
Korean: 
이 영상의 목적은
여러분이 만족할만한 곱셈 법칙의
증명을 보여 주는 것입니다.
자, 미분의 정의부터 시작해봅시다.
여기 함수 f(x)가 있습니다.
그리고 이 함수의 미분을 하려면
미분의 정의에 따라서
함수 f(x)의 미분은
h가 0으로 수렴할 때
f(x+h)-f(x)를
h로 나눌 때의 값입니다.
시각적으로는
접선의 기울기라고 할 수 있죠.
그런데 지금은
조금 더 흥미로운 것을 
하려고 합니다.
저는 미분을
f(x)만 쓰지 않고
두 개의 함수의 곱인
f(x)* g(x)를 하려고 합니다.
그리고 이에 대한 
간단한 답이 나오면
그것이 곱셈 법칙이 되는 것이지요.
자, 이 문제를 풀기 위해서

Bulgarian: 
Надявам се, че в това видео
ще ти дам задоволително доказателство на правилото за производна на произведение.
Нека започнем с дефиницията
 за производна.
Имаме функцията f(x).
Ако искаме да намерим 
производната ѝ,
по дефиниция производната
на f(х)
е границата при х,
 клонящо към 0,
на f(х +h) минус f(x),
цялото върху h.
Ако го разглеждаме графично,
това е наклонът на 
допирателната и т.н.,
но сега искам да направя
нещо малко по-интересно.
Искам да намеря производната
спрямо х не само на f(x),
а на произведението от две функции:
f(x) по g(x).
Ако измисля простичък вариант 
за това,
то ще бъде по същество 
правило за производна на произведение.
Ако приложим дефиницията
за производна,

Czech: 
Cílem tohoto videa je dát vám
uspokojivý důkaz součinového pravidla.
Začněme s definicí derivace.
Tedy pokud mám funkci f(x)
a chtěl bych získat její derivaci,
podle definice je derivace f(x) 
limitou pro 'h' jdoucí k 0
funkce f (x plus h) minus f(x),
to celé lomeno 'h'.
Pokud bychom si to chtěli představit, 
je to tangens úhlu dané funkce a tak dále,
ale já chci udělat něco 
trochu zajímavějšího.
Chci najít derivaci podle 'x'
nejen funkce f(x),
ale součinu dvou funkcí, f(x) krát g(x).
A pokud pro to naleznu jednoduchý postup,
je to v podstatě součinové pravidlo.
Pokud pouze aplikujeme definici derivace,

Thai: 
สิ่งที่ผมอยากทำในวิดีโอนี้
คือให้บทพิสูจน์ที่น่าพอใจสำหรับกฎผลคูณ
ลองเริ่มต้นด้วยนิยามของอนุพันธ์กัน
ถ้าผมมีฟังก์ชัน f ของ x
และถ้าผมอยากหาอนุพันธ์ของมัน
โดยนิยาม โดยนิยามแล้ว
อนุพันธ์ของ f ของ x คือลิมิต
เมื่อ h เข้าใกล้ 0
ของ f ของ x บวก h ลบ f ของ x
ทั้งหมดนั้น ทั้งหมดนั้นส่วน h
ถ้าเราอยากคิดเป็นภาพ
นี่คือความชันของเส้นสัมผัส อะไรพวกนั้น
แต่ตอนนี้ผมอยากทำ
ให้น่าสนใจกว่านั้นหน่อย
ผมอยากหาอนุพันธ์
เทียบกับ x ไม่ใช่แค่ f ของ x
แต่เป็นผลคูณของฟังก์ชันสองตัว
f ของ x คูณ g ของ x
และถ้าผมหาสูตรง่ายๆ สำหรับค่านี้ได้
มันก็คือกฎผลคูณนั่นเอง
ถ้าเราใช้นิยามของอนุพันธ์

English: 
- [Voiceover] What I
hope to do in this video
is give you a satisfying
proof of the product rule.
So let's just start with our
definition of a derivative.
So if I have the function F of X,
and if I wanted to take
the derivative of it,
by definition, by definition,
the derivative of F of X is the limit
as H approaches zero,
of F of X plus H minus F of X,
all of that over, all of that over H.
If we want to think of it visually,
this is the slope of the
tangent line and all of that,
but now I want to do something
a little bit more interesting.
I want to find the derivative
with respect to X, not just of F of X,
but the product of two functions,
F of X times G of X.
And if I can come up with
a simple thing for this,
that essentially is the product rule.
Well if we just apply the
definition of a derivative,

Bulgarian: 
което означава да намеря
границата
при h, клонящо към 0,
в знаменателя ще имам h.
Ще направя голяма
дробна черта.
Ще бъде голям дробен израз.
В знаменателя ще имаме h,
после ще сметна това нещо
 за х + h.
Това ще бъде f(х + h),
g(х + h), и от това ще извадя
това нещо, сметнато за f(х).
Извинявай! Това нещо, сметнато за х.
Това ще бъде f(x) по g(x).
Ще оставя голямо празно
пространство тук.
След малко ще видиш защо.
Ако пресметна това за х,
то ще бъде минус
f(x) g(x).
Дотук просто приложих 
дефиницията
за производна. Вместо да я приложа за f(x),
аз я приложих за f(x) по g(x).
Получаваме f(х + h) g(х + h),
минус f(x)g(x),
цялото върху h.

