
English: 
We already know that the
product rule tells us
that if we have the product of
two functions-- so let's say
f of x and g of x--
and we want to take
the derivative of this
business, that this is just
going to be equal
to the derivative
of the first function,
f prime of x, times
the second function, times g
of x, plus the first function,
so not even taking its
derivative, so plus f
of x times the derivative
of the second function.
So two terms, in each term
we take the derivative of one
of the functions and not the
other, and then we switch.
So over here is the
derivative of f, not of g.
Here it's the derivative
of g, not of f.
This is hopefully a
little bit of review.
This is the product rule.
Now what we're
essentially going to do
is reapply the
product rule to do
what many of your calculus books
might call the quotient rule.
I have mixed feelings
about the quotient rule.

Portuguese: 
Já sabemos
a regra do produto
se temos o produto
de duas funções-- digamos
f de x e g de x--
e queremos calcular
a derivada disso,
será igual a derivada
da primeira função, f linha de x, vezes
a segunda função g de x,
mais a primeira função
sem calcular a derivada,
mais f, de x vezes
a derivada da segunda função.
Em dois termos, em cada termo
calculamos a derivada de uma função
e não da outra, e alternamos.
Aqui a derivada é de f, não de g.
E aqui a derivada é de g, não da f.
Fazendo uma revisão.
Esta é a regra do produto.
O que essencialmente iremos fazer
é reaplicar a regra
do produto
o que muitos livros de Cálculo
chamam de regra do quociente.
Eu tenho sensações 
diferentes sobre a regra do quociente.

Korean: 
 
곱의 미분법에 따르면
두 개의 함수
f(x)와 g(x)가 있을 때
도함수를 구하게 되면
첫 번째 함수의 도함수
f'(x)와 두 번째 함수 g(x)를
곱한 것에
첫 번째 함수 f(x) 와
두 번째 함수의 도함수를 곱한 것을
더하면 됩니다
 
그러니까 두 함수 중에
하나만 도함수를 취하고
다른 것은 그 원래 형태로 두고
서로 바꿔서 진행한 후 더하면 됩니다
여기서는 f 함수의 도함수가 있습니다
여기서는 g 함수의 도함수가 있습니다
지금까지는 복습이었습니다
곱의 미분법을 복습한 것입니다
오늘 본격적으로 할 것은
곱의 미분법을
몫의 미분법에 적용시키는 것입니다
저는 몫의 미분법에 대해 
여러가지 생각을 가지고 있습니다

Czech: 
Už víme, jak zní
součinové pravidlo.
Pokud máme součin dvou
funkcí, řekněme f(x) a g(x),
a chceme jej zderivovat, bude
to derivace první funkce, f s čárkou (x),
krát druhá funkce, krát g(x), plus
první funkce, kterou nederivujeme,
takže plus f(x), krát
derivace druhé funkce.
Máme dva výrazy, v každém z nich
derivujeme jednu funkci a druhou ne,
a pak se to prohodí.
Tady máme derivaci
f, ale ne g.
Tady derivaci g, ale ne f.
Tolik k malému opakování
pravidla součinu.
Teď pravidlo součinu využijeme pro to,
čemu učebnice říkají podílové pravidlo.
Mně se to úplně nelíbí.

Thai: 
 
เรารู้แล้วว่า กฎผลคูณบอกเราว่า
ถ้าเรามีผลคูณของฟังก์ชันสองตัว -- สมมุติว่า
f ของ x กับ g ของ x -- และเราอยากหา
อนุพันธ์ของตัวนี้ นี่ก็
จะเท่ากับอนุพันธ์
ของฟังก์ชันแรก คือ f ไพรม์ของ x คูณ
ฟังก์ชันที่สอง คูณ g ของ x บวกฟังก์ชันนี้
ไม่ต้องหาอนุพันธ์ แล้วบวก f
ของ x คูณอนุพันธ์ของฟังก์ชันที่สอง
 
