
English: 
We've been doing a lot
of examples where we just
take implicit
derivatives, but we
haven't been calculating
the actual slope
of the tangent line
at a given point.
And that's what I want
to do in this video.
So what I want to
do is figure out
the slope at x is equal to 1.
So when x is equal to 1.
And as you can imagine,
once we implicitly
take the derivative
of this, we're
going to have that as
a function of x and y.
So it'll be useful
to know what y
value we get to when
our x is equal to 1.
So let's figure
that out right now.
So when x is equal to 1, our
relationship right over here
becomes 1 squared,
which is just 1
plus y minus 1 to the
third power is equal to 28.
Subtract 1 from both sides.
You get y minus 1 to the
third power is equal to 27.

Thai: 
 
เราได้ทำตัวอย่างมากมาย โดยเรา
หาอนุพันธ์โดยนัย แต่เรา
ยังไม่ได้หาควาชันของ
เส้นสัมผัสที่จุดที่กำหนดจริงๆ เลย
นั่นคือสิ่งที่ผมอยากทำในวิดีโอนี้
สิ่งที่ผมอยากทำคือหา
ความชันที่ x เท่ากับ 1
เมื่อ x เท่ากับ 1
คุณคงนึกออก เมื่อเราหาอนุพันธ์
โดยนัยของความสัมพันธ์นี้ เรา
จะได้มันเป็นฟังกืชันของ x กับ y
มันจึงดีถ้าเรารู้ว่า y
มีค่าเท่าใดเมื่อ x เท่ากับ 1
ลองหาค่านั้นตอนนี้เลย
เมื่อ x เท่ากับ 1 ความสัมพันธ์ของเราตรงนี้
กลายเป็น 1 กำลังสอง ซึ่งก็คือแค่ 1
บวก y ลบ 1 ยกกำลัง 3 เท่ากับ 28
ลบ 1 จากทั้งสองด้าน
คุณได้ y ลบ 1 ยกกำลัง 3 เท่ากับ 27

Czech: 
Sice jsme už počítali spoustu
příkladů na derivaci implicitní funkce,
avšak ještě jsme pomocí ní nenašli
směrnici tečny v daném bodě.
A to bude cílem 
tohoto videa.
Pro danou rovnici budeme chtít
najít směrnici tečny pro x rovno 1.
Všimněme si, při implicitním derivováním
budeme pracovat s funkcí proměnných x a y.
Proto je třeba zjistit, jaké hodnoty 
bude nabývat y, pro x rovno 1.
Tak to prozkoumejme.
Pro x rovno 1, zde 
obdržíme následující:
1 na druhou, což je 1, plus třetí 
mocnina rozdílu y a 1, a to celé rovno 28.
Odečtěme 1 
od obou stran.

Korean: 
 
우리는 지금까지
음함수의 도함수만 구했으며
어떤 특정한 점에서
실제 접선의 기울기는
구하지 않았습니다
그것을 이번에 해보겠습니다
x＝1 일 때
기울기를 구해봅시다
x＝1 일 때
이 음함수의 도함수를
구하면
x와 y에 대한 함수가 됩니다
따라서 x＝1일 때
y값을 구할 수 있습니다
한 번 찾아봅시다
x＝1을 대입해보면
이 식은
1＋(y－1)³＝28 이 됩니다
양변에서 1을 뺍시다
(y－1)³＝27 입니다

Bulgarian: 
Решавали сме много примери, където
намираме производни на неявни функции,
но не сме изчислявали наклон
на допирателна към 
дадена точка от функцията.
А точно това искам 
да направя в настоящия урок.
Искам да намеря наклона
на допирателната за x равно на 1.
Тоест, когато x е равно на 1.
Както можеш да си представиш, когато намерим
производната на тази неявна функция,
то тя ще бъде функция на x и y.
Следователно ще бъде полезно да знаем
каква стойност приема y, 
когато x е равно на 1.
Нека да направим това още сега.
Когато x е равно на 1, уравнението,
което ни е дадено,
става 1 на квадрат, което е равно на 1,
плюс( y – 1) на трета степен,
 и всичко е равно на 28.
Изваждаме 1 от двете страни
на уравнението.
Получаваш (y – 1) на степен 3, 
което е равно на 27.

