Proof of the Product Rule from Calculus 

This was awesome! I love learning proofs after I've been using the rule, sort of justifies it. I appreciate you, Patrick.

well, less important would be proofs to know. much more important are proof techniques: induction, proof by contradiction, etc

The reason he can do that is because they cancel out and don't actually change the equation, they just make it easier to come to a solution. He uses those 2 expressions in order to factor out common terms and achieve the final goal but as you can see @2:42 is that the blue expressions just add up to 0, therefore leaving the expression undisturbed while being able to simplify it.

what was the thought process behind adding and subtracting f(x+h)g(x)

just joking, i have heard of him. i am not interested in grants from people. then everything starts to become corporatized and you have to answer to some board of directors. no thanks, that is not for me.

never heard of him

if they want to license / use my stuff, that is fine. but i do not think they have the least amount of desire or interest.

i can run through them all if people want. it is pretty much the same idea in each!

This is the first time a proof has really clicked for me and after following along I completely understood what is happening. Thank you for the video!

Hey Patrick, I have a serious question. Have you heard of the Khan Academy? I think you and Salman could do so much good in the world since you're both really great teachers. I was thinking maybe you two could be partners and make videos that followed the typical curriculum for advanced math classes like at least college algebra to Calculus 3.

Honestly from the bottom of my heart, THANK YOU!

i just did

5+1-1+1-1+1-1=5 I can add and subtract 1 as many times as I'd like, I'm still not changing the equation.

yes it is

very well done. You explain this concept in simple and easy details. Far more understanding than most of my math professors. I appreciate the upload :)

already have

I started to have a nice grin the moment you added and subtracted f(x+h)*g(x). More proofs would be lovely!

Great presentation! I like the proof using Logs: Let y=f(x)g(x) and take the (natural) log of both sides ln(y)=ln(f(x)*g(x))=ln(f(x)) + ln(g(x)) ...Now differentiate both sides: (1/y)dy/dx=(1/f)df/dx +(1/g)dg/dx ... Solve for dy/dx ... dy/dx=y[(1/f)df/dx +(1/g)dg/dx] ...Now subs in for y ...i.e. y=f*g so dy/dx=f*g([(1/f)df/dx +(1/g)dg/dx] = g*df/dx + f/dg/dx QED

why does f prime of x doesn't include dx any more? Does anybody know? Same question for g prime of x. Why doesn't it multiply with dx anymore?

And this is why I frickin' love maths.

Why do we substract and add "f(x+h)g(x)" Like is there a reason? or only we do it to make the proof work?

Nice proof, I looked over one other video and was confused but this one explained it clearly enough to easily understand it. Have a good day if you’re reading this comment on a 10 y/o video OP

Very good method.Thanks a lot.DrRahul Rohtak.India

Man, I love this stuff. I struggled in calc in college, but watching these videos are very informative. Great channel!!

What gives you the "right" to factor out the terms f(x+h) and g(x) from the numerator h?

This is awesome, thank you so much. I wish you were my teacher...

Perfect! Thanks.

Is the proof the same with that in a complex number sense?

just blithely made f(x+h) = f(x) on the left but not elsewhere.

❤

you don't think that some are of particular fundamental value which serve as exemplars of the techniques?

Why did you add and substract that transaction, sir ?

thanks man,we can never thank you enough but trust me you'll definitely be on my graduation speech :)

Would be interesting to see you do a series on what you thought are the top ten most valuable proofs to learn in math education ? Top 20 ?

Not only that but unlike this proof it's not as cheaty. I don't like these proofs where it's like: "add this subtract this" because to come up with them you already need to know what you're looking for. The proof with differentials leads you to the product rule (and the chain rule) without requiring you to know it beforehand. As far as I know it's also the original proof by Leibniz which makes it even more interesting. :P

By the way. . .Studying Electrical Engineering in a prestigious university in Canada. Just got done 2nd year and im done studying any new math material for the rest of my life (I think). So I wanted to personally thank Patrick. Probably wouldn'tve made it through Calc I.. .II. . .III. . Differential Equations. . .Linear Algebra and Vector Calculus without his help

Your proof is not complete. You have not proved that lim f(x+h) = f(x) as h goes to zero. This is true only because differentiability implies continuity. In general, it is false.

Yes I understand that, but as I've said before, the definite integral is the anti-derivative evaluated at the upper limit subtracted by the anti-derivative evaluated at the lower limit. I am aware of all of that, but I was asking about the proof of this or the process itself from which the definite integral was derived from the infinite Riemann summation.

