How Leibniz Invented The Product Rule

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♪ Lukrembo - Flower Cup

⏰ Timestamps ⏰
0:00 - Introduction
0:27 - Deriving Product Rule
1:53 - Thanks for Watching!

[developsome]

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.

[allenanderson]

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

[Censorededs]

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

[markbracegirdle]

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

[KerrySoileau]

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

[insoleafptyptd]

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-nquh]

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

[MichaelRothwell]

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.

[MelodiCat]

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.

[vikrantbhadouriya]

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

[kinshuksinghania]

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’

[mndtr]

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

[michaelrudert]

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

[ethanluvisia]

Such a nice video man, loved every second of it

[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.

[JohnConway-dglc]

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.

[Player_is_I]

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

[bjornfeuerbacher]

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.

[bikwode]

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

[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?