
English: 
The alternate form of the
derivative of the function
f, at a number a,
denoted by f prime of a,
is given by this stuff.
Now this might look a
little strange to you,
but if you really think
about what it's saying,
it's really just
taking the slope
of the tangent line
between a comma f of a.
So let's imagine some
arbitrary function like this.
Let's say that that
is-- well I'll just
write that's our function f.
And so you could
have the point when
x is equal to a-- this is our
x-axis-- when x is equal to a,
this is the point a, f of a.
You notice a, f of a.
And then we could take the slope
between that and some arbitrary
point, let's call that x.
So this is the point x, f of x.
And notice, the numerator
right here, this
is just our change in the
value of our function.
Or you could view that as the
change in the vertical axis.

Thai: 
 
รูปทางเลือกของอนุพันธ์ของฟังก์ชัน
f ที่จำนวน a แทนด้วย f ไพรม์ของ a
กำหนดด้วยพจน์นี้
อันนี้อาจดูแปลกสำหรับคุณ
แต่ถ้าคุณคิดถึงสิ่งที่มันกำลังบอก
มันก็แค่ความชัน
ของเส้นสัมผัสระหว่างจุด (a, f ของ a)
ลองนึกภาพฟังก์ชันตามใจสักตัวแบบนี้
สมมุติว่านั่นคือ -- ผมจะ
เขียนว่านั่นคือฟังก์ชันของเรา f
แล้วคุณก็มีจุดที่
x เท่ากับ a -- นี่คือแกน x ของเรา --
เมื่อ x เท่ากับ a
นี่คือจุด (a, f ของ a)
คุณสังเกต (a, f ของ a)
แล้วเราหาความชันระหว่างจุดนั้นกับจุด
ใดๆ อีกจุด เรียกมันว่า x
นี่คือจุด (x, f ของ x)
 
และสังเกตว่า ตัวเศษตรงนี้
นี่ก็แค่การเปลี่ยนแปลงของค่าฟังก์ชันเรา
หรือคุณมองมันเป็น
การเปลี่ยนแปลงของแกนตั้งก็ได้

Korean: 
 
a에서 f ' 즉 f 함수의
도함수는
이런 형태입니다
이것은 조금 어색할 수 있습니다
하지만 그 의미를 살펴보면
그냥 f 함수에서
접선의 기울기를 구하는 것과 같습니다
임의의 함수를 봅시다
이것을
함수 f 라고 하겠습니다
여기가 x 축에서 a라고 하면
여기가 x 축에서 a라고 하면
이 점이 f(a) 라는 것을 
알 수 있습니다
(a, f(a))
저 점과 임의의 점 x 사이의
기울기를 구해봅시다
이 점의 좌표는 （x, f(x)）입니다
 
이 분자에 있는 것은
함숫값의 변화입니다
수직상의 변화라고 봐도 무방합니다

Portuguese: 
A forma alternativa da derivada da função
de f do número a
indicada por f linha de a é dada por
isto aqui
Agora isso pode parecer estranho
mas se pensar a respeito
está mostrando que é a inclinação
da linha tangente entre a e f de a.
Então vamos imaginar alguma
função arbitrária como esta,
digamos que é
Deixe-me escrever nossa função f,
e então você pode
ter o ponto onde x é igual a
este é nosso eixo x
este é o ponto: (a, f(a))
veja: a, f(a)
e então
podemos pegar a inclinação entre aquele e
outro ponto arbitrário vamos chamar de x
então este é o ponto: x, f(x)
e veja
o numerador aqui, é o valor que
fará a variação da função
ou você pode ver como a variação
no eixo vertical

Bulgarian: 
Другият вид на производната
на функция f,
за число а, означена като f'(a),
е дадена чрез този израз.
Това сега може да ти изглежда
 малко странно,
но ако наистина помислиш
 върху това, което казват,
просто представлява наклона
на допирателната в точката (a; f(a)).
Нека да си представим 
произволна функция като тази.
Нека да кажем, че е това.
Просто ще напиша, че 
това е нашата функция f.
И така, може да имаш
 точка, където
x = a. Това е нашата ос x, 
а там където x = a,
то това е точката (a; f(a)).
Забелязваш (a; f(a)).
След това можем да вземем 
наклона между тази и друга
произволна точка, която 
ще наречем x.
И така, това е точката (x; f(x)).
 
Забележи числителя ето тук. Това
е просто изменението 
в стойността на функцията.
Или може да гледаш на него
 като изменение по вертикалната ос.

