
English: 
What I want to do in this video
is start with the abstract--
actually, let me call it
formula for the chain rule,
and then learn to apply it
in the concrete setting.
So let's start off
with some function,
some expression that
could be expressed
as the composition
of two functions.
So it can be expressed
as f of g of x.
So it's a function that can
be expressed as a composition
or expression that
can be expressed
as a composition
of two functions.
Let me get that same color.
I want the colors
to be accurate.
And my goal is to take the
derivative of this business,
the derivative
with respect to x.
And what the chain
rule tells us is
that this is going to be
equal to the derivative
of the outer function with
respect to the inner function.
And we can write that as f
prime of not x, but f prime
of g of x, of the
inner function.

Korean: 
 
이번 강의에서
할 내용은
연쇄법칙에 대한
공식입니다
그리고 이것을 구체적인
문제에 적용해봅시다
먼저 임의의 함수에서부터
시작할텐데
이것은 두 함수로
이루어져 있으며
f(g(x))로
표현할 수 있습니다
f(g(x))로
표현할 수 있습니다
이것은 두 개의 함수로
이루어진
합성함수입니다
색깔을 구분지어서
표현해 봅시다
이제 우리의 목표는
이 함수를 x에 대해
미분을 취하는 것입니다
연쇄법칙에 의하면
이것은 바깥쪽 함수를
안쪽 함수에 대해
미분한 것인
f '(x)가 아니라
f '(g(x))와

Czech: 
V tomto videu bych rád začal s obecným
vzorcem pro derivaci složené funkce
a pak se podíváme, jak ho
použít na konkrétní příklad.
Mějme tedy nějakou
funkci nebo výraz,
který můžeme vyjádřit
jako složení dvou funkcí,
tedy jako
f v bodě g(x).
Máme tedy funkci nebo výraz, který lze
vyjádřit jako složení dvou funkcí.
Udělám to stejnou barvou,
aby to bylo přehledné.
Naším cílem je teď
tohle zderivovat,
tedy spočítat
derivaci podle x.
Pravidlo o derivaci
složené funkce nám říká,
že toto se rovná derivaci
vnější funkce podle vnitřní funkce,
což můžeme napsat jako f s čárkou,
ale ne v bodě x, nýbrž v bodě g(x),
což je
vnitřní funkce.

Thai: 
 
สิ่งที่ผมอยากทำในวิดีโอนี้คือเริ่มด้วย --
ขอผมเรียกมันว่าสูตรสำหรับกฎลูกโซ่ดีกว่า
แล้วเรียนวิธีใช้มันจริงๆ
ลองเริ่มต้นด้วยฟังก์ชัน
พจน์ที่สามารถเขียน
เป็นฟังก์ชันประกอบจากฟังก์ชันสองตัว
มันเขียนได้เป็น f ของ g ของ x
 
มันเป็นฟังก์ชันที่เขียนได้เป็นการประกอบ
หรือพจน์ที่สามารถเขียน
ได้เป็นฟังก์ชันประกอบของฟังก์ชันสองตัว
ขอผมใช้สีเดิมนะ
ผมอยากให้สีถูกต้อง
เป้าหมายของผมคือหาอนุพันธ์ของตัวนี้
อนุพันธ์เทียบกับ x
และสิ่งที่กฎลูกโซ่บอกเราคือว่า
อันนี้จะเท่ากับอนุพันธ์
ของฟังก์ชันนอกเทียบกับฟังก์ชันตัวใน
และเราเขียนได้ว่า f ไพรม์ของ ไม่ใช่ x
แต่เป็น f ไพรม์
ของ g ของ x, ของฟังก์ชันใน

Portuguese: 
O que quero fazer neste vídeo
é começar com a abstrata--
deixe-me chamá-la de fórmula
para a regra da cadeia,
e, em seguida, aprender a aplicá-la
na configuração prática.
Então, vamos começar
com alguma função,
uma expressão que
pode ser expressa
como a composição de duas funções.
Então, pode ser expressa
como f de g de x.
É uma função que pode ser
expressa como a composição
ou expressão que
pode ser expressa
como a composição
de duas funções.
Deixe-me usar aquela mesma cor.
Quero que as cores
sejam exatas.
E meu objetivo é calcular a
derivada deste negócio,
a derivada em relação a x.
E o que a cadeia
regra nos diz é
que isto será igual a derivada
da função externa com
relação à função interna.
E podemos escrever isto como f
linha de, não x, mas f linha
de g de x, da
função interna.

