
Thai: 
สมมุติว่า
เราได้ฟังก์ชัน f ของ x มา
และมันเท่ากับล็อกธรรมชาติ
ของ x บวก 5 ส่วน x ลบ 1
และสิ่งที่เราอยากหา
คือ f ไพรม์ของ x
และผมแนะนำให้คุณหยุดวิดีโอนี้
แล้วพยายามหาคำตอบด้วยตนเอง
มีวิธีทำอยู่สองวิธี
ผมเรียกวิธีหนึ่งว่าวิธีง่าย
และอีกวิธี คือวิธียาก
และเราจะทำทั้งสองวิธี
วิธีง่ายคือสังเกต
สมบัติลอการิทึม
นึกดูว่าล็อกธรรมชาติ
ของ a ส่วน b
นึกดูว่าล็อกธรรมชาติ ก็แค่ล็อกฐาน e
นี่ก็เท่ากับ
ล็อกธรรมชาติของ a
ลบล็อกธรรมชาติของ b
ถ้าเราใช้สมบัตินี้
ตรงนี้
แล้วเขียนพจน์นี้ให้ง่ายลง
หรืออย่างน้อยก็เขียนให้ง่ายลงในมุมมอง
เพื่อหาอนุพันธ์นี้
เราก็เขียน f ของ x ใหม่ได้

Korean: 
x에 대한 함수 f 가 있습니다
x에 대한 함수 f 가 있습니다
이 함수는 자연로그
(x-5) / (x-1)이라고 합시다
우리가 알고 싶은것은
f의 도함수입니다
이 동영상을 잠시 멈추고
혼자 생각해 보십시오
도함수를 구하기 위해
쉬운 한 가지 방법이 있고
보다 어려운 방법이 있습니다
두 가지 모두 해봅시다
쉬운 방법을 사용하려면
로그함수의 특징을 알아야 합니다
ln(a/b)에서
ln은 자연로그로
밑이 e인 로그이며
ln(a/b)는
ln a-ln b입니다
ln a-ln b입니다
이 특징을 적용하여
문제를 해결하면
표현을 단순화하거나
관점을 단순화하여
미분을 쉽게 할 수 있습니다
f를 다시 써봅시다

Czech: 
Řekněme, že máme funkci f(x),
která se rovná
přirozenému logaritmu
(x plus 5) lomeno (x minus 1).
A co chceme vyřešit je, co je f'(x).
Jaká je derivace f(x).
A vyzývám vás, pozastavte toto video
a zkuste si to nejdřív sami.
Jsou dva způsoby,
jak k tomu přistupovat.
Jeden bych nazval lehkým,
a druhý způsob těžkým.
A my si ukážeme oba.
Ten lehký je použít vlastnosti logaritmu,
vzpomenout si,
že přirozený logaritmus (a/b), jak víme,
ln je jen logaritmus o základu 'e'.
Takže toto se bude rovnat
přirozenému logaritmu 'a',
minus přirozený logaritmus 'b'.
Takže když použijeme
vlastnosti logaritmů tady,
a zjednodušíme tento výraz,
nebo asopň zjednodušíme
z pohledu derivace,
můžeme přepsat f(x).

English: 
Voiceover:Let's say that
we've got the function F of X
and it is equal to the natural log
of X plus five over X minus one.
And what we want to figure out
is what is F prime of X.
And I encourage you to pause this video
and try to figure it out on your own.
So there's two ways that
you can approach this.
I would call one way the easy way.
And the other way, the hard way.
And we'll work through both of them.
The easy way is to recognize
your logarithm properties,
to remember that the natural log
of A over B.
Remember natural log is
just log base the number E.
So this is just going to be equal to
the natural log of A
minus the natural log of B.
So if we just apply this property
right over here,
and just simplify this expression,
or at least simplify
it from a point of view
in terms of having to
take this derivative.
We can rewrite F of X.