Thai: 
นั่นหมายความว่า ผมจะหาลิมิต
เมื่อ h เข้าใกล้ 0
และตัวส่วน ผมจะมี h
และตัวส่วน ผมจะเขียนเส้นยาวๆ
มันจะเป็นพจน์ตรรกยะที่ใหญ่มาก
ในตัวส่วน ผมจะมี h
แล้วผมจะหาค่านี้ที่ x บวก h
มันจะเท่ากับ f ของ x บวก h
g ของ x บวก h และจากนั้น ผมจะลบ
อันนี้หาค่าที่ f ของ x
โทษที อันนี้หาค่าที่ x
มันจึงเท่ากับ f ของ x คูณ g ของ x
และผมจะมีที่ว่างเกินๆ ตรงนี้
คุณจะเห็นว่าทำไมในไม่ช้า
ถ้าผมแค่ ถ้าผมหาค่านี้ที่ x
อันนี้จะเป็นลบ
f ของ x, g ของ x
ที่ผมทำถึงตอนนี้ ผมแค่ใช้นิยาม
ของอนุพันธ์ แทนที่จะใช้มันกับ f ของ x
ผมใช้มันกับ f ของ x คูณ g ของ x
คุณจึงได้ f ของ x บวก h, g ของ x บวก h
ลบ f ของ x, g ของ x
ทั้งหมดนั้นส่วน h

English: 
that means I'm gonna take the limit
as H approaches zero,
and the denominator I'm gonna have at H,
and the denominator,
I'm gonna write a big,
it's gonna be a big rational expression,
in the denominator I'm gonna have an H.
And then I'm gonna evaluate
this thing at X plus H.
So that's going to be F of X plus H,
G of X plus H and from
that I'm gonna subtract
this thing evaluated F of X.
Or, sorry, this thing evaluated X.
So that's gonna be F of X times G of X.
And I'm gonna put a
big, awkward space here
and you're gonna see why in a second.
So if I just, if I evaluate this at X,
this is gonna be minus
F of X, G of X.
All I did so far is I just
applied the definition
of the derivative, instead
of applying it to F of X,
I applied it to F of X times G of X.
So you have F of X plus H, G of X plus H
minus F of X, G of X,
all of that over H.

Korean: 
미분의 정의를 사용해보겠습니다.
h가 0으로 수렴할 때
분모에는 h가 있고,
분자에는
아주 긴 유리식이네요
분모에는 h가 있습니다.
먼저 x+h에 대한 식을 써보겠습니다.
그래서 저 항들은
f(x+h)*g(x+h)가 됩니다.
그리고 f(x)에 대한
아니, x에 대한 식을 쓸 겁니다.
f(x)*g(x)를 빼게 되겠지요.
그전에 공간을 
조금 비워두고요.
이 공간을 비워둔 이유는
 곧 알게 될겁니다.
이제 x에 대한 식, 즉
f(x)g(x)를
뒤에서 뺄겁니다.
지금까지 한 것은
 미분의 정의를
f(x) 대신
f(x)*g(x)에 
적용 한 것 뿐입니다.
그래서 여기 
f(x+h)*g(x+h)가 있고
f(x)*g(x)를 빼고
분모는 h인 식이 있습니다.

Czech: 
tedy limitu pro 'h' jdoucí k 0,
dělitel bude 'h' a dělenec...
(Napíšu si to velké, půjde
o velký výraz...)
V děliteli budu mít 'h' a pak si
vyčíslím tento součin v bodě (x plus h).
Tedy f (x plus h) g (x plus h),
a od toho odečtu součin v bodě f(x).
Pardon, součin v bodě 'x'.
Bude to tedy f(x) g(x).
Navíc si zde nechám hodně volného místa,
za chvíli uvidíte proč.
Takže pokud si toto vyčíslím v bodě 'x',
bude to minus f(x) g(x).
Zatím jsem pouze aplikoval definici
derivace na f(x) krát g(x) místo f(x).
Tedy je to f (x plus h) g (x plus h) minus 
f(x) g(x), to celé lomeno 'h',

English: 
Limit as H approaches zero.
Now why did I put this
big, awkward space here?
Because just the way I've
written it write now,
it doesn't seem easy to
algebraically manipulate.
I don't know how to evaluate this limit,
there doesn't seem to be
anything obvious to do.
And what I'm about to show you,
I guess you could view it
as a little bit of a trick.
I can't claim that I would
have figured it out on my own.
Maybe eventually if I
were spending hours on it.
I'm assuming somebody was
fumbling with it long enough
that said, "Oh wait, wait.
"Look, if I just add and
subtract at the same term here,
"I can begin to
algebraically manipulate it
"and get it to what we all know
"as the classic product rule."
So what do I add and subtract here?
Well let me give you a clue.
So if we have plus,
actually, let me change this,
minus F of X plus H, G of X,
I can't just subtract, if I subtract it
I've got to add it too, so
I don't change the value
of this expression.

