The Leibniz rule for integrals: The Derivation

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German Version: • Die Leibniz Regel für ...
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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[@LuisFlores-mu7jc]

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.*

[@tangpiseth8416]

Isn't the Dominated Convergence Theorem enough for interchanging the limit and integral?

[@LuisFlores-mu7jc]

It is, but you first have to demonstrate that a dominating function exists.

[@tangpiseth8416]

@@LuisFlores-mu7jc Okay thanks! Because over here at my engineering college, we do not cover the theorems and lemma of what math majors usually go through so I do not know much about other theorems ): I want to major in math but the closest thing to a math major in my country is an engineering major.

[@dw5chaosfan]

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.

[@homosapien5684]

@@thegigachad1254 why are you so mad😂

[@adi8oii]

Thank you!! This is one thing about the video that was annoying me, but now I can go check out the wiki page!

[@IlTrojo]

I was going to point out too, then I read supergeek supergeek's and ERik's comments below: at <a href="#" class="seekto" data-time="580">9:40</a> 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!

[@barbazzfoo]

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

[@PapaFlammy69]

Glad you enjoyed it! =D

[@GeodesicBruh]

Is he a student or a professor?

[@BeattapeFactory]

he is so much more dude...so much more :')

[@aryan040103]

Student studying to be a professor IIRC.

[@lunaleonem3378]

Yes.

[@mohandas5118]

@@aryan040103 you re s

[@Mr-dq6gc]

Yes

[@FernandoVinny]

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

[@nahidhkurdi6740]

Be sure whatever alarmed you here would follow you in Analytic Number Theory, at least.

[@oneinabillion654]

Still calc all the way

[@oneinabillion654]

Flammable maths, it'll be great if you can create playlist for all ur integral techniques especially for high school students like me. I just completed A levels. I also completed calc 1 and 2. I've been interested in integration lately. I need such playlists

[@robmorgan1214]

Heh... it's all the same thing at the end of the day... only difference is notation. Look at operator notation in QM, QED and particle physics, it will make you feel better and worse about all things discrete. Look at Noether's theorem and the Lorentz group.

[@robmorgan1214]

@@jcd-k2s yes from a technical perspective no from an intuitive. Also where that analytic stuff leads is not something to sneeze at look at renormalization group flow look at universality etc. as well as the work on what is and is not possible in different dimensions/spaces. Most people get stuck on the quantum "paradoxes" that lead to all the wave particle Copenhagen handwringing. This is unnecessary if you ground your intuition in the algebra and statistics instead of the geometry. The geometry follows from the algebra and the paradoxes ARE part of that emergent geometry not the other way around. Gain an intuitive understanding of this and quantum mechanics becomes about as confusing and difficult "understand" as any other physical theory. No one wastes any time worrying about why the order of rotation of a physical object about 3 axes determines its orientation and which face it presents. Some things commute some don't and this has consequences in how they project on to the chosen basis of measurement. As for the born rule that's a different can of worms but one that's no more mysterious than the Pythagorean theorem or the boundary conditions canceling the contribution of the retarded potential, or any similar conservation law. The point is we learn techniques and many different ways to approach looking at the same problem without deference to how the mind that evolved to solve calculus and geometry problems in a 3d world can be recruited and changed into a mind capable of thinking about and "understanding" problems outside the realm of common perception othen than trough cheap projection or the parlor tricks of metaphorical representation. That's a problem straight out of differential geometry. How to make the brain think in manifolds about manifolds that exist in topological or algebraic spaces it does not currently possess. The math and pedagogy is a means to physically grow a physical representation of itself in an object embedded in a discrete 3d space... yeah that's something that makes my head explode to consider and the hard part is definitely not the differential geometry part... we get that for free from things like vision, facial recognition, hunting, throwing, and knitting skills.

[@michaldvorak8586]

At around <a href="#" class="seekto" data-time="600">10:00</a> mark, shouldn't you plug those bounds for x instead? I am confused :(

[@michaldvorak8586]

Oh, you bad boii 0 this all equals to f(c,t)db.