Bulgarian: 
 
В това видео искам да започна 
с абстрактното...
Всъщност нека го нарека 
верижното правило за диференциране
и после ще се научим да го прилагаме
 в конкретни случаи.
Да започнем с някаква функция,
някакъв израз, който
 може да се изрази
като сложна функция 
от две функции.
Може да се изрази като
 f от g от х.
Това е функция, която може 
да се изрази като сложна функция,
израз, който може да се изрази
като съставен от две функции.
Нека взема същия цвят.
Искам цветовете да са точни.
Целта ми е да сметна
 производната на това нещо,
производната спрямо х.
Верижното правило ни казва, че
това ще бъде равно на 
производната
на външната функция спрямо
вътрешната функция...
Можем да запишем това не като
f прим от х, а като f прим от g(x),
което е вътрешната функция.

Bulgarian: 
f прим от g(x) по производната
на вътрешната функция
 спрямо х.
Това може да изглежда много
 абстрактно и математическо.
Как всъщност да го приложим?
Нека пробваме с реален пример.
Да кажем, че искаме 
да сметнем производната
на корен квадратен от
 3х^2 – х.
Как можем да дефинираме f и g,
за да може това наистина да е
сложна функция от f(x) и g(x).
Можем да дефинираме f(x).
Ако дефинираме f(x) 
равно на корен квадратен от х,
а g(x) да е равно на
  3х^2 – х,
тогава какво ще е f от g(x)?

Korean: 
f '(g(x))와
안쪽 함수를 x에 대해
미분한 것의
곱으로 쓸 수 있습니다
이것은 매우 추상적이고
수학적으로 보입니다
연쇄법칙을 어떻게
적용할 수 있을까요?
실제 문제를 
풀어봅시다
√3x²-x에 미분을
취한다고 가정해 봅시다
이 때 어떻게 함수
f와 g를 정의내릴 수 있을까요?
이 함수가 정말로
f(x)와 g(x)로 이루어진
합성함수일까요?
f(x)는 √x로 정의하고
f(x)는 √x로 정의하고
g(x)는 3x²-x로 정의한다면
f(g(x))는 무엇이 될까요?

English: 
f prime of g of x
times the derivative
of the inner function
with respect to x.
Now this might seem all
very abstract and math-y.
How do you actually apply it?
Well, let's try it
with a real example.
Let's say we were trying
to take the derivative
of the square root of
3x squared minus x.
So how could we
define an f and a g
so this really is
the composition
of f of x and g of x?
Well, we could define f of x.
If we defined f of x as being
equal to the square root of x,
and if we defined g of x as
being equal to 3x squared
minus x, then what
is f of g of x?

Portuguese: 
f linha de g de x
vezes a derivada
da função interna
em relação a x.
Agora isto pode parecer muito
abstrato e matemático.
Como você realmente a aplica?
Bem, vamos tentar
com um exemplo real.
Vamos dizer que estávamos
tentando calcular a derivada
da raiz quadrada de
3x ao quadrado menos x.
Então, como poderíamos
definir um f e um g.
para que isto seja a composição
de f de x e g de x?
Bem, poderíamos definir f de x.
Se definirmos f de x como sendo
igual à raiz quadrada de x,
e se definirmos g de x como
sendo igual a 3x quadrado
menos x, então o
que é f de g de x.

Thai: 
f ไพรม์ของ g ของ x คูณอนุพันธ์
ของฟังก์ชันในเทียบกับ x
 
ทีนี้ มันอาจฟังดูเป็นนามธรรมและ
เป็นคณิตศาสตร์มาก
คุณจะนำมันไปใช้ได้อย่างไร?
ลองดูตัวอย่างจริงกัน
สมมุติว่าเราพยายามหาอนุพันธ์
ของรากที่สองของ 3x กำลังสองลบ x
เราจะนิยาม f กับ g อย่างไร
นี่ก็แค่การประกอบ
f ของ x กับ g ของ x
เรานิยาม f ของ x ได้
ถ้าเรานิยาม f ของ x ว่าเท่ากับรากที่สองของ x
และเรานิยาม g ของ x ว่าเท่ากับ 3x กำลังสอง
ลบ x แล้ว f ของ g ของ x คืออะไร?