Thai: 
ลิมิตเมื่อ h เข้าใกล้ 0
แล้วทำไมผมถึงใส่ที่ว่างแปลกๆ ตรงนี้ด้วย?
เพราะวิธีที่ผมเขียนตอนนี้
มันใช้พีชคณิตจัดการได้ไม่ง่ายนัก
ผมไม่รู้ว่าจะหาค่าลิมิตนี้อย่างไร
มันดูไม่มีอะไรชัดเจนให้ทำ
และสิ่งที่ผมกำลังจะทำให้คุณดู
ผมว่า คุณจะมองมันเป็นกลเม็ดก็ได้
ผมไม่ได้บอกว่าผมเจอวิธีนี้ด้วยตัวเอง
ถ้าผมใช้เวลาคิดเป็นชั่วโมง ก็อาจจะได้
ผมถือว่ามีคนลองผิดลองถูกมานานพอ
จนบอกว่า โอ้ เดี๋ยว เดี๋ยว
ดูสิ ถ้าฉันบวกและลบเทอมเดียวกันตรงนี้
ฉันจะเริ่มจัดการมันได้ด้วยพีชคณิต
และได้สิ่งที่เราทุกคนรู้จัก
ว่าคือ กฎผลคูณสุดคลาสสิค
แล้วผมบวกลบอะไรตรงนี้?
ขอผมให้คำใบ้หน่อย
ถ้าเรามีบวก
ที่จริง ขอผมเปลี่ยนอันนี้ดีกว่า
ลบ f ของ x บวก h, g ของ x
ผมลบเฉยๆ ไม่ได้ ถ้าผมลบมัน
ผมต้องบวกมันด้วย ผมจะได้ไม่เปลี่ยนค่า
ของพจน์นี้

Korean: 
그리고 h가 0으로 접근할 때의
극한을 찾는 거지요.
자, 왜 이 공간을
비워 두었을까요?
왜냐하면
지금 있는 형태로는
대수적으로 풀어내기가
어렵기 때문입니다.
이 극한값을 찾을
 방법이 없으므로
한눈에 보자마자 
할 수 있는건 별로 없습니다
그리고 제가 지금
 보여드리려는 것은
해결의 지름길이라고 
할 수 있겠네요.
이 방법을 제가 
직접 고안하지는 않았습니다.
만약 몇시간동안 고민을 한다면
 알아 낼 수도 있었겠지요.
제 추측으로는 어떤 사람이 
해결 방법을 오래 고민하다가
"오 잠깐만
같은 항을 더하고 빼면
이걸 대수적으로 풀 수 있고
우리가 흔히 곱셈 법칙이라고
아는 것에 도달할 수 있겠구나!" 
라고 알아냈을 겁니다.
자, 여기서는 무엇을 
더하고 빼야 할까요?
힌트를 주겠습니다.
여기 f(x+h)*g(x)를 더하면
아니,
f(x+h)*g(x) 를 빼고,
여기서 그냥 뺄 수는 없고
식의 값을 바꾸지 않기 위해서
더하기도 해야겠지요.

Czech: 
limita pro 'h' jdoucí k 0.
Teď, proč jsem si tu nechal
tolik prostoru?
Protože v tuhle chvíli se nezdá, 
že by výraz šel snadno upravit,
nevím, jak tuhle limitu vyhodnotit,
není zde žádný zřejmý postup.
To, co vám ukážu, 
je takový malý trik.
Sám bych na něj asi nepřišel,
možná pokud bych na něm strávil hodiny.
Někdo se tím zřejmě zabýval tak dlouho,
až si řekl: „Počkat, pokud si zde
přičtu a odečtu stejný výraz, 
už je možné výraz upravovat
a získat to, co všichni známe jako
klasické součinové pravidlo.“
Tedy co zde přičtu a odečtu?
Trochu vám napovím.
Pokud máme plus...
Nebo to raději změním na
minus f (x plus h) g(x).
Nemůžu to jen odečíst,
když to odečtu,
musím to také přidat, 
abych nezměnil hodnotu výrazu.

Bulgarian: 
Границата при h, клонящо към 0.
Защо оставих това голямо
странно пространство тук?
Защото записано по този начин,
не изглежда да има лесен 
алгебричен начин за преобразуване.
Не знам как да намеря
 тази граница.
Изглежда, че няма нищо очевидно, 
което да направим.
Това, което ще ти покажа,
можеш да го приемеш 
малко като трик.
Не твърдя, че щях да се сетя
 за него сам.
Може би след време, ако се занимавах
с часове с това.
Предполагам, че някой си е играл
 с това достатъчно дълго и
си е казал: "Чакай, чакай!
Ако тук просто събера и извадя
 едно и също нещо,
мога да преобразувам 
алгебрично
и да стигна до това, което
 всички знаем
като класическото правило
за производна на произведение."
Какво трябва да съберем
 и извадим тук?
Нека ти подскажа.
Ако имаме плюс...
Всъщност нека променя това.
Минус f(х + h) g(х).
Не мога просто да извадя. 
Ако го извадя,
трябва и да го прибавя, за да 
не се промени стойността
на този израз.