[@MarcoMate87]

It's all correct except for the last limit. As b(t) is supposed to be derivable (and so continuous), when dt -> 0 also db -> 0. So, the entire limit simply goes to 0. In fact, you have to do the limit not in this moment, but later, when you divide by dt: in that case, when dt -> 0, f(c,t+dt) db/dt -> f(b,t) b'(t). In the hypotheses, omitted from the beginning, we have to suppose a(t) and b(t) as derivable functions of t, and f(x,t) as an integrable function with respect of x ( for example f continuous w.r.t. x ), and derivable function with respect to t.

[@elasmarsaadallah6126]

👍🏼

[@newkid9807]

Flammable Maths I thought you were the mistake!

[@vishakp89]

The whole proof thus has an error

[@43.5]

<a href="#" class="seekto" data-time="635">10:35</a> 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

[@yiruili2453]

Thanks for pointing that out. I didn't know how to fix that, but thanks to IITrojo's comment. I think the correct one should be: the integral equals F(b+delta b, t+delta t)-F(b, t+delta t)=dF(c, t+delta t)/dx *delta b= f(c, t+delta t) delta b. Just for everyone's convenience.

[@vibhu8024]

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.

[@trttrt9838]

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

[@PapaFlammy69]

[@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.

[@bggbbdg5625]

Erin Cobb Yes, I should be a function of t only. A minor mistake.

[@murtithinker7660]

@@bggbbdg5625 Not a mistake. I is both dependent on x and t, but the integral was given in terms of x alone. The fact that the integral of I(x,t) is done over x does not mean that I is only I(x). That is why he started with dI/dt, which clearly means it is not zero. There was no mistake sir.

[@digbycrankshaft7572]

@@murtithinker7660 if the function is integrated in terms of x and the limits of integration are in terms of t then when they are substituted in for the x values the entire expression would be a function of t and so I(x,t) would actually be I(t) so the way it is written at the beginning does appear to be erroneous.

[@thomasjefferson6225]

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.

[@premdeepkhatri1441]

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

[@bobman8192]

Papa flammy’s voice was so deep damn, Also, who else is here watching prerequisite videos for log gamma video

[@susiehue9465]

at <a href="#" class="seekto" data-time="600">10:00</a> 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

[@zokalyx]

you could have looked at the video though

[@seriousbusiness2293]

Stopped the video exactly like you and was also baffled until I read your helpful comment.

[@flutterwind7686]

I am confused. What's the error

[@oni8337]

@@flutterwind7686 Error is that the function was in respect to x so the bounds that he substituted in should be in the x part

[@GAPIntoTheGame]

<a href="#" class="seekto" data-time="639">10:39</a> 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)

[@finalpurez]

I was just thinking about this too... Is there any particular reason why its like this?

[@supergeeksupergeek2355]

Excuse me, but I think I found a little error. When you use the primitive of f(x, t + delta t) to solve its integral you wrote F(x, b + delta b) - F(x, b) but you were integrating over x so the expression should be F(b + deltab, t + delta t) - F(b, t + delta t). Because at being integrating over x f(x, t + delta t) is a function of x and t + delta t would be a constant.

[@Gossamer2288]

I was just going to ask the same question. That could be a small error which does not contribute very much to the conclusion. in 9:50, you should first rewrite the integral f(x,t+delta t) as integral of f(x,t)+f'*delta t (definition of derivate) then use the mean value theorem which gives you f(c,t)*delta b +f'(c,t)*delta t* delta b, then dividing this term by delta t and sending it to zero you get f(b,t)*b'(t) +f'(b,t)* delta b, where delta b is infinitesimally small and therefore zero. So you will get the same result f(b,t)*b'(t)

[@metuphys5611]

@@Gossamer2288 yep same question on my mind as well

[@maxmuller590]

Holy shit i Watch this Video so often i Love it

[@Leakey57]

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...

[@robmorgan1214]

Great video, very clear explanation! It's just given in Strauss in the appendix as a theorem but never really proven. This is a shame since it's such an important result and people often forget to differentiate the bounds of the integral which is problematic in many situations in engineering physics and differential geometry. It's also not clear how to handle it when the derivative may not exist but the integral does (ie integrating around a pole/residue etc.) Just some ideas for future videos, maybe take this on in the context of an analytic function or a Feynman path integral, or a problem that needs renormalization... I'd prefer you didn't wave your hands while doing it but Andrew may be watching so just do what feels right!

