
Czech: 
V tomto videu si ukážeme, že implicitní
i explicitní derivování ‚dá to stejné‘.
Nebo-li dostaneme
stejný výsledek.
Mějme například rovnici: 
x krát odmocnina z y je rovna 1.
Snadno můžeme 
v závislosti na x vyjádřit y.
Podělením obou stran x dostaneme,
že 1 lomeno x je rovno odmocnině z y.
Umocněním na druhou dostaneme, 
že y je rovno 1 lomeno x na druhou.
Což je to stejné
jako x na −2.
Potom už je vypočítání 
derivace y podle x přímočaré.
Použitím pravidla o derivování mocninné
funkce pak máme −2 krát x na −3.
To bylo
celkem snadné.

Portuguese: 
O que quero mostrar
a vocês neste vídeo
é que derivação implícita produz
o mesmo resultado que, acho
que podemos dizer,
derivação explícita quando você
pode utilizar a derivação explícita.
Digamos que temos uma relação
x vezes a raiz quadrada 
de y é igual a um.
Esse aqui é bastante óbvia
para explicitar em relação 
a x, e resolver para y.
Então se dividirmos 
ambos os lados por x,
ficamos com a raiz quadrada
de y é igual a 1/x.
E elevamos ambos os lados ao quadrado,
assim y é igual a 1 sobre x ao quadrado,
o que é a mesma coisa que
x elevado a -2.
Portanto se você quer a derivada
de y em relação a x
isto é bem fácil.
Aqui apenas aplicamos 
a regra da cadeia.
Assim temos dy/dx é igual a -2x
elevado a -2 menos 1
x elevado a -3.
Então é bem simples e direto.

Thai: 
 
สิ่งที่ผมอยากแสดงให้คุณดูในวิดีโอนี้
คือการหาอนุพันธ์โดยนัยจะให้ผล
เหมือนกับ เราจะเรียกว่า
การหาอนุพันธ์โดยตรงก็ได้ โดยคุณ
หาอนุพันธ์ได้ตรงๆ
สมมุติว่าผมมีความสัมพันธ์
x คูณรากที่สองของ y เท่ากับ 1
อันนี้ค่อนข้างตรงไปตรงมา
หากจะนิยามโดยตรงในรูปของ x
คือแก้หา y
ถ้าเราหารทั้งสองข้างด้วย x 
เราจะได้รากที่สองของ y
เท่ากับ 1/x
แล้วถ้าเรายกกำลังสองทั้งสองข้าง
คุณจะได้ y เท่ากับ 1
ส่วน x กำลังสอง ซึ่งเท่ากับ x
กำลังลบ 2
แล้วถ้าคุณต้องการอนุพันธ์ของ y
เทียบกับ x อันนี้ก็ตรงไปตรงมา
นี่ก็แค่การใช้กฎยกกำลัง
เราได้ dy/dx เท่ากับลบ 2x
กำลังลบ 2 ลบ 1 -- x ยกกำลังลบ 3
มันค่อนข้างตรงไปตรงมา

Korean: 
 
제가 오늘 말하고자 하는 것은
여러분이 음함수 미분법으로
계산한 값과
그냥 미분해서 나온 값이
같을 것이라는 겁니다
x√y=1이라는 식을
세워봅시다
이 함수는 x에 대해서 미분하기에
매우 쉬운 함수입니다
양변을 x로 나눈면 우리는
√y=1/x라는 식을 얻을수 있습니다
양변을 제곱하면
y=1/x²라는 식을 얻을 수 있습니다
y=1/x²라는 식을 얻을 수 있습니다
위의 y를 x에 대해 미분하는 것은
매우 간단합니다
연쇄법칙을 사용하면 됩니다
dy/dx=-2/x³이라는 결과가 나옵니다
dy/dx=-2/x³이라는 결과가 나옵니다
간단합니다

English: 
What I want to show
you in this video
is that implicit
differentiation will give you
the same result as,
I guess we can say,
explicit
differentiation when you
can differentiate explicitly.
So let's say that I
have the relationship
x times the square root
of y is equal to 1.
This one is actually
pretty straightforward
to define explicitly in
terms of x, to solve for y.
So if we divide both sides
by x, we get square root of y
is equal to 1/x.
And then if you square both
sides, you get y is equal to 1
over x squared, which
is the same thing as x
to the negative 2 power.
And so if you want
the derivative of y
with respect to x, this
is pretty straightforward.
This is just an application
of the chain rule.
So we get dy dx is
equal to negative 2 x
to the negative 2 minus 1--
x to the negative 3 power.
So that's pretty
straightforward.

