
Bulgarian: 
Нека видим дали можем
да намерим производната
на корен четвърти 
от х^3 + 4х^2 + 7.
Първоначално може да си кажеш:
"Как да сметна производната 
на корен четвърти
от нещо? Изглежда, че имам 
сложна функция
и смятам корен четвърти 
от друг израз."
Това ще е вярно.
И ако се занимаваш 
със сложни функции,
верижното правило 
трябва да е първо в ума ти.
Първо нека направим
 този корен четвърти
малко по-удобен за нас.
Просто да осъзнаем, че
 този корен четвърти
е просто дробна степен.
Това е същото нещо като
 производната
на х^3 + 4х^2 + 7, 
цялото на степен 1/4.
Как се смята производната
 на това?
Можем да го разгледаме, както
 казах преди няколко секунди,
като сложна функция.
Какво правим първо
 с нашето х?
Взимаме всичко това

Thai: 
ลองดูว่าเราหาอนุพันธ์
เทียบกับ x ของรากที่สี่
ของ x กำลังสามบวก 4x กำลังสองบวก 7 
ได้ไหม
ตอนแรกคุณอาจบอกว่า เอาล่ะ
ฉันจะหาอนุพันธ์ของรากที่สี่
ของอะไรสักอย่างได้ยังไง ดูเหมือนว่า
ฉันจะมีฟังก์ชันประกอบ
ฉันกำลังหารากที่สี่ของพจน์อีกพจน์ใช่ไหม?
คุณถูกแล้ว
ถ้าคุณเจอฟังก์ชันประกอบ
กฎลูกโซ่ควรอยู่ตรงหน้าคุณ
แต่ก่อนอื่น ลองทำให้รากที่สี่
เป็นรูปที่น่าคิดกว่านี้หน่อย
แค่่สังเกตว่ารากที่สี่นี้
ไม่ใช่อะไรอื่นแต่เป็นเลขชี้กำลังเศษส่วน
นี่ก็เหมือนกับอนุพันธ์
เทียบกับ x ของ x กำลังสามบวก 4x กำลังสอง
บวก 7 ยกกำลัง 1/4
ยกกำลัง 1/4
ทีนี้ เราจะหาอนุพันธ์ของตัวนี้อย่างไร
เรามองอันนี้ อย่างที่ผมบอกเมื่อกี้นี้
เรามองอันนี้เป็นฟังก์ชันประกอบได้
แล้วเราทำอะไรกับ x ของเราก่อน?
เราทำทั้งหมดนี้

English: 
- [Voiceover] Let's see if
we can take the derivative
with respect to x of the fourth root
of x to the third power plus
four x squared plus seven.
And at first you might say, "All right,
"how do I take the
derivative of the fourth root
"of something, it looks like
I have a composite function,
"I'm taking the fourth root
of another expression?"
And you'd be right.
And if you're dealing
with a composite function,
the chain rule should be front of mind.
But first, let's just
make this fourth root
a little bit more tractable for us.
And just realize that this fourth root
is nothing but a fractional exponent.
So this is the same
thing as the derivative
with respect to x of x to
the third plus four x squared
plus seven to the 1/4 power,
to the 1/4 power.
Now, how do we take
the derivative of this?
Well, we can view this, as
I said a few seconds ago,
we can view this as a composite function.
What do we do first with our x?
Well, we do all of this business,

Czech: 
Řekněme, že chceme spočítat derivaci
podle x ze čtvrté odmocniny z výrazu:
x na třetí plus 4 krát
x na druhou plus 7.
Nejprve si asi řeknete: „Jak se derivuje
čtvrtá odmocnina z něčeho?“
„Vypadá to, že mám
složenou funkci,
protože mám čtvrtou odmocninu
z nějakého výrazu.“
To je pravda.
A když derivujete
složenou funkci,
měli byste myslet na pravidlo
pro derivaci složené funkce.
Nejprve si tuto čtvrtou odmocninu přepišme
tak, aby se nám s ní lépe pracovalo.
Čtvrtou odmocninu můžeme napsat
jako mocninu ve tvaru zlomku.
Toto se tedy bude rovnat
derivaci podle x z výrazu:
(x na třetí plus 4 krát x na druhou
plus 7) na (1 lomeno 4).
Jak tohle zderivovat?
Jak jsem před chvílí řekl, můžeme se na
to dívat jako na složenou funkci.
Nejprve naše 
x dosadíme sem,