[@douro20]

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

[@sshannon1948]

Again..thank you for all of your effort...it takes time, dedication and passion...

[@alancai9982]

Young Papa Flammy is so cute

[@zhaow4832]

Thank you sir. This helped a lot. I can feel your excitement as you finished the derivation :D

[@leif1075]

Mr. Flammy, A plus delta a is greater than a so you have to switch the linits of integration at <a href="#" class="seekto" data-time="259">4:19</a>. Also the other inegrand shiuld have linits of b and a pkus delta a..because delta a is between a and b..

[@user-ps1dm4fc4l]

👌😂

[@juodapimpiumusikas9495]

Correct me if I'm wrong but.. I think by the fundamental theorem of calculus: [ F(a) - F(a+del_a) ] + [ F(b) - F(a) ] + [ F(b+del_b) - F(b) ], which simplifies to -F(a+del_a)+F(b+del_b) which is the integral from a+del_a to b+del_b

[@christianjorquera]

I'm glad I discover this channel. greetings from Chili :D

[@gnikola2013]

At <a href="#" class="seekto" data-time="852">14:12</a>, the terms f(a,t)Δa and f(b,t)Δb are the results of f(c1,t)Δa and f(c2,t)Δb (you called them both c) when Δt tends to zero, so they shouldn't be placed inside of the limit again as you musn't replace an expression inside a limit whith the value to which it tends. I mean the end product is the same, because you then could've just written f(c,t)Δt/Δt and then calculate the limit and get f(b,t)db/dt, but the way you did it is a bit less rigorous. Or am I talking pure nonsense? What I'm trying to say would be really easy if I were there with you pointing at stuff the on the board lol

[@jacobharris5894]

Great video. I actually understand most of it this time.

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@spanishlanguageeducational3737]

I had to watch your video 5 times, but I finally got it. Thank you!

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@engr.rimarc.liguan1795]

Watching this late at night while wearing headset in loud volume then suddenly the sound. Haha. My sleepy feeling gone. 🤣

[@Kuratius]

Have you tried differentiating an integral using the multi-dimensional chain rule? It makes the the Leibniz Integral rule obvious.

[@MrRishi_YT]

Idk if you are going to see this comment, but Im just saying, thank you because I thought your whole channel composed of mostly high level stuff, so I was so happy when you linked this video to a newer one so that I could actually know the “why does this work?” Behind he math you where doing

[@MrRishi_YT]

Flammable Maths damn you really do care about the comments, this made me happy :)

[@CrazyY637]

Fun and cute explanation of Math. You are the beeeeesssttt!

[@bggbbdg5625]

You are so helpful in my learning of advanced maths. Thanks a lot! You are so great!

[@void2258]

I was with this until <a href="#" class="seekto" data-time="610">10:10</a>. You plugged the limits of integration with respect to x in for t in when evaluating the integral.

[@boliboba8717]

Thank you very much for this explanation! From Russia

[@restitutororbis964]

The leibniz rule is so powerful. Thank Papa leibniz for this rule and Feynmann for making it popular.

[@astroboy6608]

Other mathematicians: QED Flammable: *slams board* DONE BEI GOTT

[@PapaFlammy69]

BEI GOTT!

[@astroboy6608]

Flammable Maths bro i want that to be your official way of ending proofs, if not then I’m gonna have to steal because I’ve never laughed so hard during a math video before.

[@PapaFlammy69]

Alright, I'll try to use it more chief.

[@kevinarturourrutiaalvarez2613]

Beautiful explanation. Thank you Papa Flammy!

[@samirgeiger1042]

<a href="#" class="seekto" data-time="920">15:20</a> I dont quite get how the integrant becomes dt(f(x,t)), i assume you are using L'hopital yes? So dont you have to differentiate the whole sum in the numerator? Where did dt(f(x,t+dt)) go?

[@user-so2gs2gw9z]

thanks for your help. I'm student studying Economics in Korea. Leibinz rule for differentiation of definite integrals was big problem to me. I can now overcome that thanks to you. ^^

[@josvandeneynde5849]

Helped me a lot! Keep the math going!!

[@jebhank1620]

How does one become such a cheeky math monke 😍

[@monke9865]

At <a href="#" class="seekto" data-time="720">12:00</a> b is continous function of t so delta of b is actually b(t+delta t) - b(t) so it goes to zero as delta t goes to zero...