Portuguese: 
Seja uma função f de x
igual ao logaritmo natural de x mais cinco
dividido por x menos um, e
queremos descobrir como
seria f linha de x.
Incentivo você a pausar este vídeo
e tentar descobrir por si mesmo.
Existem duas abordagens possíveis
para este problema.
Eu chamaria uma de forma fácil
e a outra de forma difícil.
Trabalharemos com ambas.
A forma fácil é reconhecer
as propriedades dos logaritmos
para lembrar que
o logaritmo natural de a sobre b
-- lembre-se que logaritmo natural é só
logaritmo com e na base -- será igual ao
logaritmo natural de a menos o
logaritmo natural de b.
Se somente aplicarmos esta
propriedade
aqui e simplificarmos esta expressão,
ou ao menos a simplificarmos
ao ponto de termos de calcular
esta derivada

Bulgarian: 
Дадена е функцията f(х)
е равно на натурален
логаритъм
от (х + 5)/(х –1).
Искаме да намерим 
колко е f'(х).
Насърчавам те да спреш
видеото на пауза
и да опиташ самостоятелно
да го намериш.
Има два начина да
подходим към задачата.
Бих нарекъл първия
лесния начин.
А другия – трудния начин.
Ще направим и двата.
Лесният начин е да приложиш
свойствата на логаритмите,
да си спомниш свойствата на натурален
логаритъм от А върху В
Спомни си, че натурален 
логаритъм означава основа е.
Значи това ще бъде равно на
натурален логаритъм от А
минус натурален логаритъм от В.
Просто прилагаме това
свойство ето тук,
и само опростяваме израза,
или поне го опростяваме
от гледна точка на
намирането на производните
на членовете на израза.
Можем да преработим f(х).

English: 
We can write F of X as being equal to
the natural log of X plus five
minus the natural log of X minus one.
And when we take the derivative now
with respect to X,
F prime of X,
well this is going to be the derivative
of the natural log of X plus five
with respect to X plus five,
so that's going to be one over X plus five
times the derivative of X plus five
with respect to X.
I'm just applying the chain rule here,
and that's just going to be one.
So this, the derivative of that is that.
And the derivative of this,
well, let's see,
we're going to have a minus sign there
and the derivative of the natural log
of X minus one with respect to X minus one
is going to be one over X minus one
and the derivative of X minus one
with the respect to X is just one
you just multiply this by one,
it doesn't really change the value.
And we're done!
We were able to figure
out the derivative of F.

Thai: 
เราเขียน f ของ x ได้เท่ากับ
ล็อกธรรมชาติของ x บวก 5
ลบล็อกธรรมชาติของ x ลบ 1
และเมื่อเราหาอนุพันธ์ตอนนี้
เทียบกับ x
f ไพรม์ของ x
อันนี้จะเท่ากับอนุพันธ์
ของล็อกธรรมชาติของ x บวก 5
เทียบกับ x บวก 5
มันจะเท่ากับ 1 ส่วน x บวก 5
คูณอนุพันธ์ของ x บวก 5
เทียบกับ x
ผมแค่ใช้กฎลูกโซ่ตรงนี้
และนั่นจะเท่ากับ 1
อันนี้ อนุพันธ์ของค่านั้นคือตัวนั้น
และอนุพันธ์ของพจน์นี้
ลองดู
เราจะได้เครื่องหมายลบตรงนั้น
และอนุพันธ์ของล็อกธรรมชาติ
ของ x ลบ 1 เทียบกับ x ลบ 1
จะเท่ากับ 1 ส่วน x ลบ 1
และอนุพันธ์ของ x ลบ 1
เทียบกับ x ก็แค่ 1
คุณแค่คูณค่านี้ด้วย 1
มันจะไม่เปลี่ยนค่าไป
เราก็เสร็จแล้ว!
เราหาอนุพันธ์ของ f ได้แล้ว

Portuguese: 
Podemos escrever f de x como sendo igual
ao logaritmo natural
de x mais cinco, menos
o logaritmo natural
de x menos um.
Quando calculamos a derivada,
com relação a x, f linha de x,
isto será a derivada do logaritmo natural
de x mais cinco 
em relação a x mais cinco.
Isto será um sobre x mais cinco
vezes a derivada de x mais cinco
em relação a x.
Estou só aplicando a regra da cadeia aqui.
Isto será igual a um.
A derivada disto é isto,
e a derivada disto,
perceba que temos 
um sinal de menos ali,
e a derivada do logaritmo natural de x
menos um em relação a x menos um
será um sobre x menos um vezes a 
derivada de x menos um com respeito a x,
que é um, o que não muda seu valor.
Feito!
Descobrimos a derivada de f.

