignore the product dx dy. He did _not_ "insert" anything. So what is shown in the video here really has essentially nothing to do with how Leibniz actually derived the product rule.

[@borincod]

@@bjornfeuerbacher5514 indeed, haha. At least Horvath claims that. What a cumbersome derivation. Funny enough, it was Leibniz who introduced the notion function and used it along with variable. So, he understood the concept of a function as quantities depending on each other.

[@KerrySoileau]

Two of the smartest people ever to live, living at the same time.

[@animenmusic16]

Like rivals almost

[@KerrySoileau]

@@animenmusic16 Big rivals. Big dispute over which of them should get credit for inventing calculus. We still use Leibniz' notation to this day.

[@vikrantbhadouriya]

served as an excellent revision about how the product rule came into being! 🙌

[@MichaelRothwell1]

Nice video! Here's a small detail: for this to work, we should be given that u and v are differentiable at x. Hence u is also continuous at x which justifies that u(x+h)→u(x) as h→0.

[@kinshuksinghania4289]

A much more intuitive way is to consider the product uv and then trying to write the change in uv in terms of individual changes in u and v. Δ(uv) = (u+Δu)(v+Δv) - uv Expanding this we get Δ(uv) = (uv+vΔu+uΔv+ΔuΔv) - uv Δ(uv) = vΔu+uΔv+ΔuΔv In the limiting case when these changes approach infinitesimally small values, we get d(uv) = vdu+udv+dudv And then the term dudv vanishes in comparison to the other terms and we get d(uv) = vdu + udv Rearranging d(uv) = udv + vdu And if u and v are functions of another variable, say x, we can write (uv)’ = uv’ + vu’

[@bjornfeuerbacher5514]

And I'm quite sure that _this_ is how Leibniz _actually_ derived the product rule - _not_ in the way claimed in the video.

[@irappapatil8621]

It was like this:(x+dx)(y+dy)=xy+xdy+ydx+dxdy-xy. The difference is xdy+ydx+dxdy dxdy->0. d(x)=xdy+ydx

[@bjornfeuerbacher5514]

@@irappapatil8621 That's what I already wrote in another comment, lower down, 5 days ago. ;)

[@frankfahrenheit9537]

Δ(uv) = (u+Δu)(v+Δv) - uv is not intuitive. As little as the product rule itself.

[@bjornfeuerbacher5514]

@@frankfahrenheit9537 Why is that not intuitive? It can be visualized very simply by drawing a rectangle with sides u and v and then asking by how much the area of the rectangle changes when both of the side lengths change.

[@insoleafptyptd3801]

Well done on presenting such a key concept in calculus in a straight forward manner. Looking forward to seeing more quality content such as this.

[@Alex-nq7uh]

All around amazing video. Really nice production, not too long nor short. Can't fault it- thank you very much for such a great explanation! glad it popped up in my feed

[@MelodiCat753]

Perfect video that hints towards the intuition of the product rule essentially being the result of the area of rectangle with side lengths that are the functions, which connects to the fundamental theorem of calculus. Definitely going to show this to my tutoring students.

[@michaelrudert3406]

Wow, short and sharp well animated and easy to understand! Thanks for this demo👍

[@JohnConway-dg6lc]

I'm almost certain that this is not how Leibniz came up with the product rule. Leibniz used a delta-process employing infinitesimals, not this convoluted approach with limits. I'd really be interested if you could point me in the direction of your source material showing that Leibniz actually authored this approach. Thank you in advance.

[@gaurakshsinghi9219]

this is not an approach used by Leibniz it is an approach used in high school

[@mikloscsuvar6097]

This was so beautiful and well paced.

[@kashhusain8154]

Incredible.

[@mndtr0]

Also you can draw a geometrical picture to see that delta of whole function is approaches the product rule

[@fadydavis7457]

Amazing how much simple the explanation was just what type of content i wanted. Short and concise just, ( chef's kiss) (Btw pls also make long form vids)

[@tuitia]

Will do, thanks @fadydavis7457 !

[@ethanluvisia8678]

Such a nice video man, loved every second of it

[@bikwode]

This is a nice video, but it would be nice if you added a lot of details and longer videos.

[@Player_is_I]

Love math lore, hence, love this channel❤❤❤

[@whatitmeans]

From the geometric approach I always wonder why the u'v' term dissapears.... until I learned about Ito's Calculus and how that assumption is not valid for functions with non-zero quadratic variation, and in other look, it explains why functions with quadratic variation don't fulfill the Fundamental Theorem of Calculus... I never thought that Brownian motion could be such a weird thing - an interesting.