Bulgarian: 
В този урок искам 
да ти покажа,
че диференцирането на 
неявни функции ще ти даде
същите резултати както,
явното диференциране,
тоест когато можеш да диференцираш явно.
Нека да имам дадено отношението
x умножено по квадратен корен от y 
е равно на 1.
Този пример може сравнително директно
да бъде дефиниран за x, 
или да се реши за y.
Ако разделим и двете страни на x,
получаваме, че квадратен корен от y
е равно на 1 върху x.
Ако след това повдигнеш двете страни 
на квадрат, получаваш, че y е равно
на 1 върху x на квадрат, което 
е същото нещо като
x на степен минус 2.
Ако търсиш производната на y
спрямо x, това е сравнително
лесно да се намери.
Това е просто приложение 
на верижното правило.
Получаваме, че dy/dx 
е равно на минус 2x
на степен минус 2 минус 1,
или по x на степен минус 3.
Това е сравнително лесно.

English: 
But what I want to see is if
we get the same exact result
when we differentiate
implicitly.
So let's apply our
derivative operator
to both sides of this equation.
And so let me make
it clear what we're
doing-- x times the square root
of y and 1 right over there.
When you apply the
derivative operator
to the expression on
the left-hand side,
well, actually, we're going
to apply both the product
rule and the chain rule.
The product rule
tells us-- so we
have the product of
two functions of x.
You could view it that way.
So this, the product
rule tells us this
is going to be the
derivative with respect
to x of x times the
square root of y plus x,
not taking its derivative,
times the derivative

Thai: 
แต่สิ่งที่ผมอยากเห็นคือว่า เราได้ผลลัพธ์เหมือนกับ
เวลาเราหาอนุพันธ์โดยนัยไหม
ลองใช้ตัวดำเนินการอนุพันธ์
ทั้งสองข้างของสมการนี้กัน
ขอผมบอกให้ชัดว่าเรา
จะทำอะไร -- x คูณรากที่สองของ y กับ 1 ตรงนี้
เวลาคุณใช้ตัวดำเนินการอนุพันธ์
กับพจน์ทางซ้ายมือ
ที่จริง เราจะใช้กฎผลคูณกับกฎลูกโซ่
ทั้งสองด้าน
กฎผลคูณบอกเราว่า -- เรา
มีผลคูณของฟังก์ชันของ x สองตัว
คุณมองมันแบบนั้นได้
อันนี้ กฎผลคูณบอกเราว่า
อันนี้จะเท่ากับอนุพันธ์เทียบ
กับ x ของ x คูณรากที่สองของ y บวก x
ไม่หาอนุพันธ์ คูณอนุพันธ์

Portuguese: 
Mas quero ver é se chegamos
ao mesmo resultado
quando diferenciamos 
de forma implícita.
Vamos aplicar o 
operador de derivada
a ambos os lados 
dessa equação.
E deixemos claro o
quê estamos fazendo --
x vezes a raiz quadrada 
de y e 1 bem aqui.
Quando usamos o 
operador de derivada
na expressão no lado 
esquerdo da função
e, na verdade, vamos aplicar tanto
a regra dos produtos
quanto a regra da cadeia
O regra do produto 
nos diz -- então temos
o produto de duas 
funções de x.
Você pode ver 
dessa maneira.
Aqui, a regra do 
produto nos diz que
isso será a derivada 
em relação a x
de x vezes a raiz 
quadrada de y mais x,
não calculando 
a sua derivada,

Korean: 
하지만 제가 보이고자 하는 것은
함수를 그냥 미분했을 때에도 값이 
같은지 보는 것입니다
미분연산자를 양 변에
작용시켜봅시다
좌변에는 x√y를 넣고
우변에는 1을 넣읍시다
좌변을 미분할 때
좌변을 미분할 때
곱의 법칙과 연쇄 법칙을 이용하여
미분을 해 봅시다
곱의 법칙을 적용하면
곱의 법칙을 적용하면
곱의 법칙을 적용하면
곱의 법칙을 적용하면
곱의 법칙을 적용하면
(dx/dx)√y에다가
(dx/dx)√y에다가

Czech: 
Ukážeme, že stejný výsledek 
dá i implicitní derivování.
Zderivujme obě dvě 
strany rovnice podle x.
Při derivování výrazu vlevo
použijeme následující pravidla:
o derivaci součinu a 
o derivaci složené funkce.
Z pravidla o derivaci
součinu dostaneme…
Máme tady součin dvou
funkcí proměnné x,
takže dle pravidla o součinu máme derivaci
podle x funkce x krát odmocnina z y

Bulgarian: 
Но какво ще стане ако искам да проверя
дали ще получим същия резултат,
когато диференцираме функцията неявно?
Нека да запишем означението
за диференциране d/dx
в двете страни на това уравнение.
Нека да изясня какво правим.
x по квадратен корен от y и 1 ето там.
Когато приложиш правилата за намиране на производна
към израза в лявата страна 
на уравнението,
в действителност ще приложим две правила:
правилото 
за намиране производна на произведение
и верижното правило.
Правилото за намиране на производна 
на произведение ни казва,
че е дадено произведението 
на две функции на x.
Може да го разглеждаме 
по следния начин.
Правилото за намиране на 
производна на произведение ни казва,
че това ще бъде равно на 
производната спрямо х
на x по квадратен корен от (y + x)...
но не производната му... и умножено по производната