Czech: 
Můžeme psát, že f(x) je rovno
přirozenému logaritmu (x plus 5) 
minus přirozený logaritmus (x minus 1).
A když to teď zderivujeme podle x,
f s čárkou x, to bude
derivace přirozeného logaritmu (x plus 5)
podle (x plus 5).
To bude 1 lomeno (x plus 5),
krát derivace (x plus 5) podle x.
Jen používám řetízkové pravidlo.
A to bude jednoduše 1.
Takže derivace tohoto je toto.
A derivace tohoto, uvidíme.
Tady budeme mít minus,
a derivace přirozeného logaritmu
od (x minus 1) podle (x minus 1)
bude 1 lomeno (x minus 1) a pak
derivace (x minus 1) podle x je jednoduše 1.
Násobení 1 vlastně nezmění hodnotu
a jsme hotovi!
Spočítali jsme derivaci f.

Bulgarian: 
Можем да напишем, че f(х)
е равно на натурален 
логаритъм от (х + 5)
минус натурален логаритъм
от (х – 1).
Сега намираме производните
спрямо х,
f'(х),
като това ще бъде производната
на натурален логаритъм
от (х + 5) спрямо (х + 5),
значи става 1/(х + 5)
по производната на (х + 5)
спрямо х.
Просто прилагам верижното 
правило –
и това става 1.
Това е производната на тази част.
Производната на това, 
да видим,
тук ще имаме знак минус,
и производната на 
натурален логаритъм
от (х – 1) спрямо (х – 1)
е равна на 1 върху (х – 1),
а производната на (х – 1)
спрямо х е просто 1,
така че умножаваме по 1,
което не променя стойността.
И сме готови.
Намерихме производната на f.

Korean: 
x에 대한 함수 f는
ln (x+5) - ln (x-1)
라고 정의되고 있습니다
주어진 함수 f를
x에 대해 미분할 것입니다
x에 대한 f의 도함수
이제 ln (x+5)/(x-1) 에 관한
이제 ln (x+5)/(x-1)에 관한
미분을 할 것 입니다
이것은 1/(x+5) 곱하기
x+5를 x에 대해 미분한 것이죠
x+5를 x에 대해 미분한 것이죠
연쇄법칙을 적용한 것입니다
미분값이 1이므로
ln(x+5)의 미분값은 1/x+5가 되고
ln(x-1)의 미분값은
ln(x-1)의 미분값은
마이너스
1/(x-1)에
x - 1 을 미분한 것을
곱해주면 됩니다
이 값 역시 1이기 때문에
곱셈에 영향을 미치지 않습니다
따라서 뒷 항의 미분값은
1 / (x-1) 입니다
미분이 끝났습니다
함수 f의 도함수를 구했습니다

Korean: 
어려운 방법으로 넘어가겠습니다
어려운 방법으로 넘어가겠습니다
스스로 풀어보았을 때
이 방법을 사용했을 지도 모릅니다
로그의 성질을 이용하여
식을 단순화시키지 않고
연쇄법칙을 활용하여
직접 미분하는 방법입니다
직접 하면서 설명하겠습니다
이 방법에서는
함수 전체를
한 번에 미분합니다
(x+5)/(x-1)에 대해서 말이죠
(x+5)/(x-1)에 대해서 말이죠
전체를 미분하면
자연로그의 미분에 의해
(x+5)/(x-1)의 역수가 되고
(x+5)/(x-1)를 미분한 것을
다시 곱해주게 됩니다
연쇄법칙의 적용에 불과합니다
함수 전체를 미분하는데
자연로그를 먼저 미분하고
로그 안의 값을 미분하여
곱해주는 것입니다
그저 연쇄법칙입니다
차근차근 분석합시다
차근차근 분석합시다
앞의 항은
역수를 취한 것이므로
(x-1)/(x+5)와 같습니다
(x-1)/(x+5)와 같습니다