Czech: 
f s čárkou v bodě g(x) krát
derivace vnitřní funkce podle x.
Tohle možná vypadá
hodně abstraktně a formálně,
takže jak tenhle
vzorec použít?
Zkusme si to
na nějakém příkladu.
Řekněme, že hledáme derivaci druhé
odmocniny z (3 krát x na druhou minus x).
Jak zadefinovat funkce f a g tak,
aby tohle bylo složením f(x) a g(x)?
f(x) můžeme zadefinovat
jako druhou odmocninu z ‚x‘,
a když g(x) zadefinujeme jako
3 krát x na druhou minus x,
tak jak bude vypadat
f v bodě g(x)?

Portuguese: 
Bem, f de g de x será igual a--
tentarei manter todas
as cores exatas,
espero que isto ajude
na compreensão.
f de g de x é igual a-- em todos
os lugares onde você vê o x,
substitua com o g de x--
a principal raiz de g de x,
que é igual à raiz
principal de-- nós
definimos g de x bem aqui--
3x ao quadrado menos x.
Portanto, esta coisa bem
aqui é exatamente
f de g de x se definirmos
f de x desta maneira
e g de x desta maneira.
Justo.
Vamos aplicar
a regra da cadeia.
f linha de g de x
será igual a que,
a derivada de f
em relação a g?
Bem, o que é f linha de x?
f linha de x é igual a--
esta é a mesma coisa
que x à potência 1/2, portanto,
podemos aplicar a regra da potência.
Então,será 1/2
vezes x elevado a--

English: 
Well, f of g of x is
going to be equal to-- I'm
going to try to keep
all the colors accurate,
hopefully it'll help
for the understanding.
f of g of x is equal to--
where everywhere you see the x,
you replace with the g of x--
the principal root of g of x,
which is equal to the
principal root of-- we
defined g of x right over
here-- 3x squared minus x.
So this thing right
over here is exactly
f of g of x if we define
f of x in this way
and g of x in this way.
Fair enough.
So let's apply the chain rule.
What is f prime of g of
x going to be equal to,
the derivative of f
with respect to g?
Well, what's f prime of x?
f prime of x is equal to--
this is the same thing
as x to the 1/2 power, so we
can just apply the power rule.
So it's going to be
1/2 times x to the--

Thai: 
f ของ g ของ x จะเท่ากับ -- ผม
จะพยายามใช้สีให้ถูกต้อง
หวังว่ามันคงช่วยให้คุณเข้าใจขึ้น
f ของ g ของ x เท่ากับ -- ทุกที่ที่คุณเห็น x
คุณแทนที่มันด้วย g ของ x -- รากที่เป็นบวก
ของ g ของ x
ซึ่งเท่ากับรากที่เป็นบวกของ -- เรา
นิยาม g ของ x ตรงนี้ -- 3x กำลังสองลบ x
 
ค่านี่ตรงนี้ก็คือ
f ของ g ของ x ถ้าเรานิยาม f ของ x แบบนี้
และ g ของ x แบบนี้
ใช้ได้
ลองใช้กฎลูกโซ่กัน
f ไพรม์ของ g ของ x จะเท่ากับอะไร?
อนุพันธ์ของ f เทียบกับ g คืออะไร?
f ไพรม์ของ x คืออะไร?
f ไพรม์ของ x เท่ากับ -- อันนี้เท่ากับ
x กำลัง 1/2 เราจึงใช้กฎยกกำลังได้
มันจะเท่ากับ 1/2 คูณ x กำลัง --

Korean: 
f(g(x))는
이해를 돕기 위해
색깔을 구분 지어서
나타내면
f(g(x))는
f(x)의 x자리에
g(x)를 치환한 것이므로
결과적으로
√3x²-x가 됩니다
√3x²-x가 됩니다
따라서 이런 식으로
f(x)와 g(x)를 정의하면
f(g(x))를 우리가
원하는 함수로 만들 수 있습니다
 