Thai: 
เทียบกับ x ของรากที่สองของ y
ขอผมบอกให้ชัด วงเล็บนี้
 
และทางขวามือ ตรงนี้
อนุพันธ์เทียบกับ x ของค่าคงที่นี้
มันจะเท่ากับ 0
แล้วอันนี้ลดรูปเหลืออะไร?
อนุพันธ์เทียบกับ x ของ x ก็แค่ 1
อันนี้ลดรูปเหลือ 1 เราก็
เหลือแค่รากที่สองของ y ตรงนี้
อันนี้จะลดรูปเหลือรากที่สองของ y
แล้วอันนี้ตรงนี้ลดรูปเหลืออะไร?
อนุพันธ์เทียบกับ x ของรากที่สองของ y
ตรงนี้เราอยากใช้กฎลูกโซ่
ขอผมบอกให้ชัดนะ
เรามีบวก x นี่บวกอะไรก็ตามนี้
และผมจะใช้สีฟ้า
มันจะเท่ากับอนุพันธ์ของรากที่สอง
ของอะไรสักอย่าง เทียบกับอะไรสักอย่างนั้น

Korean: 
(d√y/dx)x를 더하는 것이
좌변임을 구할 수 있습니다
 
우변의 경우는
상수를 x에 대해 미분을 하는 것이므로
0입니다
이것은 무엇을 의미할까요?
x를 x에 대해 미분하는 것은 1입니다
그러므로
√y만 남습니다
√y만 남습니다
이것은 무엇을 의미할까요?
√y를 x에 대해 미분을 할 때
우리는 연쇄법칙을 쓸 것입니다
우리는 연쇄법칙을 쓸 것입니다
우리는 연쇄법칙을 쓸 것입니다
이 파란 색을 계산해봅시다
√y를 x에 대해 미분하는 것입니다
√y를 x에 대해 미분하는 것입니다

Czech: 
a k tomu přičteme součin x 
a derivace podle x z odmocniny z y.
Napravo pak dostaneme
derivaci z konstanty, což je 0.
Jak rovnici zjednodušit?
Derivace z x
podle x je 1.
Tím se nám tento výraz 
zjednoduší na odmocninu z y.
Jak zjednodušit
další výraz?
Zderivování odmocniny z x podle x uděláme
pomocí pravidla o derivaci složené funkce.
Uděláme to postupně.
Přičteme x krát
výraz v modrém.

English: 
with respect to x of
the square root of y.
Let me make it
clear, this bracket.
And on the right-hand
side, right over here,
the derivative with respect
to x of this constant,
that's just going
to be equal to 0.
So what does this simplify to?
Well, the derivative with
respect to x of x is just 1.
This simplifies to
1, so we're just
going to be left with the square
root of y right over here.
So this is going to simplify
to a square root of y.
And what does this
over here simplify to?
Well the derivative with respect
to x of the square root of y,
here we want to
apply the chain rule.
So let me make it clear.
So we have plus this x plus
whatever business this is.
And I'm going to
do this in blue.
Well, it's going to be the
derivative of the square root
of something with respect
to that something.

Bulgarian: 
спрямо x на квадратен корен от y.
Нека да оправя тази скоба.
На квадратен корен от y.
А от дясната страна, точно ето тук,
е производната спрямо x 
на тази константа,
което просто ще бъде равно на 0.
До какво се опростява 
полученият резултат?
Е, производната спрямо x на x 
e просто равна на 1.
Това се получава 1, така че просто
ще остане квадратен корен 
от y точно ето тук.
Това ще се опрости до 
квадратен корен от y.
А до какво се опростява това тук?
За производната спрямо x 
на квадратен корен от y
искаме да приложим верижното правило.
Нека да го изясня.
Получава се плюс този x плюс
 каквото е това нещо.
Ще го направя в синьо.
Това ще бъде равно на 
производната на квадратен корен
от нещо спрямо това нещо.

Portuguese: 
vezes a derivada em relação a 
x da raiz quadrada de y.
Deixe-me ser bem 
claro, esse colchete.
da raiz quadrada de y,
da raiz quadrada de y.
E no lado direito, logo aqui,
a derivada em relação 
a x dessa constante,
isso será somente 
igual a zero.
E como isso será simplificado?
Bom, a derivada em relação
x de x que é somente um.
Isso é simplificado para 
um, então vamos somente
ficar com a raiz 
quadrada de y logo aqui.
Isso será simplificado 
para a raiz quadrada de y.
E como isso logo aqui 
será simplificado?
Bom, a derivada em relação
a x da raiz quadrada de y,
aqui queremos aplicar
a regra da cadeia.
Vou deixar isso claro.
Nós temos mais esse x mais 
o que quer que isso seja.
E eu irie fazer isso em azul.
Bom, isso será a derivada 
da raiz quadrada
de algo em relação àquilo.