สองเทอม ในแต่ละเทอม เราจะหาอนุพันธ์ของ
ฟังก์ชันหนึ่ง แต่ไม่หาอีกฟังก์ชัน แล้วเราก็สลับ
ตรงนี้คืออนุพันธ์ของ f, ไม่ใช่ g
ตรงนี้คืออนุพันธ์ของ g, ไม่ใช่ f
นี่เป็นเพียงการทบทวน
นี่คือกฎผลคูณ
ทีนี้ สิ่งที่เราจะทำ
คือใช้กฎผลคูณอีกครั้งเพื่อทำ
สิ่งที่หนังสือแคลคูลัสหลายเล่ม
อาจเรียกว่ากฎผลหาร
ผมมีความรู้สึกผสมกันเวลาพูดถึงกฎผลหาร

Bulgarian: 
 
Вече знаем, че правилото за 
произведение ни казва,
че ако имаме произведение от 
две функции... Да кажем
f(x) и g(x)... и искаме да намерим
производната на това, 
тя ще бъде
равна на производната
на първата функция, 
f прим от х, по
втората функция, g(х), 
плюс първата функция,
без да взимаме нейната 
производна, т.е плюс f(x),
по производната на 
втората функция.
Две събираеми, в едното от тях взимаме производната на една
от функциите и другата, 
и после ги разменяме.
Тук е само производната на f, 
без тази на g.
Тук е производната само на g,
 без тази на f.
Надявам се, че това е малко 
като преговор.
Това е правилото за 
производна на произведение.
По същество ще приложим
 отново
правилото за произведение, 
за да получим
това, което учебниците наричат
правило за производна на частно.
Имам смесени чувства за
правилото за производна на частно.

Korean: 
과정을 조금 더 빠르게
해주기도 하지만
사실은 곱의 미분법에서
나온 것입니다
 
그리고 솔직히 말해서
몫의 미분법은 항상 잊어버려서
곱의 미분법에서 유도해내곤 합니다
 
f(x) 를 g(x)로 나눈
꼴의 식이 있다고 합시다
이것을 미분하고자 합니다
f(x) / g(x)의 도함수를 구하는 것입니다
우리가 알아야 할 것은
f(x) / g(x)라고 생각하는 대신
f(x) × (g(x)^-1) 로
생각할 수 있다는 것입니다
 
이제 우리는 곱의 미분법과
합성함수의 미분을 약간만 하면 됩니다
결과가 어떻게 될까요?
일단 곱의 미분을 먼저 합시다
첫 번째 함수의 도함수니까
f'(x) 가 될 것이고

Portuguese: 
Se você conhece a regra, 
irá fazer cálculos
mais rápidos, mas partindo da
regra do produto.
Eu sempre 
esqueço a regra do quociente,
e eu encontro a partir
da regra do produto.
Vamos ver do que estou falando.
Vamos imaginar que temos uma expressão
escrita por f de x dividida por g de x.
E que queremos calcular a derivada,
a derivada de f de x sobre g de x.
O ponto chave é reconhecer
que isto é o mesmo que a derivada de --
ao invés de escrever 
f de f de x sobre g de x,
podemos escrever como 
f de x vezes g de x elevado a menos um.
E daí podemos usar a regra do produto
com um pouco de regras.
O que isto resultará?
Vamos usar a regra do produto.
Será a derivada da primeira função
que está aqui--
vamos chamar de f linha de x--

Thai: 
ถ้าคุณรู้มัน มันอาจทำให้การคิด
เร็วกว่า แต่จริงๆ แล้วมันตรงมาจากกฎผลคูณ
 
และว่ากันตามตรง ผมมักลืมกฎผลหาร
และผมต้องพิสูจน์ใหม่จากกฎผลคูณ
ลองดูว่าเรากำลังพูดถึงอะไร
ลองนึกภาพถ้าเรามีพจน์ที่
เขียนได้เป็น f ของ x หารด้วย g ของ x
และเราอยากหาอนุพันธ์ของตัวนี้
อนุพันธ์ของ f ของ x ส่วน g ของ x
สิ่งที่เรารู้คือการสังเกต
ว่านี่ก็เหมือนกับอนุพันธ์ -- แทนที่
จะเขียน f ของ x ส่วน g ของ x
เราเขียนเป็น f ของ x คูณ 
g ของ x ยกกำลังลบ 1 ได้
 