Czech: 
Alternativní definice derivace funkce f
v bodě ‚a‘, značená f s čárkou v bodě ‚a‘,
je dána
tímto vzorcem.
Tohle možná vypadá trochu podivně,
ale když se nad tím pořádně zamyslíme,
tak jde jen o směrnici
tečny v bodě [a;f(a)].
Nakresleme si nějakou
libovolnou funkci.
Vypadá třeba takto,
pojmenuji ji funkce f.
Tady na ose x máme
bod x rovná se ‚a‘,
takže tohle
bude bod [a;f(a)].
Pak můžeme vzít směrnici přímky mezi tímto
bodem a libovolným jiným, nazvěme ho x.
Toto bude
bod [x;f(x)].
Všimněte si, že čitatel je pouze
změna funkční hodnoty.
Nebo se na to můžeme dívat jako
na změnu hodnoty na svislé ose.

English: 
So that would give you this
distance right over here.
That's what we're doing
up here in the numerator.
And then in the
denominator, we're
finding the change in
our horizontal values,
horizontal coordinates.
Let me do that in
a different color.
So the change in the horizontal,
that's this right over here.
And then they're trying to find
the limit as x approaches a.
So as x gets closer and
closer and closer and closer
to a, what's going to happen
is, is that when x is out here,
we have this secant line.
We're finding the slope
of this secant line.
But as x gets closer and
closer, the secant lines
better and better and
better approximate
the slope of the tangent line.
Where the limit, as x approaches
a, but doesn't quite equal a,
is going to be--
this is actually
our definition of
our derivative.
Or I guess the alternate form
of the derivative definition.
And this would be the slope of
the tangent line, if it exists.
So with that all
that out the way,
let's try to answer
their question.

Czech: 
Jde o tuto vzdálenost,
tu představuje náš čitatel.
Ve jmenovateli pak máme
změnu hodnoty na vodorovné ose.
Udělám to
jinou barvou.
Změna na vodorovné
ose je tohle.
A z toho chceme určit
limitu pro x blížící se k ‚a‘.
Jak se x čím dál tím víc
blíží k ‚a‘, tak…
Když je x tady, můžeme
udělat tuto sečnu.
Chceme najít
směrnici této sečny.
Ale jak je x čím dál blíž, tak se
sečna čím dál tím víc podobá tečně.
Limita pro x blížící se k ‚a‘,
ale ne rovno ‚a‘, bude…
Tohle je vlastně
naše definice derivace.
Neboli alternativní
definice derivace.
A je to také směrnice tečny,
pokud tečna existuje.
Když už je tohle jasné,
zkusme teď zodpovědět na otázku.

Portuguese: 
então isso dará esta distância aqui, e é o
que estamos fazendo aqui no numerador
e então no denominador, estamos
procurando a variação
nos nossos valores horizontais,
nas coordenadas horizontais
e isso faremos numa cor diferente
e isto te dará
a variação na horizontal,
que é isso logo aqui.
Então tentamos aplicar o limite
quando x se aproxima de a
quando x chega cada vez mais
perto de a o que acontecerá
é que quando x está aqui teremos
esta linha secante
nós encontramos a inclinação
desta secante
mas como x fica cada vez mais perto
a linha secante fica
mais e mais próxima da inclinação
da linha tangente
onde o limite de x aproxima de a,
mas não fica igual,
esta é a definição da nossa derivada
ou sua forma alternativa.
E isso seria a inclinação da linha
tangente que existe
então com isso em mente, vamos
tentar responder a questão

Thai: 
มันจะให้ระยะนี่ตรงนี้
นั่นคือสิ่งที่เรากำลังทำบนนี้ในตัวเศษ
แล้วในตัวส่วน เรา
จะหาการเปลี่ยนแปลงของแกนนอน
พิกัดแนวนอน
ขอผมใช้อีกสีนะ
 