Portuguese: 
Bom, a derivada 
da raiz quadrada
de algo em relação a algo,
ou a derivada de 
algo elevado a 1/2
em relação àquela 
coisa, será 1/2 vezes
aquilo elevado a -1/2.
Mais uma vez, isso 
logo aqui é a derivada
da raiz quadrada de 
y em relação a y.
Vimos isso 
múltiplas vezes.
Se eu fosse dizer que a 
derivada da raiz quadrada de x
em relação a x, você poderia 
ter 1/2 x elevado a -1/2.
Agora eu só estou 
fazendo isso com y's.
Mas não terminamos.
Lembre-se, nosso
operador derivativo
não parecia ser em relação a y.
É em relação a x.
Isso só indica ser
em relação a y.
Precisamos aplicar a
regra da cadeia completa
Temos que multiplicar aquilo 
vezes a derivada de y
em relação a x para 
obter a real derivada
dessa expressão 
em relação a x.

Korean: 
√y를 y에 대해 미분을 하면
√y를 y에 대해 미분을 하면
√y를 y에 대해 미분을 하면
(1/2√y)입니다
(1/2√y)입니다
다시 말을 하자면
이 식은 √y를 y에 대해 미분을 한 것입니다
이 식은 √y를 y에 대해 미분을 한 것입니다
만약 √x를 x에 대해 미분을 하게 되면
(1/2√x)를 얻게 됩니다
제가 하고자 하는 것은 y'을 구하는 것입니다
아직 안 끝났습니다
우리가 구하고자 하는 것은
y에 대해 미분을 하는 것이 아니라
x에 대해 미분을 하는 것입니다
그러니 이 식은 y에 대해 미분을 한 것이고
여기에 연쇄법칙을 적용시켜
dy/dx를 곱해주면
dy/dx를 곱해주면
x에 대해 미분한 값을 구할 수 있습니다

English: 
Well, the derivative
of the square root
of something with respect
to that something,
or the derivative of
something to the 1/2
with respect to that something,
is going to be 1/2 times
that something to the
negative 1/2 power.
Once again, this right
over here is the derivative
of the square root of
y with respect to y.
We've seen this multiple times.
If I were to say the derivative
of the square root of x
with respect to x, you would
get 1/2 x to the negative 1/2.
Now I'm just doing it with y's.
But we're not done yet.
Remember, our
derivative operator
wasn't to say with respect to y.
It's with respect to x.
So this only gets us
with respect to y.
We need to apply the
entire chain rule.
We have to multiply that
times the derivative of y
with respect to x in order
to get the real derivative
of this expression
with respect to x.

Thai: 
อนุพันธ์ของรากที่สอง
ของอะไรสักอย่าง เทียบกับอะไรสักอย่างนั้น
หรืออนุพันธ์ของอะไรสักอย่างยกกำลัง 1/2
เทียบกับอะไรสักอย่างนั้น จะเท่ากับ 1/2 คูณ
อะไรสักอย่างนั้นยกกำลังลบ 1/2
ย้ำเอีกครั้ง อันนี้ตรงนี้คืออนุพันธ์
ของรากที่สองของ y เทียบกับ y
เราเห็นอันนี้หลายครั้งแล้ว
ถ้าเราบอกว่าอนุพันธ์ของรากที่สองของ x
เทียบกับ x เราจะได้ 1/2 x ยกกำลังลบ 1/2
ตอนนี้ ผมจะทำแต่ y
แต่เรายังไม่เสร็จ
นึกดู ตัวดำเนินการอนุพันธ์
ไม่ได้บอกว่าเทียบกับ y
มันเทียบกับ x
อันนี้ได้เทียบกับ y
เราต้องใช้กฎลูกโซ่ให้ครบ
เราต้องคูณมันด้วยอนุพันธ์ของ y
เทียบกับ x เพื่อให้อนุพันธ์จริง
ของพจน์นี้เทียบกับ x

Bulgarian: 
Производната на квадратен корен
от нещо спрямо това нещо,
или производната на нещо на степен 1/2
спрямо това нещо, ще бъде равно на 1/2 по
това нещо на степен минус 1/2.
Още веднъж, това ето тук е производната
на квадратен корен от y спрямо y.
Виждали сме това множество пъти.
Ако трябваше да намеря производната на квадратен корен от x
спрямо x, то ще получа 
1/2 по x на степен минус 1/2.
Сега го правя просто с y.
Но все още не сме приключили.
Спомни си, че правилата за намиране на производна
не са за намиране на производна спрямо y.
Правилата са за намиране на производна спрямо x.
A тази е намерена само спрямо y.
Трябва да приложим 
пълното верижно правило.
Трябва да умножим това 
по производната на y
спрямо x, за да получим 
истинската производна
на този израз спрямо x.