이제 연쇄법칙을
적용해봅시다
이 문제에서
f를 g에 대해 미분한
f '(g(x))와
f '(x)는 무엇이 될까요?
√x는 x½과 동일하므로
다항식의 미분법칙을
사용하면
1/2 곱하기

Czech: 
f v bodě g(x)
se rovná...
Snažím se to psát stále stejnými barvami,
snad vám to pomůže tomu lépe porozumět.
f v bodě g(x)
se rovná...
Kdekoliv vidíme x, musíme
místo toho napsat g(x),
takže to bude
druhá odmocnina z g(x),
což se rovná
druhé odmocnině z...
g(x) máme
definováno zde.
...z 3 krát x na
druhou minus x.
Tento výraz je tedy přesně
roven f v bodě g(x),
pokud si f(x) a g(x)
zadefinujeme takto.
To bychom měli, nyní použijme
vzorec pro derivaci složené funkce.
Čemu se rovná
f s čárkou v bodě g(x)?
Tedy derivace f podle g.
Jak vypadá
f s čárkou v bodě x?
f s čárkou
v bodě x se rovná...
Tohle je totéž jako
x na jednu polovinu,
takže použijeme
vzorec pro derivaci mocniny.
Bude to (1 lomeno 2)
krát x na...

Bulgarian: 
f от g(x) ще бъде...
Ще се опитам да запазя
 точните цветове,
надявам се, че ще ти помогне
 с разбирането.
f от g(x) е равно на... 
Навсякъде където видим х,
ще заместваме с g(x)...
Корен квадратен от g(x),
който е равен на 
корен квадратен от...
Дефинирахме g(x) тук: 
3х^2 – х.
Следователно това нещо 
тук ще е точно
f от g(x), ако дефинираме 
f(x) по този начин,
а g(x) по този.
Дотук добре.
Нека приложим 
верижното правило.
На какво ще е равно
 f прим от g(x),
производната на f спрямо g?
Колко е f прим от х?
f прим от х е равно на... 
Това е същото нещо като
х на степен 1/2, затова можем 
да приложим правилото за производна от степен.
Ще получим 1/2 по х на степен...

Portuguese: 
e, em seguida, basta tirar um
do expoente, 1/2 menos 1
é menos 1/2.
E então o que é f
linha de g de x?
Em qualquer parte da 
derivada que vemos um x,
podemos substituí-lo por g de x.
Portanto, será 1/2
vezes-- ao invés
de x elevado a menos 1/2, podemos
escrever g de x elevado a 1/2.
E isto será igual a-- deixe-me
escrever isto aqui.
Será igual a 1/2 vezes
todo este negócio
elevado a menos 1/2.
3x ao quadrado menos x,
que é exatamente o que nós
precisamos resolver bem
aqui. f linha de g de x
é igual a isto.
Então esta parte aqui eu vou--
deixe-me enquadrá-lo em verde.
O que estamos tentando resolver aqui,

Korean: 
x의 1/2에서 1을 뺀
-1/2제곱이 됩니다
그러면 f '(g(x))는
무엇이 될까요?
그러면 f '(g(x))는
무엇이 될까요?
f '(x)의 x 자리에
g(x)를 치환한 것이므로
1/2 곱하기
x의 -1/2 제곱 대신
g(x)의 -1/2 제곱이 됩니다
따라서
이쪽에 적어보면
1/2 곱하기
3x²-x의 -1/2제곱이 됩니다
3x²-x의 -1/2제곱이 됩니다
이렇게 문제에 필요한
f '(g(x))를 구했습니다
지금까지 구한
f '(g(x))를
초록색 박스로 나타내면

English: 
and then we just take 1 away
from the exponent, 1/2 minus 1
is negative 1/2.
And so what is f
prime of g of x?
Well, wherever in the
derivative we saw an x,
we can replace it with a g of x.
So it's going to be
1/2 times-- instead
of an x to the negative 1/2, we
can write a g of x to the 1/2.
And this is just going
to be equal to-- let
me write it right over here.
It's going to be
equal to 1/2 times
all of this business to
the negative 1/2 power.
So 3x squared minus x,
which is exactly what we
need to solve right over
here. f prime of g of x
is equal to this.
So this part right
over here I will--
let me square it off in green.
What we're trying to
solve right over here,

Thai: 
แล้วเราลบ 1 ออกจากเลขชี้กำลัง, 1/2 ลบ 1
คือลบ 1/2
แล้ว f ไพรม์ของ g ของ x เป็นเท่าใด?
 