และตอนนี้เราใช้กฎผลคูณ
กับกฎลูกโซ่นิดหน่อยได้
อันนี้จะเท่ากับอะไร?
เราแค่ใช้กฎผลคูณ
มันคืออนุพันธ์ของฟังก์ชันแรกตรงนี้ --
มันจะเท่ากับ f ไพรม์ของ x --

Czech: 
Když jej znáte, některé
operace to možná urychlí,
ale vychází přímo
z pravidla součinu.
Já osobně podílové pravidlo vždycky
zapomenu a odvozuji si ho ze součinového.
O co tedy jde.
Představme si výraz zapsaný
jako f(x) děleno g(x).
A chceme určit jeho derivaci,
tedy derivaci f(x) lomeno g(x).
Důležité je si uvědomit, že
je to stejné jako derivace…
Místo f(x) lomeno g(x) můžeme
napsat f(x) krát g(x) na −1.
A nyní můžeme využít pravidlo součinu
spolu s pravidlem o složené funkci.
Čemu se to bude rovnat?
Prostě použijeme
pravidlo součinu.
Jde o derivaci první
funkce, tedy f'(x),

Bulgarian: 
Ако го знаеш, може да направи 
някои операции малко по-бързи,
но всъщност се извежда от 
правилото за произведение.
Честно казано винаги забравям 
правилото за частно
и просто го извеждам от
 правилото за произведение.
Да видим за какво говорим.
Нека си представим, че 
имаме израз, който
може да се запише като 
f(x) делено на g(x),
и искаме да сметнем 
производната на това,
производната на f(x) върху g(x).
Важното нещо е да осъзнаем,
че това е същото нещо като
 производната...
Вместо да запишем f(x) върху g(x),
можем да запишем това като
 f(x) по g(x) на степен –1.
Сега можем да използваме 
правилото за производна на произведение
и малко от верижното правило.
На какво ще е равно това?
Просто използваме правилото
за производна на произведение.
То е производната на първата
 функция тук...
Ще бъде f прим от х

English: 
If you know it, it might make
some operations a little bit
faster, but it really comes
straight out of the product
rule.
And I frankly always
forget the quotient rule,
and I just rederive it
from the product rule.
So let's see what
we're talking about.
So let's imagine if we
had an expression that
could be written as f
of x divided by g of x.
And we want to take the
derivative of this business,
the derivative of
f of x over g of x.
The key realization
is to just recognize
that this is the same thing
as the derivative of-- instead
of writing f of
x over g of x, we
could write this as f of x times
g of x to the negative 1 power.
And now we can use
the product rule
with a little bit
of the chain rule.
What is this going
to be equal to?
Well, we just use
the product rule.
It's the derivative of the
first function right over here--
so it's going to
be f prime of x--

Portuguese: 
vezes a segunda função, que é
g de x elevado a menos 1
mais a primeira função,
que é a f de x,
vezes a derivada
da segunda função.
E aqui vamos ter que usar 
um pouco de
composição de funções.
A derivada da parte externa,
que será algo do tipo
elevado a menos um.
será menos um vezes,
neste caso g de x elevado a menos dois.
e temos que calcular a derivada
da função mais
interior em relação
a x, que será g linha de x.
e está pronto.
Encontramos a derivada disso
usando a regra do produto
e composição de funções.
Não é a forma que você
vê quando pessoas
estão falando
sobre a regra do quociente
em seu livro de matemática.
Vamos simplificar um pouco mais isso.
Tudo isso é igual a -- podemos 
escrever este termo