Czech: 
Mohli byste se ptát, 
jaký je ten těžký způsob,
nebo možná jste to tak dělali,
když jste se o to pokoušeli sami.
To je nezjednodušit ten výraz
použitím této vlastnosti
a pokusit se tím probít
použitím řetízkového pravidla.
Takže to tak zkusíme.
Takže, f'(x) bude derivace
této celé věci podle (x plus 5)
lomeno (x minus 1).
Takže to bude 1 lomeno
[(x plus 5) lomeno (x minus 1)]
krát derivace podle 'x' z [(x plus 5)
lomeno (x minus 1)].
Je to jen řetízkové pravidlo.
Derivace toho celého
podle tohoto výrazu krát
derivace tohoto výrazu podle x.
Jen řetízkové pravidlo.
Podívejme se, tohle bude rovno...
Použiji tu nějaké barvy.
Tohle zarámečkuji v modré,
to je to samé jako
(x minus 1) lomeno (x plus 5).

Bulgarian: 
Сигурно се чудиш 
какъв е трудният начин.
Но може и да си го използвал/а,
когато опита самостоятелно
да решиш задачата.
Това е да не опростяваме израза
с помощта на това свойство
и просто да опитаме да решим това
с верижното правило.
Да опитаме и по този начин.
В този случай f'(х)
е равно на производната
на цялото това нещо
спрямо (х + 5)
върху (х – 1),
което е 1 върху
(х + 5)/(х – 1)
по производната спрямо х
от (х + 5)/(х –1).
Това е правилото за диференциране
на сложна функция или верижното правило.
Производната на целия
този израз
спрямо този израз,
по производната на този израз
спрямо х.
Просто верижното правило.
Да видим, това е равно на...
ще използвам различни цветове,
това, което ограждам в синьо,
е равно на (х –1)/(х + 5).

Thai: 
ทีนี้ วิธียาก
คุณคิดว่าคืออะไร?
คุณอาจทำ
โดยคุณลองหาด้วยตนเองก่อน
มันไม่ใช่การจัดรูปพจน์นี้ให้ง่ายลง
โดยใช้สมบัตินี้
แต่ลองทนทำไป
โดยใช้กฎลูกโซ่
ลองพยายามทำดู
ในกรณนั้น f ไพรม์ของ x
จะเท่ากับอนุพันธ์
ของทั้งหมดนี้
เทียบกับ x บวก 5
ส่วน x ลบ 1
ซึ่งจะเท่ากับ 1 ส่วน x บวก 5
ส่วน x ลบ 1
คูณอนุพันธ์ คูณอนุพันธ์
เทียบกับ x ของ x บวก 5
ส่วน x ลบ 1
นี่ก็แค่กฏลูกโซ่
อนุพันธ์ของทั้งหมดนี้
เทียบกับพจน์นี้
คูณอนุพันธ์ของพจน์นี้
เทียบกับ x
แค่กฎลูกโซ่
ลองดู
ค่านี้จะเท่ากับ
ลองใช้สีตรงนี้
อันนี้ สิ่งที่ผมทำกล่องสีฟ้า
มันก็เท่ากับ x ลบ 1
ส่วน x บวก 5

English: 
Now what's the hard way
you might be thinking?
Or maybe you did do it
when you tried to approach it on your own.
Well, that's not to
simplify this expression
using this property
and just to try and power through this
using the chain rule.
So let's try and do that.
In that case, F prime of X
is going to be the derivative
of this whole thing
with respect to X plus five
over X minus one
which is going to be one over X plus five
over X minus one
times the derivative, times the derivative
with respect to X of X plus five
over X minus one.
This is just the chain rule.
The derivative of this whole thing
with respect to this expression,
times the derivative of this expression
with respect to X.
Just the chain rule.
So let's see,
this is going to be equal to,
let's use some colors here,
this, what I'm boxing off in blue,
that's the same thing as X minus one
over X plus five.