Portuguese: 
Nós temos feito vários exemplos em que 
calculamos derivadas implícitas,
mas não calculamos a inclinação da 
tangente em um dado ponto.
Este é o tema deste vídeo.
Aqui nós queremos descobrir a inclinação
da tangente para x igual à 1.
Como você pode imaginar, calculando a 
derivada disto, teremos
esta equação em termos de x e y.
Por isto será importante sabermos o valor 
de y quando nosso x é 1.
Então vamos lá:
Quando x é igual à um, substituindo x por 
um temos um ao quadrado, que é um
mais menos um ao cubo é igual à 28.
Subtraindo um de ambos os lados
temos y menos um ao cubo igual à 27.

Korean: 
식이 예쁘게 정리된 것 같아 보입니다
양변을 ⅓ 제곱 해줍니다
y－1＝3 입니다
양변에 1을 더해주면
y＝4 가 됩니다
따라서 (1, 4) 에서의
접선의 기울기를 구해야 합니다
x＝1일 때 y＝4 입니다
여기서 접선의
기울기를 구하면 됩니다
음함수의 미분을 합시다
양변을
미분해서
도함수를 구해봅시다
여기 밑에 쓰겠습니다
x² 의 도함수는
2x 입니다
x에 대한 식이 세제곱 되어 
있는 것을 미분하면
3 과 x에 대한 식을 제곱한 것을 
곱한 것과

Bulgarian: 
Изглежда сякаш числата работят 
в наша полза.
Намираме корен трети от 
двете страни на уравнението.
Получаваш, че y – 1 е равно на 3.
Прибавяме 1 към двете страни 
на уравнението.
Получаваш, че y e равно на 4.
Следователно, ние искаме 
да намерим наклона
в точката (1; 4), която 
се намира ето тук.
Когато x e равно на 1, то y e равно на 4.
Искаме да намерим 
наклона на допирателната
точно в тази точка.
Нека да започнем да диференцираме 
неявната функция.
Ще намерим производната
на двете страни на тази връзка между двете променливи или това уравнение,
в зависимост от начина, 
по който го разглеждаш.
Нека да прескочим оранжевата част.
Производната на x^2 спрямо x.
ще бъде равна на 2x.
След това производната 
спрямо x от нещо
на трета степен ще бъде 3 пъти

Portuguese: 
Parece que os números 
foram bem escolhidos.
Extraímos a raíz cúbica de ambos lados.
Temos y menos um igual à três.
Adicionando um em ambos lados
temos y igual a quatro.
Agora o que nós estamos procurando é a 
inclinação no ponto p(1,4) que é aqui.
Quando x é um e y é quatro.
Estamos tentando descobrir a inclinação 
da linha tangente neste ponto.
Vamos começar fazendo uma 
diferencial implícita.
Vamos calcular a derivada em ambos lados 
desta relação ou equação,
dependendo de como vermos ela.
Vamos continuar os cálculos aqui abaixo.
A derivada com respeito à x de x ao 
quadrado será igual à 2x.
E a derivada em relação à x desta 
expressão ao cubo será três vezes

Czech: 
Máme třetí mocninu 
rozdílu y a 1 je rovna 27.
Proveďme
třetí odmocninu.
Dostaneme, že y minus 1 je 3,
tedy y je rovno 4.
Budeme tedy chtít nalézt
směrnici tečny v bodě [1; 4].
Což je zde.
Chceme zjistit
směrnici tečny v tomto bodě.
Začněme s implicitním derivováním
obou stran této rovnice.
Derivace x na druhou
podle x je 2x.
Derivace ‚něčeho‘ na třetí
podle x bude následující:

Thai: 
ดูเหมือนว่าตัวเลขจะออกมาสวยงามให้เรา
หารากที่สามของทั้งสองข้าง
คุณได้ y ลบ 1 เท่ากับ 3
บวก 1 ทั้งสองข้าง
คุณจึงได้ y เท่ากับ 4
เราอยากหาความชัน
ที่จุด 1 จุลภาค, 1 จุลภาค 4, ซึ่งก็คืตรงนี้
เมื่อ x เป็น 1, y เป็น 4
เราอยากหาความชันของเส้นสัมผัส
ตรงนี้
ลองเริ่มด้วยการหาอนุพันธ์โดยนัย
เราจะหาอนุพันธ์
ทั้งสองข้างของความสัมพันธ์นี้ หรือสมการนี้
ขึ้นอยู่กับว่าคุณอยากมองมันอย่างไร
แล้วลองข้ามลงมาตรงนี้ผ่านพวกสีส้ม
อนุพันธ์เทียบกับ x ของ x กำลังสอง
จะเท่ากับ 2x
แล้วอนุพันธ์เทียบกับ x ของอะไรสักอย่าง
ยกกำลัง 3 จะเท่ากับ 3 คูณ