Thai: 
คูณฟังก์ชันที่สองเฉยๆ ซึ่งก็คือ
g ของ x ยกกำลังลบ 1 บวกฟังก์ชันแรก
คือแค่ f ของ x คูณอนุพันธ์
ของฟังก์ชันที่สอง
และตรงนี้เราจะต้องใช้กฎลูกโซ่นิดหน่อย
 
อนุพันธ์ของตัวนอก ซึ่ง
เรามองเป็นอะไรสักอย่าง
ยกกำลังลบ 1 เทียบกับอะไรสักอย่างนั่น
จะเท่ากับลบ 1 คูณอะไรสักอย่างนั้น ซึ่ง
ในกรณีนี้คือ g ของ x ยกกำลังลบ 2
แล้วเราต้องหาอนุพันธ์
ของฟังก์ชันตัวในเทียบกับ
x ซึ่งก็คือแค่ g ไพรม์ของ x
แล้วคุณก็ได้แแล้ว
เราหาอนุพันธ์ของตัวนี้ได้
โดยใช้กฎผลคูณกับกฎลูกโซ่
ทีนี้ นี่คือรูปที่คุณ
อาจเห็นเวลาคนพูดถึง
กฎผลหารในหนังสือเลขของคุณ
ลองดูว่าเราจัดรูปพจน์นี้หน่อยได้ไหม
ทั้งหมดนี้จะเท่ากับ -- เราเขียนเทอมนี้

Korean: 
g(x)의 －1승을 곱하면 됩니다
g(x)의 －1승을 곱하면 됩니다
그 다음엔 f(x)에
두 번째 함수의 도함수를
곱하면 됩니다
여기서 합성함수의 미분을
이용해야 합니다
 
우선 바깥 함수
어떤 것의 －1 승 하는 함수의
도함수를 구하면 되는데
여기서는  －1 곱하기
g(x)의 －2 승이 됩니다
그리고 안의 함수의
도함수를 곱해야 하는데
여기서는
g'(x)가 됩니다
 
이것의 도함수는
곱의 미분법과 합성 함수의 미분을
이용해야 구할 수 있습니다
그런데 이것은
흔히 책에 나오는
몫의 미분 형태가 아닙니다
조금 더 단순화 해보겠습니다
이것을 다시 써보면

Czech: 
krát druhá funkce,
což je g(x) na −1,
plus první funkce,
což je jen f(x),
krát derivace druhé funkce.
A tady musíme použít
pravidlo o složené funkci.
Derivace vnější funkce,
kterou můžeme vnímat jako derivaci
něčeho na −1 podle toho něčeho.
A to bude −1 krát to něco, což
je v tomto případě g(x), na −2.
A pak musíme zderivovat
vnitřní funkci podle x,
což je prostě g'(x).
A máme to.
Spočítali jsme tuto derivaci pomocí
pravidla o součinu a o složené funkci.
Toto ale není ve tvaru,
který uvidíte,
když si najdete podílové
pravidlo ve své učebnici.
Podívejme se, jestli
to můžeme zjednodušit.
Toto celé bude rovno…

English: 
times just the second
function, which is just
g of x to the negative 1
power plus the first function,
which is just f of x,
times the derivative
of the second function.
And here we're going to have to
use a little bit of the chain
rule.
The derivative of
the outside, which
we could kind of
view as something
to the negative 1 power with
respect to that something,
is going to be negative 1
times that something, which
in this case is g of x
to the negative 2 power.
And then we have to
take the derivative
of the inside
function with respect
to x, which is
just g prime of x.
And there you have it.
We have found the
derivative of this
using the product rule
and the chain rule.
Now, this is not
the form that you
might see when
people are talking
about the quotient
rule in your math book.
So let's see if we can
simplify this a little bit.
All of this is going to be equal
to-- we can write this term