Do you have guess of why randomly using f(x+h)g(x) works? It because of trial and error just because they took the product rule and work backwards to find the prove? Like the same thing for proving the integral of sec(x) how did the person know how to use sec(x)+tan(x) / sec(x) + tan (x)

Yes: the Riemann sum definition of the integral. You are essentially adding infinitesimal areas between the graph and the axis across the given interval a to b. All of calculus comes down to the concept of infinitesimal parts -- from this, and some clever math, everything can be derived.

Actually wikipedia uses proper notation for it, also to understand the one on wikipedia you must also know that the difference quotient is the derivative with the addition of the limit as h approach 0.

Is there some sort of proof for why the definite integral of a function from a to b yields the area under its curve and is equal to its anti-derivative evaluated at B subtracted from it evaluated at A?

This proof is good only if u suppose that derivative of f,g exist on "x" (i hope my maths english is not too bad lol) ,but ur explanation is very clear :)

I know and understand the Riemann sum, but I don't understand is how they got from finding the limit of infinite rectangles to the concept of the antiderivative itself.

The limit as the number of rectangles in the summation approaches infinity IS the integral between two bounds; the antiderivative is an indefinite integral (no bounds).

My name is Sergio Sanchez and i am a Mormon lol... i been watching your videos all day in preparation for an exam and this advertisement has been stuck in my head.

i dont understand the second to last step, if h never touches 0 then y does is equal 0? Because the second to last step equals 0 which is the prime definition. helppppppp

why was the f(x+h)g(x) added then subtracted from the equation?

tell me why? i dont get it. Like why are you adding and subtracting the same thing..is this trick...is it allowed...

the most primitive tools in the hands of a nobleman shine the light of wisdom and generosity.

thanku sooo much sir....now my problen is solve thanku.....😊

You should try (imo) the differentials proof. It involves new concepts but it's much, much more intuitive.

I wish my teachers had gone over this in high school, instead of just teaching the short cut rules.

It would be great if you could upload some theoretical stuff like this ! I mean, theorem proofs, etc Thaks Patrick !

Why did you subtract and add the term? I mean. Which primciple states you can do that?

Awesome! Any chance of more advanced proofs, like proving that sup A is the lub of A?

that was sooo cool lol ty for this, wikipedia is so much harder to understand imo

Can you do a video on Laplace transforms in the near future will appreciate it ..

Wish I could switch brain with you...

Wonderful Patrick. I didn't think I'd be able to follow it but you explained it beautifully. Many thanks.

Tengkiu patrick for making us understand easier

you always safe my life

this is superb sir...

Thank You Sir I`m from South Africa

basically how does the last step lead to the proof?

what are the chances i just went over this today in class D:

thanks for this i always forgot that sub in trick and never understood it great quick yet detailed video :D

Very informative, thanks.

It would NEVER cross my mind to add and subtract that :/

can you run through proofs of all the calculus rules?

woooooooooooooow

Humus look up.fundamental theorem of calculus.proof

hey man do you still reply to comments? i just dont understand how to figure out the addition and subtraction of a term that's worth 0 you know where you added and subtracted *f(x+h)g(x)* at time frame 2:26 do you have exercises that goes deep into this and/or have explanation for techniques on this. i really want to understand this man

*ooooooh

patrickjmt you're the math man!

that's easy

Perfect

nice clear and concise video!

I have a question why you applied the limit to both sides of the sum. This near 4min

ohhhhhhhhh yeah it makes sense

💗💗💗💗💗💗

i have a really juvenile question. why is it that we can assume f(x+h) becomes f(x) when h->0 when we can also do so for basically everything else? Like g(x+h) and also the denominator h.. i understand that would make the whole equation untenable but why is it logically we can be selective about which term to bring to the limit, if you know what i mean

Excellent! Thanks.

can someone please help I didn't understand his explanation when he explains on the limit f(x+h)-f(x) over h.

life saver, thanksssss

nice. thank you

Great video. Really helped me

Thanks

Why are you able to just add and subtract f(x+h)g(x)?

thanks you sir

Patrick ! Thanks !!!

Thank yuo

Thank you!

*me* : that makes sense _*Also Me*_: (0.0) naw.

Thank you!

why do you move the limit sign to the right of the f(x+h) at 4:44 ? this makes no sense to me

beautiful.

Thank you!

This is excellent! A proof that can be presented in a high-school level class setting. Thank you very much. However, i do agree with one of the previous comments. The better and more intuitive one would be to use the partial differential chain rule which makes it pop right out. But this requires knowledge of partial differentiation which at least, my high school syllabus does not have. thank you.

Thank you. I figured it was something like this but you may wan to explain that adding the expression f(x + h)g(x)... was just adding zero to your numerator since the expression will always equal zero and that doesn't change the value of the number in the numerator since it is zero.