Portuguese: 
Agora, vamos à forma difícil, que 
talvez você tenha pensado ou
seguido quando tentou calcular
sozinho.
Não simplificamos a expressão utilizando
nenhuma propriedade e
tentamos somente aplicar
a regra da cadeia a esta potência.
Vamos tentar fazer isso.
Neste caso, f linha de x será
a derivada
de tudo isso em relação a x 
mais cinco sobre x menos um.
Será igual a um sobre x mais
cinco sobre x menos um
vezes a derivada em relação a x
de x mais cinco sobre x menos um.
Isto é só a regra da cadeia.
Isto tudo com respeito a esta expressão
vezes
a derivada desta expressão
em relação a x.
Apenas a regra da cadeia.
Isto será igual a
-- deixe-me usar algumas cores aqui--
Este termo nesta caixa azul aqui,
que é o mesmo que x menos um
sobre x mais cinco

Korean: 
역수를 취했기 때문입니다
뒤의 항은
원래 로그 안에 있던 값을
다시 미분하여 곱하는 것입니다
미분한 값을
구해보겠습니다
(x+5) / (x-1) 를 다시 쓰면
(x+5) × (x-1)의 역수 입니다
이 항에 대한 미분값이 되겠죠
이렇게 바꿔 쓴 이유는
분수의 미분법보다는
곱의 미분을 적용하는 것이
더 간단하기 때문입니다
왜 이 방법이
어렵다고 했는지
이제 아실 겁니다
많은 과정들이 필요하죠
다시 한번 적어보면
x-1/x+5 에 곱하기
곱함수의 미분을 쓰면
x+5의 미분값은 1이므로
1에 곱하기 두 번째 항인
x-1의 역수이므로
1/x-1이 되고
더하기
x-1 역수의 미분값입니다
x-1 역수의 미분값입니다
생각을 좀 해야겠네요
생각을 좀 해야겠네요
-1/(x-1)^2

Thai: 
ผมแค่หาส่วนกลับของค่านี้
แล้วมันจะเท่ากับ คูณ
ผมจะทำด้วยสีบานเย็นนะ
ไม่ นั่นไม่ใช่สีบานเย็น
มันจะเท่ากับคูณ
และผมจะเขียนมัน
เป็นอนุพันธ์เทียบกับ x
ของ x บวก 5 คูณ x ลบ 1
ยกกำลังลบ 1
ผมชอบเขียนมันแบบนั้น
เพราะผมชอบลืม
กฎผลหารอะไรนั่น
แต่ผมจำกฎผลคูณได้
สิ่งนี้
ขอผมเขียนมันใหม่นะ
ผมว่าคุณคงเข้าใจแล้ว
ว่าทำไมมันถึงเป็นวิธียาก
ขอผมเขียนอันนี้
นี่คือ x ลบ 1 ส่วน x บวก 5 คูณ
ลองใช้กฎผลคูณกัน
อนุพันธ์ของ x บวก 5
มันจะเท่ากับ 1 คูณเทอมที่สอง
คูณ x ลบ 1 กำลังลบ 1
นั่นคือ 1 ส่วน x ลบ 1
แล้วบวก
อนุพันธ์ของ x ลบ 1
กำลังลบ 1 คืออะไร
ลองดู
มันจะเท่ากับ มันจะเท่ากับ
ลบ x ลบ 1 กำลังลบ 2

Czech: 
Beru jen převrácenou hodnotu tohoto.
A pak to bude krát, a udělám to purpurově.
To není purpurová.
Takže to bude krát, a přepíšu to
jako derivaci podle x
z (x plus 5) krát (x minus 1) umocněno -1.
Rád to píšu takto,
protože vždycky zapomenu
vždycky zapomenu to pravidlo pro zlomky.
Ale pamatuji si pravidlo pro součin,
takže toto.
Přepíšu to.
Myslím, že už vídíte,
proč je to ten těžký způsob.
Takže píšu (x minus 1) lomeno (x plus 5)
krát, použijeme pravidlo pro součin...
krát derivace z (x plus 5).
To je jednoduše 1 krát ten druhý člen,
krát (x minus 1) na -1.
To je 1 lomeno (x minus -1), a pak plus...
Co je derivace (x minus 1) na -1?
Podívejme se, to bude
záporné (x minus 1) na -2.