Bulgarian: 
по втората функция, която
 е просто
g(x) на степен –1, плюс
 първата функция,
която е просто f(x), по 
производната
на втората функция.
Тук трябва да използваме 
верижното правило.
Производната на външната функция, 
която
можем да разглеждаме
 като нещо
на степен –1, спрямо това нещо, 
ще бъде –1 по това нещо,
което в този случай е g(x), 
на степен –2.
После смятаме производната на
вътрешната функция спрямо х,
което е просто g прим от х.
Готово.
Намерихме производната
 на това,
използвайки правилото за произведение 
и верижното правило.
Това не е формулировката, която
ще видиш, когато се говори
за правилото за производна на частно
 в учебниците по математика.
Да видим дали можем
 да опростим това малко.
Всичко това ще бъде равно на... 
Можем да запишем това

Thai: 
ตรงนี้เป็น f ไพรม์ของ x ส่วน g ของ x
 
และเราเขียนทั้งหมดนี้เป็น -- เรา
ใส่เครื่องหมายลบนี่ข้างหน้าได้
เราได้ลบ f ของ x คูณ g ไพรม์ของ x
 
แล้วทั้งหมดนั้นส่วน g ของ x กำลังสอง
ขอผมเขียนให้สวยหน่อยนะ
ทั้งหมดนั้นส่วน g ของ x กำลังสอง
 
และมันยังไม่ใช่รูปที่คุณมัก
เห็นในหนังสือแคลคูลัส
เวลาทำ เราต้องใช้เศษส่วนสองตัวนี้
ลองคูณทั้งเศษและส่วน
ด้วย g ของ x เราจะได้มีทุกอย่างในรูปของ g
ของ x กำลังสองเป็นตัวส่วน
ถ้าเราคูณตัวเศษด้วย g ของ x
เราจะได้ g ของ x ตรงนี้แล้ว
ตัวส่วนจะเป็น g ของ x กำลังสอง
และตอนนี้เราพร้อมจะบวกแล้ว

Korean: 
f'(x) / g(x) 가 되고
 
이것을 다시 써보면
마이너스를 앞으로 빼서
－f(x) × g(x) 를
 
g(x)² 으로 나눈 꼴이 됩니다
조금 더 깔끔하게 적겠습니다
전체를 g(x)² 로 나눠야 합니다
 
그래도 여전히 책에서 보는 식과는
거리가 있어 보입니다
그러기 위해서는
이 두 분수를 합쳐야 합니다
우선 분자와 분모에 곱해봅시다
여기서는 g(x)를 곱해야
모든 것이 g(x)²으로
나누어져 있는 꼴이 됩니다
분자에도 곱하면
여기에 g(x)가 생길 것이고
분모는 g(x)² 이 될 것입니다
이제 더하면 됩니다

Portuguese: 
logo aqui como f linha de x sobre g de x.
E poderíamos escrever tudo isso como--
poderíamos colocar
o negativo na frente
Temos menos f de x vezes g linha de x.
E tudo isso sobre g de x ao quadrado.
Melhorando um pouco a escrita.
Tudo isso sobre g de x ao quadrado.
E ainda não é a forma que você
normalmente encontra
em seu livro de cálculo.
Para fazer isso, temos que
adicionar essas duas frações.
Vamos multiplicar o 
numerador e o denominador
aqui por g de x encontrando
tudo em função de
g de x ao quadrado no denominador.
Se multiplicarmos o numerador por g de x,
teremos g de x aqui e
o denominador será g de x ao quadrado.
Estando pronto para somar.

Bulgarian: 
тук като f прим от х върху g(x).
Можем да запишем 
всичко това като...
Можем да сложим този
 минус отпред.
Получаваме –f(x) по g прим от х.
После цялото това върху
 g(x) на квадрат.
Нека запиша това 
малко по-ясно.
Цялото това върху 
g(x) на квадрат.
Все още не е във вида, който 
обикновено
се вижда в учебниците.
За да стигнем до там, просто трябва 
да съберем тези две дроби.
Нека умножим числителя и 
знаменателя тук
по g(x), за да имаме навсякъде
g(x) на квадрат в знаменател.
Ако умножим числителя по g(x),
ще получим g(x) тук и после
знаменателят ще стане 
g(x) на квадрат.
Сега сме готови за събиране.