Thai: 
บวก f ของ x บวก h, g ของ x
ผมยังไม่ได้เปลี่ยนค่า
ผมแค่บวกและลบสิ่งเดียวกัน
แต่ตอนนี้ อันนี้สามารถจัดการได้
โดยใช้พีชคณิตที่น่าสนใจ กลายเป็น
สิ่งที่เราชอบ คือกฎลูกโซ่
และหากคุณรู้สึกอยากลอง
ผมแนะนำให้คุณหยุดวิดีโอนี้
ทำต่อไป ลองสำรวจ
พจน์นี้ต่อไป
ทั้งหมดนี้จะเท่ากับ
มันจะเท่ากับลิมิต
เมื่อ h เข้าใกล้ 0
อย่างแรกที่ผมจะทำคือ
ผมจะดูที่ ผมจะดูที่ส่วนนี้
ส่วนนี้ของพจน์
ลองดู ผมจะ
แยก f ของ x บวก h ออกมา
ถ้าคุณแยก f ของ x บวก h ออกมา
ส่วนนี่ตรงนี้จะเท่ากับ
f ของ x บวก h
f ของ x บวก h
คูณ

Bulgarian: 
Плюс f(х + h) g(x).
Не променяме стойността.
Просто прибавих и извадих 
същото нещо,
но сега нещата могат 
да се преобразуват
по интересни алгебрични начини,
за да стигнем
до обичното ни правило за
производна на произведение.
Във всеки момент, в който се
вдъхновиш,
те насърчавам 
да спреш видеото.
За да продължим, нека 
продължим
да разучаваме още този израз.
Всичко това ще бъде равно
на границата
при h, клонящо към 0...
Първото нещо, което 
ще направя,
е да разгледам
тази част от израза.
По конкретно ще извадя
пред скоби f(х + h).
Ако извадя пред скоби f(х + h),
тази част тук ще бъде
f(х + h) по...

Czech: 
Tedy plus f (x plus h) g(x).
Nyní hodnota zůstala stejná,
pouze jsem přičetl a odečetl stejnou věc,
ale s výraz už lze algebraicky upravovat,
abychom se dostali k tomu,
co všichni milujeme 
na součinovém pravidlu.
Pokud vám během sledování dojde,
jak dál, klidně zastavte video.
Podívejme se blíže na tento výraz.
Celý výraz se bude rovnat limitě
pro 'h' jdoucí k 0.
Takže první věc, na kterou se zaměřím,
je tato část výrazu,
konkrétně si vytknu f (x plus h).
Tedy po vytknutí f (x plus h) bude 
tato část f (x plus h)...
f (x plus h) krát...

English: 
So plus F of X plus H, G of X.
Now I haven't changed the value,
I just added and
subtracted the same thing,
but now this thing can be manipulated
in interesting algebraic ways to get us to
what we all love about the product rule.
And at any point you get inspired,
I encourage you to pause this video.
Well to keep going, let's just keep
exploring this expression.
So all of this is going to be equal to,
it's all going to be equal to the limit
as H approaches zero.
So the first thing I'm gonna do is
I'm gonna look at, I'm
gonna look at this part,
this part of the expression.
And in particular,
let's see, I am going to
factor out an F of X plus H.
So if you factor out an F of X plus H,
this part right over here is going to be
F of X plus H,
F of X plus H,
times

Korean: 
f(x+h)*g(x)를 더합니다.
같은 것을 더하고 뺐으므로
식의 값은 전혀 
변하지 않았지만
이 식은 이제
대수적으로
풀 수 있게 되어서
곱셈 법칙에
 도달할 수 있게 됩니다.
그리고 이 강의를 보다가 
흥미가 생긴다면
영상을 잠깐 멈추고 
직접 생각 해 보기를 바랍니다.
자, 계속해서
이 식에 대해서 알아봅시다.
그래서 이 모든것이 결국
h가 0으로 접근 할 때의
극한이 되겠지요.
자, 제가 가장 먼저 할 것은
이 식의 이 부분을
살펴보겠습니다.
이제 저는
f(x+h)를 공통인수로
 빼 내겠습니다.
f(x+h)를 공통인수로 하면
이 부분이
f(x+h)
f(x+h)
곱하기

Bulgarian: 
Остава ни g(х + h).
Това е в малко по-различен
нюанс на зеленото.
g(х + h), което е това тук,
минус g(x)...
Опа, забравих скобите!
Трябва да е в друг цвят.
Имам нова програма
и ми е малко трудно
да сменям цветовете.
Извинявам се. Това не е лесно
доказателство
и поне трябва да сменям
цветовете гладко.
g(х + h)
минус g(х), което е това тук,
цялото върху това h.
Това е тази част тук,
после тази част тук,
която всъщност 
също е върху h,
затова нека го оградя
 ето така.
Тази част тук

Korean: 
왼쪽 식은
g(x+h)가 남겠지요
이건 저기 있는
g(x+h)
빼기 g(x)
g(x+h)-g(x)
괄호를 잊어버렸습니다.
이런, 색을 잘못 선택했습니다.
새로운 소프트웨어를 
사용하고 있는데
색을 바꾸기가 좀 어렵네요.
이 증명은 한눈에 보이는 
증명이 아니라서
제가 최소한 할 수 있는 것은 
색깔을 잘 사용하는 것 뿐이네요.
자, g(x+h)
빼기 저기 있는 g(x)
분모는 h.
분모는 h.
그래서 여기 이 부분과
이 부분,
그리고 아직도 h가 
분모인 이 부분
이렇게 동그라미로
표시하겠습니다.
자, 여기 이 부분은