เมื่อใดก็ตามที่เราเห็น x ในอนุพันธ์
เราจะแทนที่มันด้วย g ของ x
มันจะเท่ากับ 1/2 คูณ -- แทนที่
จะเป็น x กำลังลบ 1/2 เราเขียนว่า 
g ของ x กำลังลบ 1/2 ได้
อันนี้จึงเท่ากับ -- ขอ
ผมเขียนตรงนี้นะ
มันจะเท่ากับ 1/2 คูณ
ทั้งหมดนี้ยกกำลังลบ 1/2
3x กำลังสองลบ x ซึ่งก็คือสิ่งที่เรา
ต้องแก้ตรงนี้พอดี f ไพรม์ของ g ของ x
เท่ากับอันนี้
อันนี้ตรงนี้ ผมจะ --
ขอผมเขียนด้วยสีเขียวนะ
สิ่งที่เราพยายมแก้ตรงนี้

Czech: 
Exponent musíme
zmenšit o 1.
(1 lomeno 2) minus 1
je minus (1 lomeno 2).
Čemu se tedy rovná
f s čárkou v bodě g(x)?
Kdekoliv v této derivaci vidíme x,
musíme místo toho napsat g(x),
takže to bude
(1 lomeno 2) krát...
Místo x na minus (1 lomeno 2)
musíme napsat g(x) na minus (1 lomeno 2),
což se rovná...
Napíšu to sem.
Což se rovná (1 lomeno 2) krát tento
výraz umocněný na minus (1 lomeno 2).
Tedy 3 krát x na
druhou minus x.
Teď máme přesně to,
co jsme potřebovali znát zde.
f s čárkou
v bodě g(x) je tohle.
Tato část...
Vyznačím ji zeleně.

Bulgarian: 
Тогава изваждаме 1 от степента.
1/2 минус 1 е –1/2.
Следователно колко е 
f прим от g(x)?
Където видим в 
производната х,
можем да го заменим с g(x).
Ще бъде 1/2 по... Вместо х
на степен 1/2, можем 
да запишем g(х) на степен 1/2.
Това просто ще бъде
 равно на...
Нека го запиша тук.
Ще бъде равно на 1/2 по
всичко това на степен –1/2.
3х^2 – х, което е точно това,
което ни трябваше, 
за да решим това тук.
f прим от g(x) е равно 
на това.
Тази част тук ще...
Нека го оградя в зелено.
Опитваме се да решим тук

Czech: 
To, co jsme tady potřebovali znát,
tedy f s čárkou v bodě g(x),
se rovná
tomuto výrazu.
Je to derivace vnější funkce
f podle vnitřní funkce.
Tak si to
sem napišme.
Toto se rovná (1 lomeno 2) krát
g(x) na minus (1 lomeno 2),
tedy krát (3 krát x na
druhou minus x).
Tyhle dvě věci jsou si pro námi
definované funkce f(x) a g(x) rovny.
Když se na to podíváme obecně,
tak derivace vnější funkce...
Derivujeme něco
na (1 lomeno 2),
takže derivace toho celého
podle našeho něčeho bude:
(1 lomeno 2) krát to něco
na minus (1 lomeno 2).
To je v zásadě to,
co tu říkáme.
Nyní musíme zderivovat
naše něco podle x.
Toto je derivace
našeho něčeho podle x.
To už bude
přímočařejší.
g s čárkou
v bodě x...
Na oba členy použijeme
derivaci mocniny.
Je to rovno 6 krát x na prvou,
tedy 6 krát x, minus 1.