Czech: 
Tento výraz můžeme zapsat
jako f'(x) lomeno g(x).
f'(x) lomeno g(x).
A toto můžeme zapsat jako…
Toto minus můžeme
dát dopředu.
Dostaneme −f(x) krát g'(x).
A pak to celé lomeno
g(x) na druhou.
Napíšu to trochu lépe.
To celé lomeno
g(x) na druhou.
A tohle ještě stále není ve tvaru,
který obyčejně najdete v učebnici.
Aby to tak bylo, musíme
ještě sečíst tyto dva zlomky.
Vynásobme tedy tento čitatel
a jmenovatel tímto g(x),
abychom měli všude
g(x) na druhou ve jmenovateli.
Když tedy vynásobíme čitatel
g(x), dostaneme g(x) tady
a ve jmenovateli bude
g(x) na druhou.
A teď můžeme sčítat.

English: 
right over here as f
prime of x over g of x.
And we could write
all of this as-- we
could put this negative
sign out front.
We have negative f of
x times g prime of x.
And then all of that
over g of x squared.
Let me write this a
little bit neater.
All of that over g of x squared.
And it still isn't in the
form that you typically
see in your calculus book.
To do that, we just have
to add these two fractions.
So let's multiply the
numerator and the denominator
here by g of x so that we have
everything in the form of g
of x squared in the denominator.
So if we multiply the
numerator by g of x,
we'll get g of x right
over here and then
the denominator will
be g of x squared.
And now we're ready to add.

Bulgarian: 
Получаваме, че производната на f(x)
върху g(x) е равна на 
производната на f(x) по g(x)
минус... вече не е плюс... 
нека го запиша в бяло...
f(х) по g прим х,
цялото върху g(x) на квадрат.
Отново казвам, че винаги
можеш да изведеш това
от правилото за произведение
и верижното правило.
Понякога може да е удобно
да го помним, за да
решим някоя задача в този вид
малко по-бързо.
Ако искаме да видим връзката 
между правилото за произведение
и правилото за частно: 
производната на едната
функция по другата функция.
Вместо да добавяме
производната
на втората функция 
по първата функция,
сега изваждаме.

English: 
And so we get the
derivative of f
of x over g of x is equal to
the derivative of f of x times g
of x minus-- not plus
anymore-- let me write it
in white-- f of x
times g prime of x,
all of that over g of x squared.
So once again, you
can always derive this
from the product rule
and the chain rule.
Sometimes this might be
convenient to remember in order
to work through some problems of
this form a little bit faster.
And if you wanted to kind of see
the pattern between the product
rule and the quotient
rule, the derivative of one
function just times
the other function.
And instead of
adding the derivative
of the second function
times the first function,
we now subtract it.

Czech: 
Dostaneme tedy,
že derivace f(x) lomeno g(x) je rovna
derivaci f(x) krát g(x) minus, už ne plus,
napíšu to bílou,
minus f(x) krát g'(x),
to celé lomeno
g(x) na druhou.
Takže ještě jednou,
toto si vždy můžete odvodit z
pravidla o součinu a složené funkci.
Někdy se může hodit
si toto pamatovat,
abychom některé příklady
v tomto tvaru vyřešili rychleji.
A když se podíváme na to, čím
se liší pravidlo součinové a podílové,
tady je to derivace jedné
funkce krát druhá funkce,
ale místo přičítání derivace druhé funkce
krát první funkce to teď odčítáme.