Thai: 
คุณจะเหลือ g ของ x บวก h
g ของ มันเป็นสีเขียวต่างออกไปหน่อย
g ของ x บวก h นั่นคืออันนั้นตรงนั้น
ลบ g ของ x
ลบ g ของ x
โอ๊ะ ผมลืมวงเล็บ
โอ๊ะ นั่นมันคนละสี
ผมใช้โปรแกรมใหม่ มันทำให้ผม
เปลี่ยนสียาก
ขอโทษด้วย นี่ไม่ใช่บทพิสูจน์ที่ตรงไปตรงมา
อย่างน้อยผมควรจะเปลี่ยนสีได้เนียนกว่านี้
เอาล่ะ [หัวเราะ] g ของ x บวก h
ลบ g ของ x นั่นคืออันนั้นตรงนั้น
แล้วทั้งหมดนั้นส่วน h นี้
ทั้งหมดนั้นส่วน h
นั่นคือส่วนนี้ตรงนี้
แล้วส่วนนี่ตรงนี้
ส่วนนี่ตรงนี้ ที่จริง มันยังเป็นส่วน h
ขอผมวงกลมมันแบบนี้นะ
ส่วนนี่ตรงนี้

Czech: 
Zůstane zde g (x plus h).
Tedy 'g'... 
(Tohle je trochu jiný odstín zelené...)
g (x plus h), to je zde,
minus g(x).
Jejda, zapomněl jsem závorky.
Špatná barva!
Mám nový program 
a špatně se mi mění barvy.
Omlouvám se, tohle není
jednoduchý důkaz
a to nejmenší, co můžu udělat, 
je měnit rychleji barvy.
Dobře, tedy g (x plus h)
minus g(x), to je zde,
to celé lomeno 'h'.
To celé lomeno 'h',
to je tahle část,
pak tahle část...
Tahle část, je to pořád lomeno 'h',
takže to raději zakroužkuju takhle.

English: 
you're going to be left
with G of X plus H.
G of, that's a slightly
different shade of green,
G of X plus H, that's that there,
minus G of X,
minus G of X,
oops, I forgot the parentheses.
Oops, it's a different color.
I got a new software program
and it's making it hard
for me to change colors.
My apologies, this is not
a straightforward proof
and the least I could do is
change colors more smoothly.
Alright, (laughing) G of X plus H
minus G of X, that's that
one right over there,
and then all of that over this H.
All of that over H.
So that's this part here
and then this part over here
this part over here, and
actually it's still over H,
so let me actually circle it like this.
So this part over here

Thai: 
ผมเขียนได้เป็น
เราจะได้บวก --
ตรงนี้ขอผม
ขอผมแยก g ของ x ตรงนี้ออกมา
บวก g ของ x
บวก g ของ x คูณ f ของ x บวก h นี่
คูณ f ของ x บวก h
ลบ f ของ x นี้
ลบ f ของ x นั้น
ทั้งหมดนั้นส่วน h
ทั้งหมดนั้นส่วน h
ตอนนี้เรารู้จากสมบัติลิมิตของเรา
ลิมิตของทั้งหมดนี้
จะเท่ากับ
ลิมิตของอันนี้เมื่อ h เข้าใกล้ 0
บวกลิมิตของอันนี้เมื่อ h เข้าใกล้ 0
แล้วลิมิตของผลคูณ
จะเท่ากับผลคูณลิมิต
ถ้าผมใช้สมบัติลิมิตทั้งสองนั้น
ผมจะเขียนทั้งหมดนี้ใหม่ได้ว่าลิมิต
ขอผมหาที่หน่อย

Czech: 
Tedy tuhle část můžu přepsat jako...
Zde bude plus...
Nebo já tady raději vytknu g(x).
Tedy plus g(x)...
Plus g(x) krát tohle f (x plus h),
krát f (x plus h)...
Minus tohle f(x).
Minus to f(x),
To celé lomeno 'h'.
A z vlastnosti limit víme,
že limita celého tohoto výrazu
bude stejná jako limita tohoto
pro 'h' jdoucí k 0
plus limita tohoto pro 'h' 
jdoucí k 0.
A limita tohoto součinu bude stejná
jako součin limit.
Tedy pokud využiju obě tyto vlastnosti,
můžu celý tento výraz přepsat jako limitu...
Udělám si tady nějaký základ...

Bulgarian: 
мога да запиша като:
Ще имаме плюс...
Нека всъщност
изкарам пред скоби g(x).
Плюс g(x)
по това f(х + h)
минус това f(x),
цялото върху h.
От свойствата на границите
знаем, че
границата на всичко това
ще бъде същото нещо
като границата на това при h,
 клонящо към 0,
плюс границата на това при h, 
клонящо към 0.
А после границата на 
произведението
ще бъде същото нещо като
произведението от границите.
Ако използвам двете свойства
на границите,
мога да запиша цялото това
като границата...
Нека си направя малко
пространство.