Czech: 
To bude jako derivace z ‚něčeho‘ 
na jednu polovinu podle ‚něčeho‘.
Což bude jedna polovina krát
‚něco‘ na minus jednu polovinu.
Ještě jednou, toto je derivace
odmocniny z y podle y.
Viděli jsme
to už mnohokrát.
Třeba derivace odmocniny z x podle x je
jedna polovina x na minus jednu polovinu.
A nyní to stejné
počítáme s y.
Avšak ještě
nejsme hotovi.
Vzpomeňme si, že jsme derivovali
podle y a ne podle x.

Bulgarian: 
Нека да умножим по производната
на y спрямо x.
Не знаем на какво е равно това.
Това всъщност е нещото, което
 се опитваме да намерим.
Но, за да използваме
 верижното правило,
следва да запишем производната
 на квадратен корен от y
спрямо y умножено по 
производната на y спрямо x.
И това е равно на производната
 на този член спрямо x.
Получихме ето това от лявата страна
 на уравнението.
От дясната страна имаме просто 0.
А сега, още веднъж, може да опитаме
да намерим от уравнението 
производната на y спрямо x.
И може би най-лесно е първо
да извадим квадратен корен 
от y oт двете страни
на това уравнение.
И всъщност нека да преместя всичко това,
за да имам повече място, където да работя.
Всъщност, нека да го изрежа.
А след това да го поставя.
Нека да го преместя ето тук.
Преместихме го от тук до там.
Не спечелих много място,
но поне се надявам това да помогне.
Всъщност, така дори не ми харесва.
Нека да го оставя там, 
където беше преди.

Thai: 
ลองคูณมันด้วยอนุพันธ์
ของ y เทียบกับ x
เราไม่รู้ว่าอันนี้คืออะไร
นั่นคือสิ่งที่เราพยายามจะแก้หา
แต่เวลาใช้กฎลูกโซ่ เราแค่
บอกว่า มันคืออนุพันธ์ของรากทีสองของ y
เทียบกับ y คูณอนุพันธ์ของ y
เทียบกับ x
นี่คืออนุพันธ์ของอันนี้เทียบกับ x
เราจึงได้อันนี้ทางซ้ายมือ
ทางขวามือ เรามีแค่ 0
และตอนนี้ ย้ำอีกครั้ง เราพยายาม
แก้หาอนุพันธ์ของ y เทียบกับ x ได้
และบางที ขั้นตอนแรกที่ง่ายที่สุด
คือลบรากที่สองของ y จากทั้งสองข้าง
ของสมการนี้
ที่จริง ขอผมย้ายทุกอย่างไป
ผมจะได้มีที่ทำมากขึ้น
ขอผมตัดมัน
แล้วขอผมวางมันลงไป
ขอผมเลื่อนมันไปตรงนี้
เราไปจากตรงนี้ มาตรงนี้
ผมไม่ได้ที่เพิ่มมากนัก
แต่หวังว่ามันคงช่วยหน่อย
ที่จริง ผมไม่ชอบมันเลย
ขอผมปล่อยมันไว้ตรงที่เดิมดีกว่า

Portuguese: 
Vamos multiplicar a derivada
de y em relação a x.
Não sabemos qual será.
E é Isso que estou 
tentando resolver.
Mas para usar a
regra da cadeia, basta
dizer que sua derivada 
é a raiz quadrada de y
em relação a y 
vezes a derivada de y
em relação a x.
Essa é a derivada 
disso em relação a x.
Nós obtemos isso 
no lado esquerdo.
No lado direito, 
nós teremos um zero.
E agora, mais uma 
vez, podemos tentar
resolver isso para a derivada 
de y em relação a x.
E talvez o primeiro passo seja
subtrair a raiz quadrada 
de y dos dois lados
dessa equação.
E na verdade, deixe-me 
mover tudo isso,
então eu terei, mais uma vez, 
mais espaço para trabalhar.
Deixe-me cortar isso.
E colar.
Deixe-me mover isso logo aqui.
Foi daqui para lá.
Eu não ganhei 
muito espaço,
mas talvez isso 
ajudará um pouco.
E na verdade, eu 
nem gostei disso.
Vou deixar como 
estava antes.

Czech: 
Podle pravidla o derivaci složené funkce
to ještě vynásobíme derivací y podle x.
Derivaci y podle x ještě neznáme
a budeme ji chtít dopočítat.
Z pravidla o derivaci 
složené funkce máme,
že derivace odmocniny z y podle y
krát derivace y podle x
je rovna derivaci 
odmocniny z y podle x.
Toto tedy
máme nalevo.
Napravo
pak je 0.
Připomeňme, že chceme
vyjádřit derivaci y podle x.
Odečtěme od obou stran
rovnice odmocninu z y.
Trochu si to tady upravím,
abych měl více místa na práci.
Posunu to sem doprava.