English: 
It looks like the numbers
work out quite neatly for us.
Take the cube root
of both sides.
You get y minus 1 is equal to 3.
Add 1 to both sides.
You get y is equal to 4.
So we really want to
figure out the slope
at the point 1 comma 1 comma
4, which is right over here.
When x is 1, y is 4.
So we want to figure out the
slope of the tangent line
right over there.
So let's start doing some
implicit differentiation.
So we're going to
take the derivative
of both sides of this
relationship, or this equation,
depending on how
you want to view it.
And so let's skip down
here past the orange.
So the derivative with
respect to x of x squared
is going to be 2x.
And then the derivative with
respect to x of something
to the third power is
going to be 3 times

Czech: 
3 krát ‚něco‘ na druhou 
krát derivace ‚něčeho‘ podle x.
Čemu je rovna derivace 
toho ‚něčeho‘ podle x?
Derivaci y podle x pouze opíšeme
a derivace x podle x je 1.
Jelikož derivace konstanty je 0, 
tak napravo dostaneme 0.
Zbývá zjistit, čemu je 
rovna derivace y podle x.
2x opíšeme.
Nyní tento 
součin roznásobíme.
Dostaneme 3 krát rozdíl y a 
x na druhou krát derivace y podle x.

Korean: 
x에 대한 식을 x에 대해 미분 한 식을
곱한 꼴이 됩니다
x에 대한 이 식의 도함수는 무엇입니까?
y의 도함수는 dy / dx 가 됩니다
x를 미분하면 1입니다
여기 －1이 됩니다
우변의
항은 상수 이므로
미분하면 0이 됩니다
dy와 dx 에 대해서 식을 풀어야 합니다
2x 가 있습니다
이것을 분배해서 dy/dx 를 곱하고
－1 을 곱합니다
dy/dx 를 곱하면
여기 적겠습니다
3(y－x)²(dy/dx) 가 됩니다
－1을 곱하면

Portuguese: 
a expressão ao quadrado multiplicada 
pela derivada desta expressão
em relação a x.
Então qual é a derivada com respeito à x?
A derivada de y em relação a x é 
simplesmente dy/dx.
A derivada de x em relação a x é um.
Temos aqui então menos um.
No lado direito da equação temos zero.
A derivada de uma constante 
é igual a zero.
Ainda temos que resolver para dy/dx.
Aqui temos 2x...
e se distribuírmos essa multiplicação
a primeira equação vezes 
dy dx será igual a...
-- eu vou escrever mais aqui abaixo --
mais três vezes (y - x) ao quadrado 
multiplicado por dy/dx.

Thai: 
อะไรสักอย่างนั้นกำลังสอง คูณอนุพันธ์ของ
อะไรสักอย่างนั้น
เทียบกับ x
แล้ว อนุพันธ์ของอันนี้เทียบกับ x คืออะไร?
อนุพันธ์ของ y เทียบกับ x ก็แค่ dy/dx
แล้วอนุพันธ์ของ x เทียบกับ x ก็แค่ 1
เราจึงได้ลบ 1
แล้วทางขวามือ เราได้แค่ 0
อนุพันธ์ของค่าคงที่เท่ากับ 0
แล้วตอนนี้เราต้องแก้หา dy/dx
เราได้ 2x
แล้วถ้าเราแจกแจงพจน์นี้คูณ dy/dx
แล้วก็คูณลบ 1 เมื่อเราคูณมันด้วย dy/dx
เราจะได้ -- ผมจะเขียนมันตรงนี้นะ --
เราจึงได้บวก 3 คูณ y ลบ x กำลังสอง
คูณ dy/dx
แล้วเมื่อเราคูณมันด้วยลบ 1

English: 
that something squared times
the derivative of that something
with respect to x.
And so what's the derivative
of this with respect to x?
Well the derivative of y with
respect to x is just dy dx.
And then the derivative of x
with respect to x is just 1.
So we have minus 1.
And on the right-hand
side we just get 0.
Derivative of a constant
is just equal to 0.
And now we need to
solve for dy dx.
So we get 2x.
And so if we distribute this
business times the dy dx
and times the negative 1, when
we multiply it times dy dx,
we get-- and actually I'm
going to write it over here--
so we get plus 3 times y
minus x squared times dy dx.
And then when we multiply
it times the negative 1,