English: 
I can write as.
So then we're going to have plus...
actually here let me,
let me factor out a G of X here.
So plus G of X
plus G of X times this F of X plus H.
Times F of X plus H
minus this F of X.
Minus that F of X.
All of that over H.
All of that over H.
Now we know from our limit properties,
the limit of all of this business,
well that's just going
to be the same thing
as the limit of this as H approaches zero
plus the limit of this
as H approaches zero.
And then the limit of the product
is going to be the same thing
as the product of the limits.
So if I used both of
those limit properties,
I can rewrite this whole
thing as the limit,
let me give myself some real estate,

Korean: 
이렇게 나타낼 수 있습니다.
먼저
이 부분의 공통인수인
g(x)를 선택해
더하기 g(x)
곱하기 여기 있는 f(x+h)
곱하기 여기 있는 f(x+h)
빼기 이 부분의 f(x)
빼기 f(x).
그리고 이 식의 분모는 h.
분모는 h.
극한의 성질을 사용하면
여기 있는 것들의 극한은
그저
이것이 h가 0에 
접근 할 때의 극한 더하기
여기 있는 식의 h가 0에 
접근 할 때의 극한일 뿐입니다.
그리고 곱한 것의 극한은
극한값들을 곱한 것과 같습니다.
그래서 앞에서 말한 두개의 
극한의 성질을 이용하면
이 모든것을
(잠시 공간을 좀 챙겨두고요)

Bulgarian: 
Границата при h, клонящо към 0, 
на f(х + h)
по границата
при h, клонящо към 0, цялото това
g(х + h) минус g(x),
цялото върху h.
Виждаш накъде отивам.
Много вълнуващо.
Плюс границата...
Нека го запиша малко
по-ясно.
Плюс границата при h, 
клонящо към 0, на g(x),
нашето хубаво кафяво g(x),
по... Сега имаме произведение.
Границата

Korean: 
f(x+h)의 h가 0으로 
접근 할 때의 극한값에
f(x+h)의 h가 0으로 
접근 할 때의 극한값에
곱하기
여기 있는
g(x+h)-g(x)
g(x+h)-g(x)
에 분모 h
이제 여러분은 이 문제의 해법을 
눈치 챘을 수도 있겠네요.
좋습니다.
더하기
더하기
여기에다가 조금 더 
깨끗하게 쓰도록 하죠.
h가 0으로 접근할 때의
 g(x)의 극한값
h가 0으로 접근할 때의
 g(x)의 극한값
곱하기
h가 0으로 접근 할 때의
h가 0으로 접근 할 때의
f(x+h)-f(x)에

Czech: 
Limita pro 'h' jdoucí k 0 f (x plus h),
f (x plus h) krát...
... krát limita pro 'h' jdoucí k 0 
celého tohoto výrazu,
g (x plus h) minus g(x),
minus g(x)...
To celé lomeno 'h'.
Zřejmě už tušíte, kam to směřuje.
Velmi vzrušující.
Plus limita...
Napíšu to trochu čitelněji...
... plus limita pro 'h' jdoucí k 0 
g(x)
(naše pěkně hnědě zbarvené g(x)),
krát (protože tu je součin)
limita...
... limita pro 'h' jdoucí k 0
f (x plus h),

English: 
the limit as H approaches
zero of F of X plus H,
of F of X plus H times,
times the limit
as H approaches zero,
of all of this business,
G of X plus H minus G of X,
minus G of X,
all of that over H,
I think you might see where this is going.
Very exciting.
Plus,
plus the limit,
let me write that a
little bit more clearly.
Plus the limit as H
approaches zero of G of X,
our nice brown colored G of X,
times, now that we have our product here,
the limit,
the limit
as H approaches zero of F of X plus H.

Thai: 
ลิมิตเมื่อ h เข้าใกล้ 0 ของ f ของ x บวก h
ของ f ของ x บวก h คูณ
คูณลิมิต
เมื่อ h เข้าใกล้ 0 ของทั้งหมดนี้
g ของ x บวก h ลบ g ของ x
ลบ g ของ x
ทั้งหมดนั้นส่วน h
คุณคงเห็นแล้วว่าจะเป็นยังไงต่อ
น่าตื่นเต้นมาก
บวก
บวกลิมิต
ขอผมเขียนให้ชัดกว่านี้ห่อย
บวกลิมิตเมื่อ h เข้าใกล้ 0 ของ g ของ x
g ของ x สีน้ำตาลสวย
คูณ ตอนนี้เรามีผลคูณตรงนี้
ลิมิต
ลิมิต
เมื่อ h เข้าใกล้ 0 ของ f ของ x บวก h

Bulgarian: 
при h, клонящо към 0, на f(х + h)
минус f(x),
цялото върху h.
Нека поставя скоби, където
е нужно.
Тук,
тук,
тук,
тук.
Записах просто,
че границата на този сбор,
ще бъде сборът на границите.
Ще бъде границата на това
плюс границата на това.
После границата на 
произведенията ще бъде
същото нещо като 
произведението от границите.
Просто използвах свойствата 
на границите.
Нека сега пресметнем.
Каква е границата...
Ще ги направя в различни
цветове.
Какво е това нещо тук?
Границата при h, клонящо към 0, на f(х + h).
Това ще бъде f(x).
Това е вълнуващата част.
Какво е това?
Границата при h, клонящо към 0, на g(х + h)
минус g(х), върху H.
Това е просто
дефиницията на производната ни.
Това е производната на g.
Това ще бъде
производната на g(x),