Korean: 
d/dx [(∜(x³+4x²+7)]를 구해봅시다
d/dx [(∜(x³+4x²+7)]를 구해봅시다
d/dx [(∜(x³+4x²+7)]를 구해봅시다
어떤 식의 네제곱근은어떻게
미분할까요?
어떤 식의 네제곱근은어떻게
미분할까요?
이 식은 마치 합성함수처럼 보이고
실제로도 그렇습니다
이 식은 마치 합성함수처럼 보이고
실제로도 그렇습니다
이 식은 마치 합성함수처럼 보이고
실제로도 그렇습니다
또 합성함수를 다루기 위해서는
연쇄법칙을 염두에 두어야 합니다
또 합성함수를 다루기 위해서는
연쇄법칙을 염두에 두어야 합니다
하지만 우선 이 네제곱근을
다루기 쉽게 해봅시다
하지만 우선 이 네제곱근을
다루기 쉽게 해봅시다
이 네제곱근은 그저
분수 지수에 지나지 않습니다
이 네제곱근은 그저
분수 지수에 지나지 않습니다
그러므로 이 식은
d/dx[(x³+4x²+7)¼]입니다
그러므로 이 식은
d/dx[(x³+4x²+7)¼]입니다
그러므로 이 식은
d/dx[(x³+4x²+7)¼]입니다
그러므로 이 식은
d/dx[(x³+4x²+7)¼]입니다
이제 이 식의 미분은
어떻게 구할 수 있을까요?
이 식은 아까 언급했듯이
합성함수로 볼 수 있습니다
이 식은 아까 언급했듯이
합성함수로 볼 수 있습니다
우선 x에 대해 무엇을 먼저 해야 할까요?
x에 대한 이 식을 u(x)로 치환하고

Korean: 
x에 대한 이 식을 u(x)로 치환하고
u(x)의 값을 ¼승 해주면 됩니다
u(x)의 값을 ¼승 해주면 됩니다
그리고 미분 시에는 이 바깥쪽
함수를 u(x)에 대한 식으로 보고
그리고 미분 시에는 이 바깥쪽
함수를 u(x)에 대한 식으로 보고
u(x)에 대한 미분값과
곱해주면 됩니다
u(x)에 대한 미분값과
곱해주면 됩니다
u(x)에 대한 미분값과
곱해주면 됩니다
이제 해보도록 합시다
지금 녹색으로 칠해지는
바깥쪽 함수, 즉 ¼승의
지금 녹색으로 칠해지는
바깥쪽 함수, 즉 ¼승의
지금 녹색으로 칠해지는
바깥쪽 함수, 즉 ¼승의
지금 녹색으로 칠해지는
바깥쪽 함수, 즉 ¼승의
지금 녹색으로 칠해지는
바깥쪽 함수, 즉 ¼승의
u(x)에 대한 미분을 구할 겁니다
u(x)에 대한 미분을 구할 겁니다
우선 멱의 법칙을
이용할겁니다
¼승을 앞으로 가져오게 되면
이 식은 1/4{u(x)} ¼-1이 됩니다
¼승을 앞으로 가져오게 되면
이 식은 1/4{u(x)} ¼-1이 됩니다
¼승을 앞으로 가져오게 되면
이 식은 1/4{u(x)} ¼-1이 됩니다
¼승을 앞으로 가져오게 되면
이 식은 1/4{u(x)} ¼-1이 됩니다
보시다시피 여기서 사용한 건
멱의 법칙밖에 없습니다
보시다시피 여기서 사용한 건
멱의 법칙밖에 없습니다
이제 이 u(x)에 대한
다항식의 미분값을 구해봅시다
이제 이 u(x)에 대한
다항식의 미분값을 구해봅시다

Bulgarian: 
и го наричаме u(x).
После каквото получим за u(x)
го повдигаме на четвърта степен.
Тази производна можем
да сметнем като...
Можем да разглеждаме това
 като външната функция
спрямо u(x) и тогава 
да умножим това
по производната на u
спрямо х.
Хайде да го направим.
На какво ще е равно това?
Ще вземем нашата 
външна функция,
която ще оцветя в зелено.
Взимам нещо на степен 1/4
и ще намеря 
производната му спрямо
вътрешната функция, 
т.е. спрямо u(x).
Просто ще използвам правилото
за намиране на производна от степен.
Просто ще изкарам 1/4 отпред
и ще получа 1/4 по
това, спрямо което смятам производната,
на степен 1/4 минус 1.
Просто използвам правилото
 за производна от степен.
Тук нямам х.
Сега смятаме производната
 спрямо u(x),
т.е. спрямо този полиномен израз.