English: 
I'm just taking the reciprocal of this.
And then it's going to be times,
and I'll do this in magenta.
No that's not magenta.
So that's going to be times
and I'm going to rewrite it
as the derivative with respect to X
of X plus five times X minus one
to the negative one power.
And I like to write it that way
because I always forget the whole
quotient rule thing.
But I remember the product rule.
So this thing
so let me rewrite it,
I think you already appreciate
why this is the hard way.
So let me write this
this is X minus one over X plus five times
so let's apply the product rule,
The derivative of X plus five,
well that's just one times the second term
times X minus one to the negative one,
so that's one over X minus one.
And then plus,
what's the derivative of X minus one
over negative one.
Well, let's see,
that's going to be, that's going to be
negative X minus one to
the negative two power.

Bulgarian: 
Просто взимам 
реципрочното на това.
И това е умножено по всичко,
което е в цикламено.
Не, това не е в цикламено.
Значи това е по...
ще го преработя като
производната спрямо х
на (х + 5) по (х – 1)
на степен –1.
Харесва ми да го запиша 
по този начин,
защото винаги забравям
правилото за коефициентите.
Но помня правилото
за производението.
И сега това,
ще го преработя,
сигурно вече разбираш
защо това е трудният начин.
Ще запиша това като
(х –1) върху (х + 5) по...
прилагаме правилото
за произведение.
Производната на (х + 5),
това е просто 1 по втория член,
по (х – 1) на степен –1,
значи това е 
върху (х – 1).
После плюс,
колко е производната
на (х – 1) на степен –1.
Да видим,
това е равно на
–(х – 1) на степен –2.

Portuguese: 
-- Só estou calculando o recíproco disto--
vezes -- usarei magenta para isso,
isto não é magenta--
vezes, eu vou reescrever isto
como uma derivada em relação
a x de x mais cinco vezes x menos um
elevado a menos um.
Eu prefiro escrever desta forma
porque eu sempre
me esqueço da regra do quociente
mas eu lembro da regra do produto,
então isto
-- deixe-me reescrevê-la para
você entender porque esta
é a forma difícil--
isto é x menos um
sobre x mais cinco vezes,
vamos aplicar a regra do produto,
derivada de x mais cinco,
que é igual a um, vezes o segundo termo
que é x menos um elevado a menos um.
Isto é um sobre x menos um, mais
-- qual a derivada de x menos um 
elevado a menos um?--
Isto será
menos x menos um elevado
a menos dois,

Portuguese: 
posso dizer x negativo menos um
elevado a menos dois
--ou eu poderia só escrever
assim--
vezes a derivada de x menos um
com relação a x, que é um,
vezes isto, x mais cinco.
Tudo que usei aqui foi 
a regra do produto
Derivada disto é um, vezes aquilo,
que nos dá isto aqui, e
então calculei a derivada disto,
que está bem aqui.
Um negativo sobre
x menos um ao quadrado,
ou x negativo menos um
elevado a menos dois.
Vezes esta primeira expressão aqui,
Esta é a derivada,
vamos ver se conseguimos
simplificar as coisas.

Thai: 
ผมบอกได้ว่าลบ 1, x ลบ 1
กำลังลบ 2
ผมเขียนมันแบบนี้ได้
แล้วคูณอนุพันธ์ของ x ลบ 1
เทียบกับ x
นั่นจะเท่ากับ 1
แล้วคูณอันนี้
x บวก 5
ที่จริง
คูณ คูณ x บวก 5
ที่ผมทำตรงนี้ แค่กฎผลคูณ
อนุพันธ์ อนุพันธ์ของอันนี้
คือ 1 คูณตัวนั้น
แล้วมันจะให้ค่านั่นตรงนั้น
แล้วผมหาอนุพันธ์ของตัวนี้
ซึ่งก็คือค่านี่ตรงนี้
ลบ 1 ส่วน x ลบ 1
แล้ว ส่วน x ลบ 1 กำลังสอง
หรือคุณบอกได้ว่า ลบ x ลบ 1
ยกกำลังลบ 2
คูณพจน์แรกนี่ตรงนี้
มันมีอนุพันธ์นั่น
ทีนี้ ลองดูว่าเราจัดรูปหน่อยได้ไหม
ถ้าเรา ลองดู
ถ้าเรา ถ้าเรา
ขอผมเขียนทุกอย่างใหม่

English: 
So I can say negative one, X minus one
to the negative two.
I can just write it like this.
And then times the
derivative of X minus one
with respect to X.
Well that's going to be one.
And then times this.
X plus five.
So actually,
so times, times, X plus five.
So all I did here, product rule.
Derivative, derivative of this
is one times that.
And that gave us that over there.
And then I took the derivative of this
which is this right over here.
Negative one over X minus one
and the, over X minus one squared.
Or you can say negative X minus one
to the negative two power
times this first expression over there.
So there's that derivative.
Now let's see if we can simplify things.
So if we, let's see,
if we were to, if we were to
let me just rewrite everything.