Thai: 
เราได้อนุพันธ์ของ f
ของ x ส่วน g ของ x เท่ากับ
อนุพันธ์ของ f ของ x คูณ g
ของ x ลบ -- ไม่ใช่บวกแล้ว -- ขอผมเขียน
ด้วยสีขาว -- f ของ x คูณ g ไพรม์ของ x
ทั้งหมดนั้นส่วน g ของ x กำลังสอง
ย้ำอีกครั้ง คุณหาอันนี้ได้
จากกฎผลคูณและกฎลูกโซ่
บางครั้ง การจำอาจช่วย
ให้แก้ปัญหาในรูปนี้ง่ายขึ้น
และถ้าคุณอยากเห็นรูปแบบระหว่างผลกฎคูณ
กับผลหาร อนุพันธ์ของ
ฟังก์ชันหนึ่งคูณอีกฟังก์ชันหนึ่ง
และแทนที่จะบวกอนุพันธ์
ของฟังก์ชันที่สองคูณฟังก์ชันแรก
ตอนนี้เราลบมันแทน

Korean: 
f(x) 나누기 g(x) 꼴의
도함수는
f'(x)g(x) －f(x)g'(x) 가 분자가 되고
g(x)² 가
분모가 되는 형태입니다
항상 이렇게 곱의 미분과 합성함수의
미분을 통해 유도할 수 있습니다
이런 꼴의 문제를 풀기에는
이렇게 하는 것이 조금 더
빠를 수도 있습니다
곱의 미분과 몫의 미분과의
차이를 보고 싶다면
그냥 하나의 도함수와 다른 함수를 곱하고
처음 함수와 두 번째 함수의 도함수를
곱한 것을 더하지 말고
빼면 됩니다

Portuguese: 
Derivando f de x
em relação a g de x é igual a
derivada de f de x vezes g
de x menos-- não é mais--
vou escrever
na cor branca-- f de x 
vezes g linha de x,
tudo isso sobre g de x ao quadrado.
Mais uma vez, você pode sempre 
derivar isso
pela regra do produto e
composição de funções
vamos relembrar a ordem
para trabalhar através de alguns
problemas deste tipo um pouco mais rápido.
E se você quer ver o padrão
entre a regra do produto
e a regra do quociente,
a derivada de uma função
somente multiplica a outra função.
E ao invés de adicionar a derivada
da segunda função multiplicado
pela primeira função,
E então subtraímos

Czech: 
A celé je to ještě lomeno
druhou funkcí na druhou.
Cokoli bylo ve jmenovateli,
je teď na druhou.
Takže když derivujeme
tuto funkci a jmenovatel,
tady je odčítání
a celé je to ještě lomeno
druhou funkcí na druhou.

Portuguese: 
E tudo isso sobre a segunda
função ao quadrado.
Entretanto estava no denominador, 
tudo isso ao quadrado.
Quando encontramos a derivada
da função no denominador aqui acima,
existe uma subtração, 
e nós vamos também colocar tudo
sobre a segunda função ao quadrado.
[legendado por Renata do Vale]

Thai: 
และทั้งหมดนั้นมีส่วนฟังก์ชันที่สองกำลังสอง
อะไรก็ตามที่อยู่ในตัวส่วน ทั้งหมดนั้นกำลังสอง
เมื่อเราหาอนุพันธ์
ของฟังก์ชันในตัวส่วนบนนี้
มันมีเครื่องหมายลบ แล้วเราก็ใส่ทุกอย่าง
ส่วนฟังก์ชันที่สองกำลังสอง

English: 
And all that is over the
second function squared.
Whatever was in the denominator,
it's all of that squared.
So when we're taking
the derivative
of the function in the
denominator up here,
there's a subtraction, and then
we are also putting everything
over the second
function squared.

Korean: 
그리고 전체를
두 번째 함수의 제곱으로 나누면 됩니다
이 도함수의
분자를 보면
여기 마이너스가 있고
전체를 두 번째 함수의 제곱으로
나눈 꼴이 됩니다

Bulgarian: 
И всичко е върху 
втората функция на квадрат.
Каквото е имало в знаменател, 
цялото е на квадрат.
Когато говорим за производната
на функцията в знаменател 
тук горе,
има изваждане, а после 
слагаме всичко
върху втората функция
на квадрат.