Czech: 
takže si to označme
třeba jako u(x),
a výslednou hodnotu u(x) pak
umocníme na tuhle mocninu.
Derivaci tedy
spočítáme tak,
že zderivujeme tuto
vnější funkci podle u(x)
a pak to vynásobíme
derivací ‚u‘ podle x.
Tak pojďme na to.
Tohle se
tedy rovná...
Vnější funkci, kterou právě
zvýrazňuji zelenou,
tedy funkci, která mi něco
umocní na (1 lomeno 4),
zderivuji podle vnitřní
funkce, tedy podle u(x).
K tomu použiji
derivaci mocniny.
Jednu čtvrtinu
napíšu dopředu,
takže to bude (1 lomeno 4) krát
to, podle čeho derivuji,
umocněno na
(1 lomeno 4) minus 1.
Jen jsem použil vzorec
pro derivaci mocniny.
Nemám tady x, dělám
totiž derivaci podle u(x),
tedy podle
tohoto polynomu,

English: 
and we can call this u of x.
And then whatever we get for u of x,
we raise that to the fourth power.
So the way that we would
take the derivative,
we would take the derivative of this,
you could view it as the outer function
with respect to u of x
and then multiply that
times the derivative
of u with respect to x.
So let's do that.
So what this is going to be,
this is going to be equal to,
so we're gonna take our outside function,
which I'm highlighting in green now,
so, or I take something to the 1/4,
I'm gonna take the derivative
of that with respect
to the inside, with respect to u of x.
Well, I'm just gonna
use the power rule here.
I'm just gonna bring that 1/4 out front,
so it's gonna be 1/4 times
whatever I'm taking the
derivative with respect to,
to the 1/4 minus one power.
Look, all I did was use
the power rule here.
I didn't have an x here.
Now I'm taking the derivative
with respect to u of x,
with respect to this
polynomial expression here.

Thai: 
เราเรียกอันนี้ว่า u ของ x ได้
แล้วเมื่อใดก็ตามที่เราได้ u ของ x
เราก็ยกกำลังมันด้วย 1/4
วิธีที่เราหาอนุพันธ์
เราจะหาอนุพันธ์ของตัวนี้
คุณมองมันเป็นฟังก์ชันนอก
เทียบกับ u ของ x ก็ได้ แล้วคูณมัน
กับอนุพันธ์ของ u เทียบกับ x
ลองทำกันดู
สิ่งนี้จะเท่ากับ
อันนี้จะเท่ากับ
เราจะหาฟังก์ชันนอก
ซึ่งผมเน้นด้วยสีเขียวตอนนี้
ผมหาอะไรสักอย่างยกกำลัง 1/4
ผมจะหาอนุพันธ์ของมันเทียบกับ
ตัวใน เทียบกับ u ของ x
ผมจะใช้กฎยกกำลังตรงนี้
ผมจะนำ 1/4 มาไว้ข้างหน้า
มันจะเท่ากับ 1/4 คูณ
อะไรก็ตามที่ผมหาอนุพันธ์อยู่
ยกกำลัง 1/4 ลบ 1
ดูนะ ที่ผมทำคือใช้กฎยกกำลังตรงนี้
ผมไม่มี x ตรงนี้
ทีนี้ ผมจะหาอนุพันธ์เทียบกับ u ของ x
เทียบกับพจน์พหุนามนี่ตรงนี้