การเปลี่ยนแปลงของแกนนอน มันคืออันนี้ตรงนี้
แล้วเขาพยายามหาลิมิตเมื่อ x เข้าใกล้ a
เมื่อ x เข้าใกล้ a มากขึ้น มากขึ้น และมากขึ้น
สิ่งที่จะเกิดขึ้นคือว่า เมื่อ x อยู่ข้างนอกนี้
เรามีเส้นตัด
เรากำลังหาความชันของเส้นตัดนี้
แต่เมื่อ x ใกล้เข้ามาเรื่อยๆ เส้นตัด
จะประมาณความชันของเส้นสัมผัส
ได้ดีขึ้นเรื่อยๆ
โดยลิมิต เมื่อ x เข้าใกล้ a 
แต่ไม่เท่ากับ a ทีเดียว
จะเท่ากับ -- อันนี้ที่จริงแล้ว
คือนิยามอนุพันธ์ของเรา
หรือผมจะเรียกว่า รูปทางเลือกของ
นิยามอนุพันธ์ก็ได้
และอันนี้จะเป็นความชันของเส้นสัมผัส ถ้ามีจริง
พักเรื่องนั้นไว้
ลองตอบคำถามกันดีกว่า

Korean: 
이 간격을 말합니다
그것이 여기서 분자가 뜻하는 바입니다
분모에서는
수평축 상의 변화를 다루고 있습니다
수평축 상의 변화를 다루고 있습니다
다른 색깔로 하겠습니다
 
수평축 상의 변화는 여기있습니다
그리고 x가 a 로 갈때의
극한 값을 찾아야 합니다
x가 a로 가까워지면서
x가 여기있을 때
선을 얻을 수 있습니다
우리는 이 선의 기울기를 찾아야 합니다
그런데 x가 점점 가까워지면
선의 기울기는
접선의 기울기가 됩니다
x가 a가 되진 않지만 a에 가까워지면
이것은
도함수의 정의가 됩니다
도함수의 또 다른 정의라고 볼 수 있습니다
접선이 존재한다면
그 선의 기울기가 됩니다
우선 다 제쳐두고
문제를 풀어보도록 합시다

Bulgarian: 
А това ще ти даде разстоянието
 точно ето тук.
Това е, което се случва тук
 в числителя.
След това в знаменателя
търсим изменението за 
хоризонталните стойности,
т.е. в хоризонталните координати.
Нека да го направя с различен цвят.
Това ще ти даде...
Изменението по хоризонталата
 е ето това тук.
След това се опитват да намерят 
границата, когато x клони към a.
Когато x се приближава 
все повече и повече до а,
това, което се случва, е, 
че когато x е тук,
имаме тази секуща.
Намираме наклона на тази секуща.
Когато обаче x се приближава 
все повече до a, то секущата
все по-добре и по-добре 
се доближава до
наклона на допирателната.
Там където е границата, когато 
x клони към a, но не е точно равно на a,
това всъщност ще бъде
нашето определение за производна.
Или другият (алтернативният) вид
 на определението за производна.
Това ще бъде наклонът на 
допирателната, ако тя съществува.
И така, с това, което 
открихме дотук,
нека да се опитаме да отговорим 
на поставения въпрос.

Portuguese: 
Com a forma alternativa da
derivada para ajudar,
resolva a seguinte expressão de limite
identificando a função f e o número a.
Então aqui querem encontrar a inclinação
da linha tangente em cinco
aqui ele querem achar a inclinação
da linha tangente em a
então está claro que
a é igual a cinco
e que f de a é igual a 125.
Agora, e sobre f de x?
Bem, aqui está o limite de f(x)-f(a),
e aqui está o limite de
x ao cubo menos 125, e isso faz sentido
se f de x é igual a x ao cubo
então faz sentido que f de cinco
f de cinco será igual a cinco ao cubo
que será igual a 125, e estamos também
chegando no limite de x
quando se aproxima de a que é
x aproximando de cinco

Korean: 
도함수의 정의를 이용해서
저 식이 말이 되도록
함수 f 를 정의하고  a의 값을 
정해야 합니다
여기서 x＝5일 때
접선의 기울기를 구해야 합니다
이 식은 x＝a일 때 접선의
기울기를 구하는 것입니다
a＝5 임은 거의 확실합니다
그리고 f(5)＝125 일 것입니다
f(x)는 무엇일까요?
여기서 이것은 f(x)－f(a) 입니다
여기서는 x³ －125 입니다
이것은 말이 됩니다
f(x)가 x³ 이면
f(5)＝5³ ＝125 입니다
f(5)＝5³ ＝125 입니다
그리고 x가 a로 갈 때의 극한값을 보면

Bulgarian: 
С помощта на другия вид на производната
има смисъл от следния израз
 за граница,
като определим функцията f 
и числото a.
И така, ето тук, искат от нас 
да намерим наклона на допирателната
в точката x = 5.
Тук искаха да намерим наклона
 на допирателната в точка a.
Сега е ясно, че точката a = 5.
И, че f(a) = 125.
А какво става с f(x)?
Е, ето тук имаме 
границата за f(x) – f(a).
Ето тук е границата, дадена
 като x^3 – 125.
И това има смисъл.
Ако f(x) = x^3,
тогава е логично, че f(5) = 5^3,
което ще бъде 125.
А ето тук също търсим границата, 
когато x клони към a.