Bulgarian: 
Значи –1(х –1) на степен –2.
Мога да го напиша 
просто ето така.
И после по производната на (х – 1)
спрямо х.
Това ще бъде 1.
После по това,
по (х + 5).
Всъщност, да, по (х + 5).
Просто приложих правилото за 
производна на произведение.
Производната на това
е едно, по ето това.
И получихме това тук.
После производната на това,
която е ето това тук.
–1/(х – 1) на квадрат или
можем да кажем върху
(х – 1) на степен –2,
по този първия израз тук.
Значи това е тази 
производна.
Сега да видим можем ли
да опростим нещата.
Да видим, ако трябваше...
всъщност ще препиша всичко.

Korean: 
-1/(x-1)^2
-1/(x-1)^2
이렇게 적을 수 있습니다
여기에 곱하기 x-1의 미분값을 하면 됩니다
물론 x에 대한 미분값이죠
이것은 1이 될 것입니다
그다음 x+5를 곱해야 합니다
그다음 x+5를 곱해야 합니다
그러니까
여기에 x+5를 곱하죠
곱함수를 미분한 형태입니다
x+5의 미분값에
x-1의 역수를 곱하고
결과는 이러합니다
x-1의 역수를
미분하면 이렇게 됩니다
-1/(x-1)^2
-1/(x-1)^2
-1/(x-1)^2
-1/(x-1)^2
곱하기 x+5를 해줬습니다
따라서 결과를 얻을 수 있습니다
이것을 한번 단순화 시켜보겠습니다
 
이것을 한번
다시 적어보겠습니다

Czech: 
Můžu říct -1 lomeno (x minus 1) na -2
nebo to můžu napsat takto.
A pak krát derivace (x minus 1) podle x.
To bude 1. A pak krát toto, (x plus 5).
Krát (x plus 5).
Použil jsem jen pravidlo pro součin.
Derivace tohoto je 1, krát toto
a to nám dalo toto tady.
A pak jsem vzal derivaci tohoto,
což je tohle.
-1 lomeno (x minus 1) na druhou,
nebo můžete říct -(x minus 1) na -2.
Krát tento první výraz támhle,
takže to je ta derivace,
a teď se podívejme,
jestli to jde zjednodušit.

Thai: 
อันนี้เท่ากับ x ลบ 1 ส่วน x ลบ 5
คูณ 1 ส่วน x ลบ 1
ลบ x บวก 5
ส่วน x ลบ 1 กำลังสอง
ทีนี้ ลองคิดดูว่าเกิดอะไรขึ้น
เมื่อเรากระจายพจน์นี้
เมื่อคุณแจกแจงพจน์นี้คูณพจน์นั้น
ตัวเศษนี้หักล้างกับตัวส่วนนั้น
แล้วเราจะได้ จะได้
ลองดู
1 ส่วน x บวก 5
แล้วเมื่อคุณแจกแจงมันตรงนี้
x บวก 5 จะหักล้าง
x บวก 5
และ x ลบ 1 ตัวหนึ่ง
จะหักล้าง
กับ x ลบ 1 ตัวหนึ่งนี้
แล้วคุณจะเหลือ
แค่ x ลบ 1 ตัวหนึ่ง
เป็นตัวส่วน
แล้วคุณได้ f ไพรม์ของ x
เท่ากับค่านี้
โชคดีของเรา
เราได้คำตอบเหมือนกันทั้งสอวิธี
แต่อย่างที่เราเห็น วิธีง่าย
นั้นง่ายกว่าวิธียากเยอะมาก