Korean: 
f(x+h)-f(x)에
f(x+h)-f(x)에
이 식에
분모를 h로 놓습니다.
잠깐 괄호를 넣도록 하겠습니다.
자,
이것,
이것,
그리고 이것.
여기서 한 것은 고작
더한 식의 극한값은
(그것은 극한값들을 더한 것과 같겠지요)
이것과
이것의 극한값을 더한 것과 같습니다.
그리고 곱한 식의 극한값은
극한값들의 곱과 같게 됩니다.
그냥 극한의 성질만 
사용한 것 뿐입니다.
자, 이제는 풀어 봅시다.
색을 다르게 하고
여기 있는 것의
극한 값은 무엇일까요?
f(x+h)의 h가 0으로 
접근할 때의 극한값 말입니다.
그냥
f(x)가 되겠지요.
자, 이제부터 본론입니다.
여기 이건 뭘까요?
(g(x+h)-g(x))/h 에서
h가 0으로 접근 할 때의
 극한값 말입니다.
그건 그저
미분의 정의일 뿐입니다!
그것은 g(x)를 미분한 거지요.
그래서 여기 이건
g(x)의 미분식인

Thai: 
ของ f ของ x บวก h ลบ f ของ x
ลบ f ของ x
ทั้งหมดนั้น
ทั้งหมดนั้นส่วน h
ขอผมใส่วงเล็บตรงนี้ให้เหมาะสม
แล้ว
นั่น
นั่น
นั่น
ทั้งหมดที่ผมทำตรงนี้ ลิมิต
ลิมิตของผลบวก
มันจะเท่ากับผลบวกของลิมิต
มันจะเท่ากับลิมิตของอันนี้
บวกลิมิตของอันนั้น
แล้วลิมิตของผลคูณจะ
เท่ากับผลคูณของลิมิต
ผมแค่ใช้สมบัติลิมิตพวกนั้นตรงนั้น
แต่ตอนนี้ ลองหาค่าพวกมันดู
ลิมิตเป็นเท่าใด
ผมจะใช้คนละสีนะ
อันนี้ตรงนี้คืออะไร?
ลิมิตเมื่อ h เข้าใกล้ 0 ของ f ของ x บวก h
มันจะเท่ากับ
f ของ x
ทีนี้ อันนี้น่าตื่นเต้น
อันนี้คืออะไร?
ลิมิตเมื่อ h เข้าใกล้ 0 ของ g ของ x บวก h
ลบ g ของ x ส่วน h
มันก็คือ
นิยามอนุพันธ์ของเรา
นั่นคืออนุพันธ์ของ g
อันนี้จะเท่ากับ
อันนี้จะเท่ากับอนุพันธ์ของ g ของ x

English: 
Of F of X plus H minus F of X,
minus F of X,
all of that,
all of that over H.
And let me put the parentheses
where they're appropriate.
So that,
that,
that,
that.
And all I did here, the limit,
the limit of this sum,
that's gonna be the sum of the limits,
that's gonna be the limit of this
plus the limit of that,
and then the limit of
the products is gonna be
the same thing as the
product of the limits.
So I just used those
limit properties here.
But now let's evaluate them.
What's the limit,
and I'll do them in different colors,
what's this thing right over here?
The limit is H approaches
zero of F of X plus H.
Well that's just going to be
F of X.
Now, this is the exciting part,
what is this?
The limit is H approaches
zero of G of X plus H
minus G of X over H.
Well that's just our,
that's the definition of our derivative.
That's the derivative of G.
So this is going to be,
this is going to be the
derivative of G of X,

Czech: 
f (x plus h) minus f(x),
minus f(x)...
To celé lomeno 'h'.
Pro čitelnost přidám pár závorek tady,
tady, tady a tady.
Vše, co jsem udělal, 
je, že jsem přepsal
limitu tohoto součtu 
na součet těchto limit,
tedy limita tohohle plus limita tohohle,
a poté jsem si přepsal limitu součinu
jako součin limit.
Tedy jsem využil těchto vlastností limit.
Tak si je vyhodnoťme.
Co je limitou...
(Udělám to různými barvami.)
Co je limitou tohoto výrazu?
Jde o limitu pro 'h' jdoucí k 0 
f (x plus h).
No to bude jednoduše f(x).
Teď k té zajímavější části,
co je tohle?
Limita pro 'h' jdoucí k 0 g (x plus h)
minus g(x), to celé lomeno 'h'.
To je prostá definice derivace,
jde o derivaci g.
Tohle bude derivace g(x),

Bulgarian: 
което ще бъде g прим от х.
g'(х).
Умножаваме тези двете
и после имаме плюс...
Каква е границата при h,
 клонящо към 0, на g(x)?
Дори няма h тук,
следователно това 
ще бъде просто g(x).
Следователно плюс g(x)
по границата...
Да видим, това е кафяво,
а последното ще направя
 в жълто.
По границата при h, 
клонящо към 0...
Вече сме много близко.
Барабанният звук трябва
 да започва.
Границата при h, клонящо към 0, 
на f(х + h)
минус f(x), върху h.
Това е дефиницията на производната
f(x).
Това е f прим от х.
По f'(х).
Готово.
Производната на f(x) по g(x)
е това.
Ако исках да го запиша в малко
по-сбита форма,
то ще е равно на f(x)