Czech: 
Pomocí alternativní definice derivace
vysvětlete následující limitu tak,
že zjistíte, co je funkce f
a co je bod ‚a‘.
Tady chtějí najít
směrnici tečny v bodě 5,
zatímco zde chtěli najít
směrnici tečny v bodě ‚a‘,
takže je docela zřejmé,
že ‚a‘ je rovno 5
a že f v bodě ‚a‘
je rovno 125.
Čemu je rovno f(x)?
Tady je limita z
f(x) minus f(a)
a toto je limita z výrazu
x na třetí minus 125.
A to dává smysl.
Pokud je f(x) rovno x na třetí,
pak dává smysl, že f(5) je 5 na třetí,
což je 125.
Zde máme limitu
pro x blížící se k ‚a‘

English: 
With the Alternative Form
of the Derivative as an aid,
make sense of the
following limit expression
by identifying the function
f and the number a.
So right here, they want to find
the slope of the tangent line
at 5.
Here they wanted to find the
slope of the tangent line at a.
So it's pretty clear
that a is equal to 5.
And that f of a is equal to 125.
Now what about f of x?
Well here, it's a limit
of f of x minus f of a.
Well here it's the limit as
x to the third minus 125.
And this makes sense.
If f of x is equal
to x to the third,
then it makes sense that f of 5
is going to be 5 to the third,
is going to be 125.
And we're also taking up here
the limit as x approaches a.

Thai: 
ใช้รูปทางเลือกของอนุพันธ์เป็นตัวช่วย
ทำความเข้าใจพจน์ลิมิต
ต่อไปนี้โดยระบุฟังก์ชัน f และจำนวน a
ตรงนี้ เขาอยากหาความชันของเส้นสัมผัส
ที่ 5
ตรงนี้ เขาอยากหาความชันของเส้นสัมผัสที่ a
เห็นได้ชัดว่า a เท่ากับ 5
และ f ของ a เท่ากับ 125
แล้ว f ของ x ล่ะ?
ตรงนี้ มันคือลิมิตของ f ของ x ลบ f ของ a
ตรงนี้ มันคือลิมิตเมื่อ x กำลังสามลบ 125
และมันสมเหตุสมผล
ถ้า f ของ x เท่ากับ x กำลังสาม
แล้วมันก็สมเหตุสมผลที่ f ของ 5
จะเท่ากับ 5 กำลังสาม
จะเท่ากับ 125
แล้วเรามาบนนี้ ลิมิตเมื่อ x เข้าใกล้ a

Thai: 
ตรงนี้เราจะหาลิมิตเมื่อ x เข้าใกล้ 5
นี่ก็คืออนุพันธ์ของฟังก์ชัน
f ของ x เท่ากับ x กำลังสาม
ขอผมเขียนมันลงไปด้วยสีเขียวนะ
x กำลังสามที่จำนวน a เท่ากับ 5
แล้วเราก็นึกภาพได้
ลองพยายามวาดกราฟมันดู เราจะได้นึกออก
ที่จริง ผมจะทำตรงนี้ โดย
ผมจะได้มีสีแยกชัดเจนหน่อย
สมมุติว่านั่นคือแกน y
สมมุติว่านี่คือแกน x
ผมจะไม่วาดมันตามสัดส่วนนะ
สมมุติว่าค่านี่ตรงนี้คือ 125
หรือ y นี่คือเมื่อ y เท่ากับ 125
นี่คือเมื่อ x เท่ากับ 5 พวกมันไม่ได้
วาดตามสัดส่วนชัดเจน
แต่ฟังก์ชันจะเป็นแบบนี้
เรารู้ว่า x กำลังสามเป็นอย่างไร
มันจะเป็นแบบนี้

English: 
Here we're taking the
limit as x approaches 5.
So this is the derivative
of the function
f of x is equal
to x to the third.
Let me write that down
in the green color.
x to the third at the
number a is equal to 5.
And so we can imagine this.
Let's try to actually graph it,
just so that we can imagine it.
Actually, I'll do
it out here, where
I have a little bit better
contrast with the colors.
So let's say that is my y-axis.
Let's say that
this is my x-axis.
I'm not going to quite
draw it to scale.
Let's say this right
over here is the 125.
Or y, this is when y equals 125.
This is when x is equal
to 5, so they're clearly
not at the same scale.
But the function is going
to look something like this.
We know what x to
the third looks like,
it looks something like this.