Korean: 
그러니 dy/dx를 곱합시다
그러니 dy/dx를 곱합시다
우리는 이것이 무엇인지 모릅니다
이것이 우리가 구해야 할 것입니다
연쇄법칙을 쓰면
이식은 √y를 y에 대해 미분한 값에
y를 x에 대해 미분한 값을
곱한 값이 됩니다
이것은 그러므로 위 식을 x에 대해 미분한 값입니다
우리는 이 식을 좌변에 얻었습니다
우변에는 0이 있습니다
다시 한 번 우리는
dy/dx를 구할 수 있습니다
쉬운 방법은
√y를 두 식에서
뽑아내는 것입니다
넓은 공간에서 쓰기 위해서
이것들을 옮기겠습니다
이것들을 옮기겠습니다
이것들을 옮기겠습니다
여기로 옮겼습니다
여기로 옮겼습니다
큰 공간을 얻지는 못했지만
조금은 넓어졌습니다
마음에 안 듭니다
그냥 다시 원래대로 하겠습니다

English: 
So let's multiply
times the derivative
of y with respect to x.
We don't know what that is.
That's actually what
we're trying to solve for.
But to use the
chain rule, we just
have say it's the derivative
of the square root of y
with respect to y times
the derivative of y
with respect to x.
This is the derivative of
this thing with respect to x.
So we get this on
the left-hand side.
On the right-hand
side, we just have a 0.
And now, once again,
we can attempt
to solve for the derivative
of y with respect to x.
And maybe the
easiest first step is
to subtract the square
root of y from both sides
of this equation.
And actually, let me move
all of this stuff over,
so I have, once again,
more room to work with.
So let me cut it, actually.
And then let me paste it.
Let me move it over,
right over here.
So we went from there to there.
I didn't gain a
lot of real estate,
but hopefully this
helps a little bit.
And actually, I
don't even like that.
Let me leave it
where it was before.

Bulgarian: 
Тогава, ако извадим
 квадратен корен от y
от двете страни на уравнението...
 ще се опитам да го опростя,
докато го решавам...получаваме
 ето този израз, който ще запиша като x
умножено по...Просто ще бъде
 равно на x в числител,
разделено на 2 по квадратен корен от y.
 y на степен 1/2
е просто квадратен корен
 от y в знаменателя.
И 1/2, просто поставям 
двойката в знаменател,
а след това умножено по dy/dx,
което е производната на y спрямо x.
И това ще бъде равно на
минус квадратен корен от y.
Просто извадих квадратен корен 
от y от двете страни.
Всъщност ето това е нещо, което може
да искам да копирам и поставя ето тук.
Първо копирам, а след това поставям.
Нека да се върнем тук горе, просто за да 
продължим с опростяването
и решението на уравнението за dy/dx.

Korean: 
양변에서 √y를 빼내서
간단하게 만들어 보면
간단하게 만들어 보면
x/2√y
곱하기 dy/dx입니다
곱하기 dy/dx입니다
곱하기 dy/dx입니다
곱하기 dy/dx입니다
위 식의 값은
-√y와 같은 값을 가집니다
방금 √y를 양 변으로 빼냈습니다
이것을 복사해 붙이겠습니다
이것을 복사해 붙이겠습니다
이것을 복사해 붙이겠습니다
위에서 계속
dy/dx를 구하겠습니다
dy/dx를 구하려면

Czech: 
Vlastně to radši dám sem.
Pak dostaneme nalevo x lomeno
2 odmocniny z y krát derivace y podle x.
A na pravé straně máme
minus odmocninu z y.
Překopírujme si
nyní rovnici sem.
Pokračujme ve vyjadřování
derivace y podle x.

Portuguese: 
Se subtrairmos a 
raiz quadrada de y
dos dois lados -- e eu 
tentarei simplificar enquanto
calculamos -- obtemos isso aqui, 
que podemos reescrever como x
vezes -- bom, será somente 
x no numerador dividido
por dois vezes a raiz quadrada 
de y. y elevado a -1/2
é somente a raiz quadrada 
de y no denominador.
E 1/2, eu coloco o 
dois no denominador
ali -- vezes dy/dx 
vezes a derivada de y
em relação a 
x será igual à
raiz quadrada 
negativa de y.
Eu só subtraí a raiz 
quadrada de y dos dois lados.
E na realidade, isso é 
algo que eu talvez
eu queira copiar 
e colar logo aqui.
Eu copio e colo.
Voltemos para cá, só 
para continuar a simplificação
resolvendo para dy/dx.