Portuguese: 
f linha de g de x,
nós já descobrimos
que é exatamente esta coisa aqui.
Assim, a derivada de f da
função externa com relação
a função interna.
Deixe-me escrevê-la.
É igual a 1/2 vezes g de
x elevado a menos 1/2,
vezes 3x ao quadrado menos x.
Isto é exatamente isso com
base em como definimos
f de x e como definimos g de x.
Conceitualmente, se
está só olhando
isto, a derivada
da coisa exterior,
você está elevando algo à potência 1/2.
Assim, a derivada
dessa coisa toda
com relação a seu
algo será 1/2 vezes
aquele algo elevado a menos 1/2.
Isto é essencialmente
o que estamos dizendo.
Agora temos que tomar a
derivada de nosso algo
em relação a x.
E isto é mais direto.
g linha de x-- nós só usamos
a regra de potência para cada
um destes termos-- é
igual a 6x elevado a um,
ou apenas 6x menos 1.

English: 
f prime of g of x,
we've just figured out
is exactly this thing
right over here.
So the derivative of f of the
outer function with respect
to the inner function.
So let me write it.
It is equal to 1/2 times g
of x to the negative 1/2,
times 3x squared minus x.
This is exactly this
based on how we've defined
f of x and how we've
defined g of x.
Conceptually, if
you're just looking
at this, the derivative
of the outer thing,
you're taking something
to the 1/2 power.
So the derivative
of that whole thing
with respect to your something
is going to be 1/2 times
that something to the
negative 1/2 power.
That's essentially
what we're saying.
But now we have to take the
derivative of our something
with respect to x.
And that's more straightforward.
g prime of x-- we just use
the power rule for each
of these terms-- is
equal to 6x to the first,
or just 6x minus 1.

Korean: 
다음과 같습니다
다음과 같습니다
즉, 바깥쪽 함수를
안쪽 함수에 대해 미분한
f '(g(x))는
1/2 곱하기
3x²-x의 -1/2제곱입니다
이것은 정확히
우리가 정의한
f(x)와 g(x)에 기반하여
얻어낸 결과입니다
개념적으로, 이 부분만
본다면
바깥쪽 함수는
어떤 함수에 1/2제곱을
취한 것이므로
전체를 어떤 함수에 대해
미분하면
1/2곱하기
어떤 함수의 
-1/2 제곱이 됩니다
이것이 바로
우리가 말하는 바입니다
이제 우리는
어떤 함수를 x에 대해
미분해야 합니다
어떤 함수를 x에 대해
미분해야 합니다
이것은 더 간단합니다
g '(x)는
다항식의 미분법칙에 의해
6x-1입니다
6x-1입니다

Thai: 
f ไพรม์ของ g ของ x เราเพิ่งหาไป
ว่าเท่ากับสิ่งนี่ตรงนี้พอดี
อนุพันธ์ของ f ของฟังก์ชันนอกเทียบกับ
ฟังก์ชันใน
ขอผมเขียนนะ
มันเท่ากับ 1/2 คูณ g ของ x กำลังลบ 1/2
คูณ 3x กำลังสองลบ x
นี่ก็คืออันนี้ จากที่เรานิยาม
f ของ x และที่เรานิยาม g ของ x
ตามหลักการแล้ว ถ้าคุณแค่ดู
อันนี้ อนุพันธ์ของตัวข้างนอก
คุณกำลังนำอะไรสักอย่างมายกกำลัง 1/2
อนุพันธ์ของทั้งหมดนั้น
เทียบกับอะไรสักอย่างนั้น จะเท่ากับ 1/2 คูณ
อะไรสักอย่างนั้นยกกำลังลบ 1/2
นั่นคือสิ่งที่เรากำลังบอก
แต่ตอนนี้ เราตอ้งหาอนุพันธ์
ของอะไรสักอย่างนั้น
เทียบกับ x
 
และมันตรงไปตรงมากกว่า
g ไพรม์ของ x -- เราแค่ใช้กฎยกกำลังสำหรับ
แต่ละเทอม -- เท่ากับ 6x กำลัง 1
หรือก็แค่ 6x ลบ 1

Bulgarian: 
f прим от g(x), тъкмо
 разбрахме,
че то е точно това нещо тук.
Производната на f от
 външната функция спрямо
вътрешната функция.
Нека го запиша.
Това е равно на 
1/2 по g(x) на степен –1/2
по 3х^2 – х.
Това е точно това, според това
 как сме дефинирали,
f(х) и как сме дефинирали g(x).
Като цяло, ако разглеждаме
само това, производната
 на външното нещо,
ще смятаме нещо 
на степен 1/2.
Следователно производната
 на цялото нещо
спрямо нашето нещо ще бъде
 1/2 по
това нещо на степен –1/2.
По същество това казваме.
Но сега смятаме производната 
на нашето нещо
спрямо х.
Това е по-лесно.
g прим от х... Просто използваме
 правилото за степента за всеки
от тези изрази... Равно е на
 6х на първа
или просто 6х минус 1.