[@SK-qc2hb]

Papa: Leibnitz trick Me: what? (after the video) HE'S A CHEATER, THAT IS THE FEYNMAN TRICK!! Feynman's book (surely you're joking Mr Feynman) : so people like to call it after me. Me: I'm an ass.

[@ENTMusic-cj7wt]

This is truly beautiful

[@HUEHUEUHEPony]

yes there's an error at <a href="#" class="seekto" data-time="600">10:00</a> where he substitutes the variables, read the comments to find out how it is solved :o

[@lightspd714]

Best explanation I’ve seen on this subject!

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@codyfan4070]

Around the <a href="#" class="seekto" data-time="300">5:00</a> minute mark, how does the linearity thing work for integrals? Why can you split the integral up like you did? Can someone pls explain?

[@writedan]

I think I(t) not I(x,t) since since x is integrated out

[@MadSideburns]

Who's here because feeling guilty of not knowing the Leibniz rule after having whatched today's video (26^{th} June 2018)?

[@kairostimeYT]

me

[@restitutororbis964]

MadSideburns Me, but I mean its understandable. Us young mathematicians and young physicist barely got into advanced Maths. The leibniz rule is probably from now on my favorite rule due to how powerful it is. Any integral that cant be solved with any sub or series, easily solved by the leibniz rule, its just so elegant and powerful. Papa flammy ' s proof is also my favorite.

[@restitutororbis964]

Well I myself dont know much about the rule, i know the process and proof and its use, but its great and one importance is that, any integral that was "impossible" to solve could now be solved with this technique which could POSSIBLY come useful when studying physics if you do stumble across something that is "impossible" to compute. I REALLY doubt it though since physics uses the lightest of mathematics and doesn´t go too deep into a certain technique. Like Feynman said, "The physicist is always interested in the special case", so a professor giving a lecture will never cover a general case on when you can use this since you will most likely never use it. This technique works really well if you are doing a hard problem just for the fun of it, it leans more into the pure maths side of importance.

[@restitutororbis964]

Ardian Np, I myself dont know physics(yet) at all since im doing self study. Im currently finishing up differential equations and calculus 3 which are the prerequisites to even learn Classical mechanics or really any undergrad physics course.

[@MadSideburns]

@@restitutororbis964 I'll come back here and tell you some cases in which the Leibniz rule is useful when I will study Physics II (in Italy this is the name of the module about electromagnetism). I bet there will be plenty of them. See you in a couple months ma bois.

[@Zonnymaka]

I feel you pain, dude...it's not so easy to record a flawless session. Keep up the good work tho! (Please, leave the "Umm, what can we do now" catchphrase to RedPen! I'm sure you'll come up with something original which will fit your carachter!)

[@blackpenredpen]

Zonnymaka eh! I don't even notice that myself lol.

[@nicholasleclerc1583]

Zonnymaka .......*Isn’t it ?* ; D

[@apolloniuspergus9295]

Bei Gott, what a coincidence.

[@edwinlin7348]

Thanks for all your hard work :D. I'm really looking forward to fresnel integrals > f(b,t)delta(b), why does that delta(b) remain unchanged. Wouldn't that go to 0 as well if we take the limit as delta(t) goes to 0, since that's what allows us to change c --> b in the first place?

[@ajinkyapawar8662]

Wow, sheldon started a KZbin channel!

[@prosnape2704]

LOVE your video!Keep up with your good work! :3

[@MathPhysicsFunwithGus]

No words can describe how beautiful I found this video, amazing derivation! I aspire to explain concepts half as well! Thank you!

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐...

[@iamgroot3615]

Just think of the integral as an infinite sum. The derivative of the sum is the same as the sum of the derivatives

[@nicholasleclerc1583]

At <a href="#" class="seekto" data-time="631">10:31</a>, why does Int{from b to b + delta-b} f(x,t + delta-t) dx = F(x,b + delta-b) - F(x,b) and not F(b + delta-b,t + delta-t) - F(b,t + delta-t) ?

[@mathunt1130]

Write the integral as: I(a(t),b(t),t) and use the chain rule to obtain: I'(a(t),b(t),t)=\partial_{a}Ida/dt+\partial_{b}Idb/dt+\partial_{t}I(a(t),b(t),t) and simply write down the terms.