Thai: 
แล้ว ถ้าเราลบรากที่สองของ y
จากทั้งสองข้าง -- และผมพยายามลดรูป
เมื่อผมทำไป -- เราได้อันนี้ ซึ่งเราเขียนใหม่ได้เป็น x
คูณ -- นี่ก็คือ x ในตัวเศษหารด้วย
2 คูณรากที่สองของ y. y กำลังลบ 1/2
ก็แค่รากที่สองของ y ในตัวส่วน
และ 1/2 ผมแค่ใส่ 2 ในตัวส่วน
ตรงนั้น -- คูณ dy/dx คูณอนุพันธ์ของ y
เทียบกับ x จะเท่ากับ
ลบรากที่สองของ y
ผมแค่ลบรากที่สองของ y จากทั้งสองด้าน
และที่จริง นี่คือสิ่งที่ผมน่าจะ
ลอกและวางตรงนี้
ลอกแล้วก็วาง
ลองกลับไปบนนี้ จัดรูปกันต่อ
แก้หา dy/dx

English: 
So then, if we subtract
the square root of y
from both sides-- and
I'll try to simplify
as I go-- we get this thing,
which I can rewrite as x
times-- well, it's just going
to be x in the numerator divided
by 2 times the square root
of y. y to the negative 1/2
is just the square root
of y in the denominator.
And 1/2, I just put the
2 in the denominator
there-- times dy dx
times the derivative of y
with respect to x
is going to be equal
to the negative
square root of y.
I just subtracted the square
root of y from both sides.
And actually, this is
something that I might actually
want to copy and paste up here.
So copy and then paste.
So let's go back up here, just
to continue our simplification
solving for dy dx.

English: 
Well, to solve
for dy dx, we just
have to divide both sides by
x over 2 times the square root
of y.
So we're left with dy dx--
or dividing both sides
by this is the same
thing as multiplying
by the reciprocal of
this-- is equal to 2
times the square root of y
over x-- over my yellow x--
times the negative
square root of y.
Well, what's this
going to simplify to?
This is going to be
equal to y times--
the square root of y times the
square root of y is just y.
The negative times the
2, you get negative 2.
So you get negative 2y over x
is equal to the derivative of y
with respect to x.
Now you might be
saying, look, we just
figured out the
derivative implicitly,
and it looks very different than
what we did right over here.

Czech: 
Stačí rovnici vynásobit převrácenou
hodnotou zlomku x lomeno 2 odmocniny z y.
Nalevo máme
derivaci y pod x.
Vpravo bude 2 krát odmocnina z y lomeno x
a to celé krát minus odmocnina z y.
A jak se to
zjednoduší?
Jelikož odmocnina z y krát odmocnina
z y je y, tak dostaneme následující:
−2 krát y lomeno x.
Což je rovno 
derivaci y podle x.
Můžete namítnout, že implicitní derivace
vypadá dosti jinak než ta dříve vypočtená.

Portuguese: 
Bom, para resolver 
para dy/dx nós só
temos que dividir 
ambos os lados por
x sobre dois vezes a 
raiz quadrada de y.
Então ficamos com dy/dx -- ou 
dividindo ambos os lados
pelo mesmo que multiplicar
pelo recíproco disso -- é igual a dois
vezes a raiz quadrada de y 
sobre x -- sobre o meu x amarelo --
vezes a raiz quadrada negativa de y.
Bom, como isso será simplificado?
Isso será igual a y vezes --
a raiz quadrada de y vezes a 
raiz quadrada de y ou apenas y
O negativo vezes o dois, você obtém -2.
Você obtém -2y sobre 
x igual à derivada de y
em relação a x.
Agora você pode 
estar dizendo, nós
acabamos de resolver 
a derivada implicitamente,
e parece ser muito diferente 
do que temos logo aqui.

Korean: 
양 변을 x/2√y로 나누면 됩니다
양 변을 x/2√y로 나누면 됩니다
dy/dx가 좌변에 남고
위 식의 역수인
2√y/x를
양 변에 곱하면
2√y/x 곱하기 -√y입니다
이것이 무엇을 의미할까요?
이것은
-2y/x와 같습니다
-2y/x와 같습니다
-2y/x와 같습니다
dy/dx는 -2y/x와 같습니다
결국 우리는
dy/dx를 구했습니다
이 식은 우리가 처음 구한 식과는 달라 보입니다

Bulgarian: 
За да го решим за dy/dx, просто
трябва да разделим двете страни на
 x/2 по квадратен корен от y.
Оставаме с dy/dx или с
 разделянето на двете страни
на този израз е същото 
като да умножим
по реципрочната му стойност, 
т.е. равна е на
2 по квадратен корен от y/x...
върху моя x с жълт цвят -
и умножено по минус 
квадратен корен от y.
До какво ще се опрости този израз?
Това ще бъде равно на y, умножено по
квадратен корен от y, 
което е равно просто на y.
Има отрицателни знак, т.е. минус 1, 
по 2 и се получава минус 2.
Получаваш минус 2 по y/x, което е 
равно на производната на y спрямо x.
Сега може би си казваш: Ние просто
намерихме производната чрез 
неявно диференциране,
а изглежда много различна, отколкото
 това, което направихме ето тук.