Portuguese: 
Então, esta parte bem
aqui será 6x menos 1.
Só para deixar claro,
isto bem aqui
é isto aqui e nós
estamos multiplicando.
E terminamos.
Aplicamos a regra da potência.
Então, só para
rever, é a derivada
da função externa
com relação a interna.
Então, ao invés de ter 1/2x 
elevado a menos 1/2,
será 1/2 g de x elevado a menos 1/2,
vezes a derivada da função
interna com relação
a x, vezes a derivada
de g com relação
a x, que está bem ali.
Legendado por [Raul Guimaraes]
Revisado por [Sérgio Fleury]

Korean: 
따라서 이 부분이
6x-1이 됩니다
명확하게 나타내기 위해
파란색으로 표현해봅시다
이제 끝입니다
우리가 한 것은
다항식의 미분밖에 없습니다
다시 한번 복습해보면
바깥쪽 함수를
안쪽 함수에 대해 미분한
1/2곱하기
x의 -1/2제곱 대신
1/2 곱하기
g(x)의 -1/2제곱과
안쪽 함수를
x에 대해 미분한
6x-1의 곱이 됩니다
6x-1의 곱이 됩니다
 

English: 
So this part right over here
is just going to be 6x minus 1.
Just to be clear,
this right over here
is this right over here
and we're multiplying.
And we're done.
We have just applied
the power rule.
So just to review,
it's the derivative
of the outer function
with respect to the inner.
So instead of having
1/2x to the negative 1/2,
it's 1/2 g of x to
the negative 1/2,
times the derivative of the
inner function with respect
to x, times the derivative
of g with respect
to x, which is right over there.

Czech: 
Tato část se tedy rovná
6 krát x minus 1.
Aby to bylo jasné,
tak tohle se rovná tomuhle.
Tím násobíme.
A máme hotovo, použili jsme
vzorec pro derivaci složené funkce.
Ještě jednou zopakuji, že to je derivace
vnější funkce podle vnitřní funkce,
takže místo (1 lomeno 2) krát
x na minus (1 lomeno 2) tady bude:
(1 lomeno 2) krát
g(x) na minus (1 lomeno 2),
a vynásobíme tím derivaci
vnitřní funkce podle x,
tedy krát derivace g podle x,
což je tento výraz.

Bulgarian: 
Тази част тук ще бъде 
просто 6х – 1.
За да е ясно, това нещо тук
е това нещо тук 
и го умножаваме.
И сме готови.
Приложихме правилото за 
намиране на производна от степен.
Да направим обобщение. 
Това е производната
на външната функция 
спрямо вътрешната.
Вместо да имаме 
1/2х на степен –1/2,
имаме 1/2g(x) на степен –1/2
по производната на 
вътрешната функция спрямо х,
т.е. по производната на
 g спрямо х,
което е това тук.
 

Thai: 
ส่วนนี่ตรงนี้จะเท่ากับ 6x ลบ 1
เพื่อบอกให้ชัด ค่านี่ตรงรนี้
คืออันนี้ตรงนี้ และเรากำลังคูณอยู่
เราก็เสร็จแล้ว
เราได้ใช้กฎลูกโซ่ไป
เพื่อเป็นการทบทวน มันคืออนุพันธ์
ของฟังก์ชันนอกเทียบกับฟังก์ชันใน
แทนที่จะได้ 1/2 x กำลังลบ 1/2
มันคือ 1/2 g ของ x กำลังลบ 1/2
คูณอนุพันธ์ของฟังก์ชันในเทียบกับ
x คูณอนุพันธ์ของ g เทียบกับ
x ซึ่งก็คือตรงนี้
 