Korean: 
여기서는 x가 5로
갈 때의 극한값을 구해봅시다
그러니까 이건
f(x)＝x³ 의 도함수입니다
초록색으로 적겠습니다
a＝5 일 때 x가 a 즉 5로 갑니다
생각해봅시다
그래프를 그리면
더 쉬울 것 같습니다
여기서 그리면 더
보기 쉬울 것 같습니다
이것이 y축입니다
이것이 x 축입니다
간격이 좀 이상할 수 있습니다
여기가 125입니다
y＝125 가 되는 것입니다
x＝5입니다
x축 간격과 y축 간격이
다릅니다
그래도 함수는 대략적으로
이렇게 생겼을 것입니다
y=x³ 그래프는
이렇게 생겼습니다

Portuguese: 
então, esta é a derivada da função f de x
que é igual a x ao cubo, deixe-me escrever
na cor verde
x ao cubo com o número x igual
com o número a igual a cinco,
então podemos pensar
vamos tentar fazer um gráfico para tentar
imaginar, na verdade tentarei
fazer aqui, onde temos um contraste
um pouco melhor com as cores
digamos que este é meu eixo y
e isto é meu eixo x.
Não vou me preocupar com a escala,
digamos que aqui
é o ponto 125 ou y é igual a 125
que é quando x é igual a cinco
e não estão na mesma escala.
Mas a função será parecida
será parecida com isso, você sabe
como x ao cubo se parece
será parecido a algo como isso,
vamos desenhar um pouco,

Czech: 
a tady je limita
pro x blížící se k 5.
Toto je tedy derivace funkce
f(x) rovná se x na třetí.
Napíšu to zeleně.
x na třetí a
bod ‚a‘ se rovná 5.
Můžeme si to i
představit.
Pro představu si
to nakreslíme.
Udělám to tady, kde
budou barvy lépe vidět.
Toto je y-ová osa.
Tohle je x-ová osa.
Asi to nebude nakreslené
úplně v měřítku.
Tady bude 125, tedy bod
y rovná se 125.
Zde bude bod x rovno 5, takže
to opravdu nemám ve stejném měřítku.
Daná funkce bude
vypadat nějak takto.

Bulgarian: 
Тук имаме границата, когато 
x клони към 5.
Следователно това е 
производната на функцията
f(x) и е равна на x^3.
Нека да запиша това отдолу 
със зелен цвят.
x^3 за числото a, което 
е равно на 5.
И така, може да си представим това.
Нека да се опитаме да го начертаем, 
просто за да си го представим.
Ще го направя ето тук,
където имам малко по-добър 
контраст на цветовете.
Нека да кажем, че това
 е моята ос y.
Да кажем, че това е 
моята ос x.
Няма да я начертая 
точно в мащаб.
Нека да кажем, че това 
ето тук е числото 125.
Или y, т.е. това е, 
когато y = 125.
Това е, когато x = 5, така че
определено не са 
в един и същ мащаб.
Функцията обаче ще изглежда 
като нещо такова.
Знаем как изглежда x^3.
Изглежда като нещо такова.

Korean: 
 
a＝5입니다
여기는 (5, 125)입니다
우리는 저 점과
곡선 위의 임의의 점 사이의
기울기를 구해야 합니다
여기는
(x, x³) 입니다
f(x) ＝x³ 이 됩니다
f(x) ＝x³ 이 됩니다
이 그래프는 y=x³ 입니다
이 식은 두 점 사이의
기울기를 나타냅니다
 
x가 5로 갈 때의 극한을 구해보면
즉 x가 5에
가까워지면

Thai: 
 
ตรงนี้ a ของเราคือ 5
จุดนี่ตรงนี้คือ (5, 125)
แล้วเรากำลังหาความชันระหว่างจุดนั้น
กับค่า x ตามใจค่าหนึ่ง
หรือผมควรบอกว่า จุดตามใจบนเส้นโค้ง
จุดนี่ตรงนี้คือจุด
เราเรียกมันว่า (x, x กำลังสาม)
เรารู้ว่า f ของ x เท่ากับ x กำลังสาม
ขอผมบอกให้ชัดนะ
นี่คือกราฟของ y เท่ากับ x กำลังสาม
แล้วพจน์นี้ ตรงนี้ ทั้งหมดนี้ นี่
คือความชันระหว่างสองจุดนี้
 