Korean: 
g'(x)가 됩니다.
g'(x)
그래서 이 두 항을 곱하고
다음에는 덧셈 부호가 있고,
g(x)에서 h가 0으로 수렴할 때의 극한값은 뭘까요?
h가 있지도 않으므로
그냥 g(x)로 남습니다.
자, g(x)
곱하기
이게 갈색이니까
이건 노란색으로 하겠습니다
곱하기
자 이제 정말 얼마 
남지 않았습니다.
마지막까지 힘내봅시다.
(f(x+h)-f(x))/h 에서
h가 0으로 접근 할 때의
 극한값이 됩니다.
그런데 이 식은 f(x)의
미분의 정의로군요!
이건 f'(x) 입니다.
f'(x)를 곱합니다.
자, 끝났군요.
f(x)*g(x)의 미분은 이겁니다.
조금 더 간략하게 쓰면
이건
f(x)*g'(x)
f(x)*g'(x)

English: 
which is going to be G prime of X.
G prime of X.
So you're multiplying these two
and then you're going to have plus,
what's the limit of H
approaches zero of G of X?
Well there's not even any H in here,
so this is just going to be G of X.
So plus G of X
times the limit,
so let's see, this one is in brown,
and the last one I'll do in yellow.
Times the limit as H approaches zero,
and we're getting very close,
the drum roll should be starting,
limit is H approaches zero of F of X
plus H minus F of X over H.
Well that's the definition
of the derivative
of F of X.
This is F prime of X.
Times F prime of X.
So there you have it.
The derivative of F of
X times G of X is this.
And if I wanted to write
it a little bit more
condensed form, it is equal to,
it is equal to F of X
times the derivative
of G with respect to X

Thai: 
ซึ่งก็คือ g ไพรม์ของ x
g ไพรม์ของ x
คุณจะคูณสองตัวนี้
แล้วคุณจะได้บวก
ลิมิตเมื่อ h เข้าใกล้ 0 ของ g ของ x เป็นเท่าใด?
มันไม่มี h ในนี้เลยด้วยซ้ำ
มันจึงเป็น g ของ x
บวก g ของ x
คูณลิมิต
ลองดู อันนี้สีน้ำตาล
และอันสุดท้าย ผมจะใช้สีเหลือง
คูณลิมิตเมื่อ h เข้าใกล้ 0
เราใกล้มากแล้ว
เริ่มรัวกลองได้แล้ว
ลิมิตเมื่อ h เข้าใกล้ 0 ของ f ของ x
บวก h ลบ f ของ x ส่วน h
นั่นคือนิยามของอนุพันธ์
ของ f ของ x
นี่คือ f ไพรม์ของ x
คูณ f ไพรม์ของ x
คูณก็ได้แแล้ว
อนุพันธ์ของ f ของ x คูณ g ของ x คืออันนี้
และถ้าผมอยากเขียนมันให้
แน่นขึ้นหน่อย มันเท่ากับ
มันเท่ากับ f ของ x
คูณอนุพันธ์ของ g เทียบกับ x

Czech: 
tedy g'(x).
Máme tedy součin těchto dvou výrazů
a k tomu je pak plus...
Co je limitou pro 'h' jdoucí k 0 g(x)?
Zde ani není žádné 'h', 
bude to tedy g(x).
Takže plus g(x) krát limita...
(Tahle byla hnědou, ta poslední žlutou...)
... krát limita pro 'h' jdoucí k 0,
a už se blížíme ke konci, 
fanfáry prosím,
limita pro 'h' jdoucí k 0 
f (x plus h) minus f(x),
to celé lomeno 'h'.
To je definice derivace f(x).
Tedy f'(x).
Hotovo. Derivace f(x) krát g(x) je toto.
Pokud bych to tedy chtěl zapsat stručněji,

Czech: 
jde o f(x) krát derivace g(x) podle 'x'
plus g(x) krát derivace f(x) podle 'x'.
Jinými slovy, tohle je první funkce krát
derivace druhé funkce
plus druhá funkce krát derivace první.
Tohle je důkaz, nebo jeden z důkazů
součinového pravidla.

Thai: 
คูณอนุพันธ์ของ g เทียบกับ x
บวก g ของ x
บวก g ของ x คูณอนุพันธ์ของ f เทียบกับ x
f เทียบกับ x
หรือวิธีคิดอีกอย่างคือว่า
นี่คือฟังก์ชันแรก คูณอนุพันธ์
ของฟังก์ชันที่สอง บวกฟังก์ชันที่สอง
คูณอนุพันธ์ของฟังก์ชันแรก
นี่คือบทพิสูจน์ บทพิสูจน์อันหนึ่ง
มันมีวิธีพิสูจน์อื่นสำหรับกฎผลคูณอีก

Bulgarian: 
по производната g'(х)
плюс g(x)
по производната f'(х).
Разгледано по друг начин,
това е първата функция
по производната
на втората плюс 
втората функция
по производната на първата.
Това е доказателството...
или поне едно от доказателствата,
всъщност има и други... на правилото
 за производна на произведение.

English: 
times the derivative
of G with respect to X
plus G of X,
plus G of X times the derivative
of F with respect to X.
F with respect to X.
Or another way to think about it,
this is the first function
times the derivative
of the second plus the second function
times the derivative of the first.
This is the proof, or a proof,
there's actually others
of the product rule.

Korean: 
f(x)*g'(x)
더하기
g(x)*f'(x)
가 됩니다.
조금 다른 시각에서 보면
앞의 함수 곱하기 
뒤의 함수의 미분
더하기
뒤의 함수 곱하기 앞의 함수의 미분
이라고 할 수 있습니다.
이건 곱셈 법칙의 여러 증명 중
하나입니다.