[@akshaypr9164]

Flammable math: Can we interchange the limit and the integral? Me: Should I consider Riemann integration or Lebesgue integration??? .....and after a while I cringe in monotone convergence theorem, change of variables in lebesgue integration ,fatou's lemma and dominated convergence theorem....and the list continues..

[@domenicapincay8313]

Incredible, thank you for this amazing video. Greetings from Ecuador.

[@PURULUMK]

Really liked it! Nice work! 💪💪

[@vinuthomas2814]

I would not have thought to say a(t+deltaT)=a+deltaA: interesting. I was a little confused when the integral was split, but I see now the idea is the integral over [a+deltaA, b+deltaB] = integral over [a,b+deltaB] - integral over [a, a+deltaA]

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo ...💐

[@vinuthomas2814]

@@beoptimistic5853 He starts off by saying volume is a triple integral - off to a bad start...

[@beoptimistic5853]

@@vinuthomas2814 for phycisian a mathematics is only tools but we should open our eyes to see it, i think it s verry good for scientist people how want convert a virtual and theory to a reality ; it s amzaing tool (he really hard and experienced scientist ) to resolve equation is the most easy but the important we use for what? . If you have chance you one day know what you do whith if no like a majority's good look

[@yaschendlima4746]

for the bounds of integration why do you separate [(a + delta a) to (b + delta b)] as [(delta a) to a] + [a to b] + ..., instead of [a to (delta a)] + ...

[@anushkasingh9332]

I am confused as well

[@MrKelvinJane]

Around <a href="#" class="seekto" data-time="935">15:35</a>, regarding the first integral, when the limit of an integral is the integral of the limit, what theorem is used there?

[@harukananami6615]

Siu Kwan Yuen Actually ， should use lebesgue dominated convergence theorem. To be honest， math people never interchaning limit with integral without justify. This video is for engineering probably

[@joefrancis2406]

toooo maddddd keep these vids coming ur too sick

[@user-up9kl5px5k]

Smooth and interesting explanation Continue

[@benjaminbrady2385]

I am this videos 69420th viewer and I think that's beautiful

[@yadakkpolyt8132]

Hi Flamy!! Cool video!

[@WIDSTIGETHEVLOGGER]

How does he know that I’m watching this in the morning when he says good morning😰

[@michaelboyd8546]

Your videos are great, it would be awesome to see a collaboration video with Dr. Peyam and BlackPenRedPen over something really awesome. Keep it up!

[@luisdaniel9542]

Comments that aged poorly

[@AnkitYadav-zg5zd]

MISTAKE!!! AT EXACTLY <a href="#" class="seekto" data-time="586">9:46</a>. Intigration has the variable x so limits will be placed for x not for t.

[@whatby101]

Does rewriting b(t+delta t) as b + delta b only work with linear transformations, meaning delta b = b(delta t), or is it not like that in this case?

[@GiornoYoshikage]

No, Δb is only a compact way to write b(t+Δt) - b(t). It doesn't matter what b(t) is except it's differentiable

[@adaptercrash]

Double dragon equals double dragon and I just leave it on, that's really old. Each one is different and inverted unto each other in subdivisions to find answers of their lives. It's also a knowledge theory .

[@kwameawereohemeng3931]

Thanks for the proof. A follow up question is what if the limits of integration involve infinity? Also, are there restrictions on when the integral and differentiation operators cannot be interchanged? Thanks.

[@antonioromerio5555]

If a or b go to infinity you can do this exact process. Just have on the outside a limit saying that a or b goes to infinity or minus infinity. Then when the process is finished you can distribute the limit back into the answer.

[@kwameawereohemeng3931]

@@antonioromerio5555 okay thanks. Do you have an idea of my other question?

[@antonioromerio5555]

@@kwameawereohemeng3931 I believe any such restrictions should be the same as those already placed upon a definitely integrable function. f(x,t) has to be continuous on [a,b] and definitely needs to be differentiable with respect to t. a(t) and b(t) must also be differentiable, but these things all just follow from the result

[@yahyasheikhnejad1877]

the upper and lowe bound will be substituted for x in the integral, not t!!

