The Leibniz rule for integrals: The Derivation

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Today's video is going to be exciting! W are going to derive the Leibniz Rule for integrals in its whole form! It's one of the most powerful tools of integration, so be prepared! :)^

Quick note: The Integral I(x,t) is just in terms of t, so it's I(t) thanks for noticing! Even though, it doesn't quite matter, since we are just differentiating by t^^

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[dwchaosfan]

For anyone who is watching now and wondering about whether the limit and the integral can be exchanged: wikipedia article on Leibniz Integral Rule, under the section "proof of basic form", has the details. You'll need analysis to fully understand, but basically assuming the partial derivative exists and is continuous, it holds.

[LuisFlores-mujc]

Flammable Math: Can we interchange the limit and the integral? We are going to assume that we can.

Me: *Cringes in Monotone Convergence Theorem, Fatou’s Lemma, and Dominated Convergence Theorem.*

[IlTrojo]

I was going to point out too, then I read supergeek supergeek's and ERik's comments below: at 9:40 you plug the boundary values into the t-part of f(x, t) instead than into the x-part. Then sending things to zero collapses the mistake.
Great channel anyway, keep going!

[vibhu]

I love how you set it up so perfectly that one could easily derive the formula for when a and b are functions of t from here.

[barbazzfoo]

thorough ground-up explanation that tied in the relevant theorems and definitions. thank you so much :)

[.]

10:35 there is confusion about the integral of f to F is subject to x (first variable), then the central theorem derivative is about second variable t

[michaldvorak]

At around 10:00 mark, shouldn't you plug those bounds for x instead? I am confused :(

[FernandoVinny]

After this video I'm convinced that what I like is Algebra, Number Theory and Discrete Maths

[thomasjefferson]

Papa Flamey, I love you diversification of the channel. You taught my Norwegian wife how to make American pancakes. She only knew how to make the inferior European version. Now you're helping me with my homework for this week. Thank you for the value you have provided me.

[Nusoa-hrpk]

even after 7 year this video is just insane, very helpful thank you ❤❤

[premdeepkhatri]

Excellent Understanding of mathematics by this Young boy. Thank youn for explaining Leibniz rule of Integration.

[trttrt]

Ich hab das so lange gesucht! Dankee! Es war seeehr nützlich :)

[RealMcDudu]

Note that delta-b is actually (b(t+delta(t)) - b(t)) and same for a.
Also there's probably some conditions on which you can replace the integral and the limit on the 1st term, most likely that it converges. Since you require the definite integral between a and b, this is most likely to happen.

[ThAlEdison]

I thought because the integral was a definite integral in terms of x, that I() was only dependent on t, not both x and t.

[HUEHUEUHEPony]

yes there's an error at 10:00 where he substitutes the variables, read the comments to find out how it is solved :o

[Leakey]

Well done sir, even your dog would have understood your exposition. Thank you for reminding me just how much tremendous fun exploring calculus can be. I wish someone could make the dry internals of algebra as exciting...

[susiehue]

at 10:00 when it said F(x, db)-F(x, b) i spent a solid half hour trying to figure out how that was possible. turns out it was a mistake on his part lol

[bobman]

Papa flammy’s voice was so deep damn,

Also, who else is here watching prerequisite videos for log gamma video

[GAPIntoTheGame]

10:39 When you are integrating f(x, t+delta t) with respect to x why are you substituting the boundaries on the t variable? If you are integrating with respect to x the boundaries should be added to x, right?(Even if the upper bound and lower bound are in terms of t)

[douro]

It's commonly called "Feynman's Technique" because it was him who popularised it in his lectures on teaching science and maths.