Thai: 
เวลาแก้หา dy/dx เราก็แค่
ต้องหารทั้งสองข้างด้วย x ส่วน 2 คูณรากที่สอง
ของ y
เราก็เหลือแค่ dy/dx -- หรือหารทั้งสองข้าง
ด้วยอันนี้ เท่ากับการคูณ
ด้วยส่วนกลับของอันนี้ -- เท่ากับ 2
คูณรากที่สองของ y ส่วน x -- ส่วน x สีเหลือง --
คูณลบรากที่สองของ y
อันนี้จะลดรูปเหลืออะไร?
อันนี้จะเท่ากับ y คูณ --
รากที่สองของ y คูณรากที่สองของ y ก็แค่ y
ลบคูณ 2 คุณจะได้ลบ 2
คุณจึงได้ลบ 2y ส่วน x เท่ากับอนุพันธ์ของ y
เทียบกับ x
ทีนี้ คุณอาจบอกว่า ดูสิ เราเพิ่ง
หาอนุพันธ์โดยนัยไป
และมันดูต่างจากสิ่งที่เราทำไปตรงนั้น

Thai: 
ตอนที่เราใช้แค่กฎยกกำลัง เราได้ลบ 2 x
ยกกำลังลบ 3
 
ประเด็นตรงนี้ คือสังเกตว่าตัวนี่ตรงนี้
เราแก้ได้โดยตรงในรูปของ -- เรา
แก้หา y ได้
เราแค่แทนค่ามันกลับไป
ตรงนี้แล้วดูว่า พวกมันเท่ากันพอดีไหม
ถ้าเราแทน y เท่ากับ 1 ส่วน x กำลังสอง
คุณจะได้ dy/dx อนุพันธ์ของ y เทียบกับ
x เท่ากับลบ 2 คูณ 1 ส่วน x กำลังสอง
แล้วทั้งหมดนั้นส่วน x ซึ่งเท่ากับลบ 2
ส่วน x กำลังสอง ซึ่งก็คือสิ่งที่เรามีตรงนี้พอดี
ลบ 2 x ยกกำลังลบ 3
 

Portuguese: 
Quando usamos a regra da 
potência nós obtivemos -2x
elevado a -3.
A ideia aqui é perceber que isso aqui
pode ser resolvido 
explicitamente em termos
de -- poderíamos resolver para y.
Fazemos somente essa substituição
logo aqui para ver que esses são 
exatamente a mesma coisa.
Então se fizermos a substituição y 
igual a um sobre x ao quadrado,
nós teríamos dy/dx, a 
derivada de y em relação a
x, igual ao nosso -2 vezes 
um sobre x ao quadrado,
e então tudo isso sobre 
x, que é igual a -2
sobre x ao cubo, o que é 
exatamente o que temos aqui,
-2x elevado ao cubo.
Legendado por: [Musa Morena Marcusso Manhães]
Revisado por: [Rodrigo Melges]

Bulgarian: 
Когато просто приложихме правилото 
за намиране на производна на степен,
получихме минус 2 по x 
на степен минус 3.
x на степен минус 3.
Ключовото нещо тук е да разбереш, 
че при това нещо ето тук,
можехме да решим явно,
т.е. можехме да намерим y.
Може да направим това заместване
ето тук, за да видим, че резултатите от двете
 решения са точно едно и също нещо.
И така, ако направим заместването 
y е равно на 1/x^2,
ще получим dy/dx, т.е. 
производната на y спрямо x,
която е равна на –2 по  1/x^2,
и тогава всичко става върху x, 
което е равно на –2
върху x на трета степен. Това е точно 
резултатът, който имаме ето тук,
т.е. –2 по x на минус трета степен.

English: 
When we just used the power
rule, we got negative 2 x
to the negative third power.
The key here is to realize
that this thing right over here
we could solve explicitly
in terms of-- we
could solve for y.
So we could just make
this substitution
back here to see that these
are the exact same thing.
So if we make the substitution
y is equal to 1 over x squared,
you would get dy dx, the
derivative of y with respect
to x, is equal to our negative
2 times 1 over x squared,
and then all of that over x,
which is equal to negative 2
over x to the third, which is
exactly what we have over here,
negative 2 x to the
negative third power.

Czech: 
Což bylo
−2x na −3.
Všimněme si, že zde 
jsme již vyjádřili y podle x.
Proveďme proto substituci 
y rovno 1 lomeno x na druhou.
Dostaneme, že derivace y podle x
je rovna následujícímu:
−2 krát 1 lomeno x na druhou
a to celé děleno x.
Což je −2x na −3, a tím jsme
dostali stejný výsledek jako dříve.

Korean: 
우리가 곱의 법칙을 이용했을 떄에는
-2/x³을 구했습니다
 
하지만 우리가 깨달아야 하는 것은
우리가 y를 x에 대한 식으로 고칠 수
있다는 것입니다
고로 우리는 이 두 식이
같은 식이라는 것을 구할 수 있습니다
y=1/x²을 이용하면
dy/dx는
-2(1/x²)나누기
x와 같음을 알 수 있습니다
정리하면 -2/x³이고
위에서 구한 식과
 일치함을 알 수 있습니다
 