[@vynneve]

Thank you!!! This proof was annoying me because I couldn't find a nice explanation for it. Finally get it now. I literally have an undergrad in maths and have never seen this, strange. But ya it's really cool. thanks again

[@tlenf]

<a href="#" class="seekto" data-time="840">14:00</a> How do we take the derivative inside the integral? After all, the limits are still a function of t. It may be illogical but I just don't get it. As far as I know, one can bring the differential operator inside the integral (integration wrt 'x') when the limits of integration are independent of 't'.

[@rezamiau]

ABSOLUTELY BRILLIANT !! but I'm really curious about how quickly you swap the board and you don't hurt your fingers. that's incredible! be careful man, we need you.

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo ...💐

[@choi-r7108]

Good day sir, can i ask something? I'am pretty desperate here So, i'm on my thesis, and i need this concept of interchange between derivative and integral. Thanks to your explanation, i understand this concept better. The thing is, is there any valid reference for this theorem (book or paper)? I just can't find it for my bibliography or literature review..... Please help me.

[@user-om2kg3vh8s]

integral_a^(b+delta b) f( x, t+delta t) dx = F( b+ delta b, t+delta t ) - F(b, t+delta t)

[@theInternet633]

That sounds really complicated. Isn't the Leibniz rule for derivates for indefinite integrals not just equivalent to the commutative property of the partial derivative? After all integration must also be commutative if the derivative is. That property is also formally proven in Fubini's theorem. Both follow from the linearity of the operator. And if it holds for indefinite Integrals it must also hold for Integrals with bounds independant of the variable

[@Killerstin2]

When you show that taking the limit when delta t approaches 0 allows for us to write c=b and c=a, why then take the limit of the same argument once more in the next step?

[@gian2kk]

wat

[@electrotrout87]

exactly

[@brendanlee1707]

I don’t understand <a href="#" class="seekto" data-time="661">11:01</a> , if we r integrating w respect to x, why is it that the boundaries are inputted to the t variable of primitive F? Shouldn’t it be F(b + db, t+ dt) - F(b, t+dt)?

[@manitheman9863]

<a href="#" class="seekto" data-time="765">12:45</a> you sounded like Sheldon from big bang theory😂❤

[@mattjs07]

Awesome video ! Thanks a lot !

[@PapaFlammy69]

=)

[@sshannon1948]

Thank you for your presentation style..sehr gut ..immer...

[@kquat7899]

Very well explained.

[@rockapedra1130]

Very cool ... loved it!

[@sarojmandal981]

Sir, a great derivation. Please also state under what conditions the Leibnitz's rule may fail. In this derivation mean value theorem was used. There must be some assumptions.

[@RealMathematician21stCentury]

It's not "Leibniz's rule" at all! Rather it is very simple calculus. You have a common function f(x,t) and perform the reverse operation on each side. To go to the lengths of proving it as he has done is rather comical.

[@minjiechen5994]

that really helped me! thanks

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@superengineer4448]

Awesome, well explained. Thanks!

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@r0cksp0t]

Why do the limits change from a(t) to a(t)+Delta a instead of changing just to a(t+delta t) ?

[@leonobiang333]

He just skipped a step. a(t+ delta t) = a + delta a

[@aryansachan6934]

Fantastic

[@6octaveoctopus]

at <a href="#" class="seekto" data-time="592">9:52</a> why didn't you integrate with respect to x? You put the upper and lower bounds of the integral in the t slot.

[@bzboii]

HELLLO FELLOW MATHEMATICIANS

[@sshannon1948]

So in the case of the integral, the first part of your result, to actually evaluate it, you would first take the partial of f(x,t) with respect to t and get f'(x,t) and then you would integrate with respect to x so that result is f(x,t) back again and then you would evaluate it at its limits b(t) and a(t), substituting in for x and get f[b(t),t] - f[a(t),t]? So to be able to do the integral, one must be able to integrate with respect to x then..is that correct? If this is not correct, would it be possible for you to direct me to an online example of an integration of this sort where both limits are functions of t. I do not want to bombard you with questions but things like this continue to bother me if I don't completely understand them. I try to leave no stone unturned in my learning.

[@sshannon1948]

No worries..I found an example but with the initial differentiation done with respect to x rather than with respect to t..... www.utdallas.edu/~pervin/ENGR3300/LeibnizRule.pdf