Portuguese: 
Se nós -- deixe-me reescrever tudo--
Isto é igual a x menos um sobre
x mais cinco
vezes um sobre x menos um,
menos x mais cinco sobre
x menos um ao quadrado.
Pense no que acontece
quando distribuímos isto.
Quando você distribui isto vezes isto,
Este numerador cancela
aquele denominador,
teremos um sobre
x mais cinco.
e, quando você distribui
isto aqui,
x mais cinco irá cancelar
x mais cinco e,
e um dos x menos um
irá cancelar este x menos um.
E você ficará com só um destes
x menos um no denominador.
f linha de x será igual a isto,
e por sorte, encontramos
a mesma resposta das duas maneiras,
mas vimos que a forma fácil é
muito mais fácil que a forma difícil.

Czech: 
Takže kdybychom, podívejme, kdybychom...
raději to všechno přepíšu.
Takže tohle je (x minus 1)
lomeno (x plus 5)
krát 1 lomeno (x minus 1).
Minus (x plus 5) lomeno
((x minus 1) na druhou).
Teď se zamysleme, co se stane,
když to roznásobíme.
Takže když pronásobíte toto s tímto.
Tento čitatel se pokrátí 
s tím jmenovatelem,
a tak dostaneme 1 lomeno (x plus 5).
A když to pronásobíte s tímto,
to (x plus 5) se vykrátí s (x plus 5)
a jeden z těch (x minus 1) se vykrátí
s jedním z těch (x minus 1).
Takže nám zůstane jen jednou
(x minus1) ve jmenovateli.
Takže f'(x) je rovno tomuto,
a máme štěstí, dostali jsme
stejnou odpověď oběma způsoby.
Ale jak vidíme, ten lehký způsob
byl mnohem lehčí, než ten těžký.

English: 
So this is equal to X
minus one over X plus five
times one over X minus one,
minus X plus five,
over X minus one squared.
Now let's think about what happens
when we distribute this.
So when you distribute this times that,
this numerator cancels
with that denominator
and so we're going to get, going to get,
let's see,
one over X plus five.
And then when you distribute it over here
the X plus five is going to cancel
the X plus five.
And one of the X minus ones
is going to cancel
one of these X minus ones.
And you're going to be left
with just one of those X minus ones
as the denominator.
And so you get F prime of X
is equal to this.
And lucky for us,
we got the same answer either way.
But as we see, the easy way
is much easier than the hard way.

Korean: 
이것은 (x-1)/(x+5) 곱하기
1/(x-1) 한 다음
빼기 (x+5)/(x-1)^2
빼기 (x+5)/(x-1)^2
이제 이것을 전개하면
어떻게 될지 생각해봅시다
이것을 이렇게 전개하게 되면
x-1항은 소거가 됩니다
계속 계산해 봅시다
계속 계산해 봅시다
1/x+5로 얻을 수 있습니다
다음으로 이렇게 전개를 해보면
x+5는 소거가 될 것이고
x+5는 소거가 될 것이고
하나의 x-1도 소거가 됩니다
하나의 x-1도 소거가 됩니다
하나의 x-1도 소거가 됩니다
이제 남은 것은
하나의 x-1만
분모에 남습니다
그래서 우리는 f의 x에 대한 도함수를
이렇게 얻을 수 있고
다행히도
같은 결과를 얻었습니다
그러나 여러분이 보고 계신 것처럼
쉬운 방법이 훨씬 낫습니다

Bulgarian: 
Значи това е равно на
(х – 1)/(х + 5)
по 1/(х – 1),
минус (х + 5)
върху (х – 1)^2.
Сега да видим какво се случва, 
когато разкрием скобите.
Когато умножим това
по това,
този числител се съкращава
с този знаменател,
и тогава ще получим,
да видим,
1/(х + 5).
После като умножим
ето тук,
(х + 5) се съкращава с
това (х + 5).
(х – 1) се съкращава
с едно от тези (х – 1).
И накрая ни остава
само едно от тези (х – 1)
в знаменателя.
И така получаваме,
че f'(х) е равно на това.
За наше щастие получихме
еднакъв отговор и по двата начина.
Но както виждаш,
лесният начин
е много по-лесен от
трудния. :-)

Portuguese: 
[Legendado por: Luiz Marangoni]
[Revisado por: Rodrigo Melges]