Bulgarian: 
по това нещо, повдигнато на квадрат и 
умножено по производната на това нещо
спрямо x.
На какво е равна производната
на това нещо спрямо x?
Производната на y спрямо x 
е просто dy/dx.
След това производната на x 
спрямо x е просто 1.
Следователно получаваме минус 1.
От дясната страна 
получаваме просто 0.
Производната на константа 
е равна на 0.
Сега следва да решим уравнението
 за dy/dx.
И така, записваме 2x.
Ако разкрием скобите и умножим 
този израз тук по dy/dx –1,
когато умножим това по dy/dx,
получаваме... ще го запиша ето тук...
получаваме плюс 3 по (y – x)^2,
 умножено по dy/dx.
След това, когато умножаваме по минус 1,

Thai: 
เราจะได้ลบ 3 คูณ y ลบ y ลบ x กำลังสอง
แล้วแน่นอน ทั้งหมดนี้จะเท่ากับ 0
ตอนนี้ที่เราต้องทำคือนำอันนี้
มาใส่ทางขวามือ
เราจะลบมันจากทั้งสองด้านของสมการนี้
ทางซ้ายมือ -- ที่จริงทุกอย่าง
ที่ไม่ใช่ dy/dx ผมจะเขียน
ด้วยสีเขียว -- ทางซ้ายมือ
เราเหลือแค่ 3 คูณ y ลบ x กำลังสองคูณ dy/dx
อนุพันธ์ของ y เทียบกับ x
เท่ากับ -- ผมแค่ลบอันนี้ทั้งสองข้าง --
เท่ากับลบ 2x บวกอันนี้
ผมเขียนมันได้เป็น 3 คูณ y ลบ x กำลังสองลบ 2x
เราก็บวกอันนี้ทั้งสองข้าง
และเราลบอันนี้ทั้งสองข้าง
ลบ 2x
แล้วเวลาแก้หา dy/dx เรา
ทำมาหลายครั้งแล้ว
เวลาแก้หาอนุพันธ์ของ y เทียบกับ x
อนุพันธ์ของ y เทียบกับ x

Korean: 
－3(y－x)² 이 됩니다
이것의 결과는 0이
될 것입니다
우리는 이제 이것을
우변으로 넘겨야 합니다
양변에서 이것을 
빼겠습니다
dy/dx 는 보라색으로 쓰고
나머지는
초록색으로 쓰겠습니다
좌변에는
3(y－x)² (dy/dx) 가 됩니다
x에 대한 y의 도함수를
구하기 위해서 양변에서 이것을
빼주면
－2x에 이것을 더한 것이 됩니다
즉 3(y－x)²－2x 가 됩니다
이것은 양변에 더해주고
이것은 양변에서 빼줍니다
－2x
dy/dx 에 대해서 풀려면
앞서 했듯이 하면 됩니다
x에 대한 y의 도함수는
x에 대한 y의 도함수는

English: 
we get negative 3 times y
minus y minus x squared.
And then of course, all of
that is going to be equal to 0.
Now all we have
to do is take this
and put it on the
right-hand side.
So we'll subtract it from
both sides of this equation.
So on the left-hand side--
and actually all the stuff
that's not a dy dx
I'm going to write
in green-- so on
the left-hand side
we're just left with 3 times
y minus x squared times dy dx,
the derivative of
y with respect to x
is equal to-- I'm just going to
subtract this from both sides--
is equal to negative
2x plus this.
So I could write it as 3 times
y minus x squared minus 2x.
So we're adding
this to both sides
and we're subtracting
this from both sides.
Minus 2x.
And then to solve
for dy dx, we've
done this multiple
times already.
To solve for the derivative
of y with respect to x.
The derivative of
y with respect to x

Czech: 
A od toho odečteme
3 krát rozdíl y a x na druhou.
To celé je
pak rovno 0.
Nyní dostaňme toto, odečtením od
obou stran rovnice, na druhou stranu.
Nalevo pak zůstane 3 krát rozdíl y a 
x na druhou krát derivace y podle x.
Napravo pak bude 3 krát
rozdíl y a x na druhou minus 2x.
Vyjádření derivace y podle x
jsme počítali již mnohokrát.
Vyjádříme derivaci y vzhledem k x.