[@RabahHamidi-zt6vg]

Very helpful,waiting for long vids!

[@tuitia]

@@RabahHamidi-zt6vg on the way!

[@chharlessweeney]

Thanks ... and well done.

[@judekirkcruz7177]

Wow...

[@lauralhardy5450]

That frigging background music, worse than the youtube ads

[@rorororo6084]

i was expecting to see the original approch taken by Leibniz to solve this problem, not the the proof that we had in high school calculus class!

[@bjornfeuerbacher5514]

No, that is not at all how Leibniz actually invented the product rule! This is the proof shown in high school textbooks today; Leibniz did do it in quite another way! He wrote that d(xy) = (x+dx)(y+dy) - xy = x dy + y dx + dx dy, and then argued that the product dx dy can be ignored.

[@epicchocolate1866]

There are zero proofs in high school textbooks

[@bjornfeuerbacher5514]

@@epicchocolate1866 So you know all high school textbooks which are used all over the world?

[@Pedritox0953]

Great video!

[@WofWca]

The four melodransitions are a little hard to follow. Maybe put both on the screen at the same time?

[@bjornfeuerbacher5514]

What are "melodransitions"?

[@WofWca]

@@bjornfeuerbacher5514 Lmao. I meant just "transitions". I bet I sounded smart there 😎

[@Loots1]

insane

[@Amit_Pirate]

Thank you for teaching me this useful info. Can I also know how the formula for integration with upper and lower limits was derived?

[@tuitia]

That one's hard. I'm still working on the script. But it will be on the way!

[@Amit_Pirate]

@@tuitia Thank you so much!

[@bjornfeuerbacher5514]

Actually, it's not hard at all. You just have to think of integration as a kind of summation. That's _exactly_ Leibniz's concept! Before he started working on calculus, he worked on series and stressed that if you sum up differences of successive terms of a sequence, you'll simply get the difference of the last term and the first term, i. e. if you have s sequence (a_n), calculate the differences d_n = a_n - a_n-1 and then sum up these differences, d_1 + d_2 + ... + d_n, the result will be a_n - a_0.

[@Amit_Pirate]

@@bjornfeuerbacher5514 that's mind blowing! Thank you for the amazing explaination :)

[@confidosine]

great video

[@neuralwarp]

"Haytch" ?

[@Tabu11211]

Who is here before 1m subs?!

[@billcook4768]

Invented???

[@user-ir8nd6mj2b]

0:19 ehe hehe e

[@derciferreira2523]

New is the only inventor of Calculus.

[@quasarsupernova9643]

Total nonsense. Calculus was invented in India by the Kerala school of Mathematics starting from Madhava. This knowledge was transported to middle east and Europe by Jesuit preists which then found its way into libraries of Europe ...

[@bjornfeuerbacher5514]

And your evidence for these claims is...?

[@quasarsupernova9643]

@@bjornfeuerbacher5514 Anyone who asks these questions in 2024 has a broken wifi connection or has visual impairment ...

[@bjornfeuerbacher5514]

@@quasarsupernova9643 So you conveniently ignore that most of your claim above contradict what actual historians of mathematics say? :D

[@bjornfeuerbacher5514]

@@quasarsupernova9643 Maybe you simply are confused what "inventing calculus" actually means... Yes, Madhava discovered lots of results centuries before they were known in Europe. But no, that does _not_ mean that he "invented the calculus". He missed _lots_ of very important results which were absolutey _crucial_ for calculus - like the Fundamental Theorem of Calculus itself.

[@quasarsupernova9643]

@@bjornfeuerbacher5514 Keep splitting hairs. Infinite series and limits are the backbone of calculus the rest comes later ...

[@bernaridho]

I am sick of x, h, u and v. Invent names. Those single letters are sickening and archaic.

[@bjornfeuerbacher5514]

You would prefer to call a variable e. g. Tom instead of calling it x?

[@bernaridho]

@@bjornfeuerbacher5514 May be.

[@bjornfeuerbacher5514]

@@bernaridho And what exactly is the advantage of that?

[@davidbrown8763]

Because of the annoying, unnecessary, distracting background noise, I cannot share this video. Such a pity, because it is a very instructive video.

[@bjornfeuerbacher5514]

Actually, it is not very instructive. Instead of actually showing Leibniz' method, it shows the standard proof which can be found in all high school text books.