และเมื่อเราหาลิมิตเมื่อ x เข้าใกล้ 5
ตรงนี้ นี่คือ x ของเรา เมื่อ x
เข้าใกล้ 5 มากขึ้น มากขึ้น และมากขึ้น

Czech: 
Víme, jak funkce x na
třetí vypadá, vypadá nějak takto.
Naše ‚a‘ je rovno 5.
Tady je
bod [5;125].
A my máme směrnici přímky
mezi tímto bodem a libovolným bodem x.
Spíš bych měl říct libovolným
bodem na této křivce.
Třeba tento bod,
to bude náš bod [x;x na třetí].
Víme, že f(x) je
rovno x na třetí.
Ujasním to.
Toto je graf funkce
y rovná se x na třetí.
Celý tento výraz je směrnice
přímky mezi těmito dvěma body.
Když hledáme limitu
pro x blížící se k 5...
Tohle je
teď naše x.
Jak se x čím dál
tím víc blíží k 5,

English: 
So here, our a is equal to 5.
This point right
over here is 5, 125.
And then we're taking the
slope between that point
and an arbitrary x-value.
Or I should say an arbitrary
other point on the curve.
So this right over here
would be the point,
we could call that
x, x to the third.
We know that f of x is
equal to x to the third.
And let me make it clear.
This is a graph of y is
equal to x to the third.
And so this expression, right
over here, all of this, this
is the slope between
these two points.
And as we take the
limit as x approaches 5,
so right now this
is our x, as x gets
closer and closer
and closer to 5,

Bulgarian: 
Ще изглежда като нещо такова.
Ето тук нашето a = 5.
Тази точка ето тук е (5; 125).
След това намираме наклона 
между тази точка
и друга произволна стойност за x.
Или трябва да кажа друга 
произволна точка от кривата.
Това точно ето тук 
ще бъде точката,
бихме могли да я наречем (x; x^3).
Знаем, че f(x) = x^3.
Нека да изясня нещо.
Това е графика на y = x^3.
Следователно този израз 
точно ето тук, т.е. всичко това,
е наклонът между 
тези две точки.
Това е наклонът 
между тези две точки.
И търсим границата за x, 
клонящо към 5,
така че сега това е нашето x,
и, когато се приближава 
все повече и повече до 5,

Portuguese: 
sabemos que será algo como isso.
Então este é nosso a é igual a cinco.
Este ponto aqui é: (5, 125)
Então vamos pegar a inclinação
entre aquele ponto e
um valor arbitrário de x,
ou devo dizer
um ponto arbitrário qualquer na curva,
então este aqui será o ponto
chamado: x, x ao cubo.
Nós sabemos que f de x
é igual a x ao cubo.
Quero deixar isso claro que o gráfico
de y é igual a x ao cubo
e então esta expressão aqui
tudo isso aqui é a inclinação
entre dois pontos.
E se pegamos o limite quando x
se aproxima de cinco,
então agora este é nosso x
quando x fica mais próximo de cinco,

English: 
the secant lines are
going to better and better
approximate the slope of the
tangent line at x equals 5.
So the slope of a tangent line
would look something like that.

Thai: 
เส้นตัดจะประมาณความชัน
ของเส้นสัมผัสที่ x เท่ากับ 5 ได้ดีขึ้นเรื่อยๆ
ความชันของเส้นสัมผัสจึงเป็นแบบนั้น

Portuguese: 
esta linha secante fica mais próxima
da inclinação da linha tangente
no ponto onde x igual a cinco.
Vamos ver a linha tangente um pouco
parecida muito mais a algo como isso.
Legendado: [ Luiz Pasqual ]
Revisado por [ Marcos Pereira ]

Korean: 
x＝5에서의
접선의 기울기와 비슷해집니다
접선은 대충 이렇게 생겼을 것입니다

Czech: 
sečna bude lepší a lepší
aproximací tečny v bodě x rovno 5.
Směrnice tečny by tedy
vypadala nějak takto.

Bulgarian: 
секущите линии все по-добре 
и по-добре
ще се доближават до наклона 
на допирателната, когато x = 5.
Следователно наклонът на допирателната 
ще изглежда като нещо такова.