Bulgarian: 
получаваме минус 3 по (y – x)^2.
След това, разбира се, 
всичко това е равно на 0.
Всичко, което сега 
трябва да направим,
е да вземем ето този израз и да го прехвърлим 
от дясната страна на уравнението.
Следователно ще го извадим 
от двете страни на уравнението.
От лявата страна,  и всъщност, всичко,
което не е dy/dx, ще запиша
в зелен цвят. От лявата страна
остава само 3 по (y – x)^2,
 умножено по dy/dx,
т.е. производната 
на y спрямо x, която...
просто ще извадя това от 
двете страни на уравнението:
е равна на минус 2x плюс този израз.
Мога да го запиша като 
3 по (y – x)^2 минус 2x.
Прибавяме този израз към
 двете страни на уравнението
и го изваждаме от двете страни
 на уравнението.
Минус 2x.
След това следва да намерим 
от уравнението dy/dx,
което сме правили вече множество пъти.
Следва да получим на какво 
е равна производната на y спрямо x.
Производната на y спрямо x

Portuguese: 
E quando multiplicamos por menos um, 
temos -3 vezes (y - x) ao quadrado.
E toda esta expressão será igual à zero.
Agora pegamos estes dois termos e passamos
para o lado direito da igualdade.
Para isto, subtraimos os termos 
em ambos os lados.
-- Agora tudo que não é dy/dx 
vou reescrever em verde --
Ao nosso lado esquerdo sobrou:
três vezes (y - x) ao quadrado dy/dx
-- que é a derivada de y em relação a x --
é igual a:
-- eu vou subtrair isto de ambos lados --
-2x mais isto
3 vezes (y - x) ao quadrado menos 2x.
Estamos adicionando isto em ambos lados
e subtraindo isto de ambos lados.
Menos 2x.
E resolvemos para dy/dx, como 
já fizemos várias vezes.
Agora resolveremos a derivada 
de y em relação a x

Korean: 
3(y－x)² －2x 를
3(y－x)² 으로 나눈 값이 됩니다
지금은 놔두겠습니다
x에 대한 y의 도함수는
무엇입니까?
x＝1이고 y＝1일 때
접선의 기울기는
얼마 입니까?
x＝1과 y＝4를 이 식에
대입해 봅시다
3(4－1)²  빼기
2가 됩니다
이 전체를 3(4－1)²
로 나눠줍니다
제곱하면
9입니다
3곱하기 9는 27 입니다
분자는 27－2＝25

Czech: 
To je rovno 3 krát y minus
x na druhou minus 2x,
kde to celé vydělíme 3 krát
rozdíl y a x na druhou.
Pro nalezení derivace y podle x
v daném bodě stačí substituci do výrazu,
kde položíme 
x rovno 1 a y rovno 4.
V čitateli dostaneme 3 krát rozdíl
4 a 1 na druhou minus 2 krát 1.
Ve jmenovateli pak máme 
3 krát rozdíl 4 a 1 na druhou.

Thai: 
จะเท่ากับ 3 คูณ y ลบ x กำลังสองลบ 2x
ทั้งหมดนั้นส่วนอันนี้, 3 คูณ y ลบ x กำลังสอง
และเราปล่อยมันแบบนี้ไปก่อนได้
แล้วอนุพันธ์ของ y เทียบกับ x เป็นเท่าใด?
ความชันของเส้มสัมผัสเมื่อ x เป็น 1
และ y เท่ากับ 4 คืออะไร?
เราต้องแทน x เท่ากับ 1 และ y
เท่ากับ 4 ลงในพจน์นี้
มันจะเท่ากับ 3 คูณ 4 ลบ 1 กำลังสอง
ลบ 2 คูณ 1
ทั้งหมดนั้นส่วน 3 คูณ 4 ลบ 1 กำลังสอง
ซึ่งเท่ากับ 4 ลบ 1 ได้ 3
คุณยกกำลังสองมัน
คุณได้ 9
9 คูณ 3 ได้ 27
คุณจะได้ 27 ลบ 2 ในตัวเศษ

English: 
is going to be equal to 3 times
y minus x squared minus 2x.
All of that over this stuff,
3 times y minus x squared.
And we can leave it
just like that for now.
So what is the derivative
of y with respect to x?
What is the slope of the
tangent line when x is 1
and y is equal to 4?
Well we just have to substitute
x is equal to 1 and y
equals 4 into this expression.
So it's going to be equal
to 3 times 4 minus 1 squared
minus 2 times 1.
All of that over 3
times 4 minus 1 squared,
which is equal to
4 minus 1 is 3.
You square it.
You get 9.
9 times 3 is 27.
You get 27 minus 2
in the numerator,

Bulgarian: 
ще бъде равна на 
3 по (y – x)^2 минус 2x.
Всичко това е върху този израз, 
т.е. 3 по (y – x)^2.
Засега може да го оставим така.
И на какво е равна 
производната на y спрямо x?
Каква е стойността на наклона
 на допирателната, когато x = 1,
а y е равно на 4?
Е, просто трябва да заместим
 x равно на 1 и
у е равно на 4 в този израз.
Следователно ще бъде равно 
на 3 по (4 – 1) на квадрат
минус 2 по 1.
Целият този израз е върху 
3 по (4 – 1) на квадрат,
което е равно на 4 – 1 и е равно на 3.
Повдигаш на квадрат.
Получаваш 9.
9 по 3 е равно на 27.
Получаваш 27 минус 2 в числител,

Portuguese: 
que é igual a três vezes (y - x) 
ao quadrado menos 2x.
Tudo isto sobre este termo, 3 vezes 
(y - x) ao quadrado.
Nós podemos deixar isto assim por agora.
Qual é a derivada de y em relação a x?
Qual é a inclinação da tangente quando 
x é um e y é igual a quatro?
Para isto basta substituir x por um e 
y por quatro nesta expressão.
Isto será igual a três vezes (4 - 1) ao 
quadrado menos dois vezes um.
Tudo isto sobre 3 vezes (4 - 1) ao 
quadrado que é igual a:
quatro menos um é três,
elevando ao quadrado temos nove.
nove vezes três é 27.
Temos então 27 menos dois no numerador, 
que é igual a 25.

Bulgarian: 
което ще бъде равно на 25.
А в знаменателя получаваш 
3 по 9, което е  равно на 27.
Следователно наклонът 
е равен на 25/27.
Стойността му е почти 1, но не точно.
И точно по този начин изглежда 
на ето тази графика.
И, за да се уверя, че знаеш
откъде разполагам с тази графика:
Това е от приложението Wolfram Alpha.
Трябваше да ти кажа още от началото.
Както и да е. Надявам се,
 че задачата ти е харесала.

English: 
which is going to
be equal to 25.
And in the denominator, you
get 3 times 9, which is 27.
So the slope is 25/27.
So it's almost 1, but not quite.
And that's actually what it
looks like on this graph.
And actually just
to make sure you
know where I got this graph.
This was from Wolfram Alpha.
I should have told you
that from the beginning.
Anyway, hopefully
you enjoyed that.

Portuguese: 
E no denominador, temos três vezes nove, 
que é 27.
A inclinação é 25/27.
É quase um, mas não chega a ser um.
E este gráfico mostra como esta curva 
se parece...
E só para você saber onde eu 
encontrei este gráfico,
foi do Wolfram Alpha.
Eu deveria ter dito isto no início.
De qualquer forma, espero 
que você tenha gostado.
Legendado por [ José Irigon ]
Revisado por [Rodrigo Melges]

Thai: 
ซึ่งเท่ากับ 25
และในตัวส่วน คุณจะได้ 3 คูณ 9 ซึ่งก็คือ 27
ความชันจึงเท่ากับ 25/27
มันเกือบเท่ากับ 1 แต่ไม่ถึง
นั่นคือสิ่งที่เป็นบนกราฟนี้
และเพื่อให้แน่ใจว่าคุณ
รู้ว่าผมได้กราฟมาจากไหน
มันมาจาก Wolfram Alpha
 
ผมควรบอกคุณตั้งแต่แรก
เอาล่ะ หวังว่าคุณคงสนุกนะ

Korean: 
가 됩니다
분모는 3과 9를 곱해서
27이 됩니다
기울기가 25/27 입니다
1은 아니지만 거의 1 입니다
이 그래프에서도 그렇게 보입니다
이 그래프의
출처는
Wolfram Alpha 입니다
 
시작할 때 말씀 드릴걸
그랬습니다
시청해주셔서 감사합니다

Czech: 
V čitateli dostaneme 
27 minus 2 což je 25.
Ve jmenovateli
pak obdržíme 27.
Hledaná směrnice tečny
je 27 lomeno 25.
Tedy téměř 1.
Což je i patrné
na obrázku.
Nakonec poznamenejme, že obrázek grafu
křivky je z programu Wolfram Alpha.
Snad jste se něco naučili.