Thai: 
ผมแค่โยน u ของ x ลงในนี้ก็ได้ถ้าต้องการ
ขอผมทำอย่างนั้นดีกว่า
นี่จะเท่ากับ x กำลังสาม
บวก 4x กำลังสองบวก 7
แล้วผมจะคูณมัน
นี่คือกฎลูกโซ่
ผมหาอนุพันธ์ของตัวนอก
เทียบกับตัวใน แล้วผมคูณมัน
ด้วยอนุพันธ์ของตัวใน
แล้วอนุพันธ์ของ u ของ x คืออะไร?
u ไพรม์ของ x ลองดู เราจะใช้
กฎยกกำลังหลายครั้ง
มันจะเท่ากับ 3x กำลังสอง
บวก 2 คูณ 4 ได้ 8x
กำลัง 2 ลบ 1 ก็คือกำลัง 1 ยกกำลัง 1
เราเขียนแค่ 8x ได้
แล้วอนุพันธ์เทียบกับ x ของ 7
อนุพันธ์เทียบกับ x ของค่าคงที่
ก็แค่ 0
นั่นคือ u ไพรม์ของ x
แล้วผมจะคูณมันด้วย u ไพรม์ของ x
ซึ่งก็คือ 3x กำลังสองบวก
3x กำลังสองบวก 8x
แล้ว ผมก็ทำความสะอาดอันนี้หน่อย

Czech: 
takže sem
musím napsat u(x).
Musím sem napsat x na třetí
plus 4 krát x na druhou plus 7.
Tohle teď
musím vynásobit...
Podle pravidla o derivaci složené funkce
mám zderivovat vnější funkci podle vnitřní
a pak to vynásobit
derivací vnitřní funkce.
Čemu se tedy
rovná derivace u(x)?
‚u‘ s čárkou v bodě x...
Jen několikrát použijeme
vzorec pro derivaci mocniny.
...se rovná
3 krát x na druhou plus
2 krát 4 je 8, tohle krát
x na (2 minus 1), tedy x na prvou,
což mohu napsat
jako 8 krát x.
Derivace podle x ze 7...
Derivace podle x
z konstanty je 0.
Takto tedy vypadá u(x) s čárkou,
kterou teď musím vynásobit,
takže to bude krát (3 krát
x na druhou plus 8 krát x).

Korean: 
이제 여기에 u(x)로 표현되었던 식을
다시 적어보면
이제 여기에 u(x)로 표현되었던 식을
다시 적어보면
1/4(∜x³+4x²+7)1/4-1이 됩니다
1/4(∜x³+4x²+7)1/4-1이 됩니다
이제 이걸 곱하는 게 바로
연쇄 법칙입니다
이제 이걸 곱하는 게 바로
연쇄 법칙입니다
밖의 식의 미분값을
안쪽 식의 미분값과 곱할 겁니다
밖의 식의 미분값을
안쪽 식의 미분값과 곱할 겁니다
밖의 식의 미분값을
안쪽 식의 미분값과 곱할 겁니다
이때 u(x)의 미분값은 뭘까요?
다시 멱의 법칙을 사용하면
상수는 미분 시 어차피 0이 되므로
다시 멱의 법칙을 사용하면
상수는 미분 시 어차피 0이 되므로
2×4=8이고 2-1=1이므로
2×4=8이고 2-1=1이므로
2×4=8이고 2-1=1이므로
(3x²+8x)가 됩니다
(3x²+8x)가 됩니다
(3x²+8x)가 됩니다
(3x²+8x)가 됩니다
이제 3x²+8x인 u'(x)를
곱해봅시다
이제 3x²+8x인 u'(x)를
곱해봅시다
이제 3x²+8x인 u'(x)를
곱해봅시다
이제 3x²+8x인 u'(x)를
곱해봅시다
식을 다시 깔끔하게
적어보겠습니다

Bulgarian: 
Така че мога просто да сложа
 това u(x) тук, ако искам.
Всъщност нека го направя.
Това ще бъде 
х^3 + 4х^2 + 7.
После ще умножа това,
което вече е част от 
верижното правило.
Сметнах производната на
 външната функция
спрямо вътрешната и тогава 
умножавам това
по производната на вътрешната.
Каква ще производната на u(x)?
u прим от х. Да видим. 
Просто ще използваме
правилото за производна от степен
 няколко пъти.
Ще бъде 3х^2 плюс,
2 по 4х е 8х
на степен 2 минус 1, което е 1, 
т.е. на първа степен.
Мога да запиша само 8х
и тогава производната 
спрямо х на 7.
Производната на константа
ще бъде просто 0.
Това е u прим от х.
Сега ще умножа по 
u прим от х,
което е 3х^2 + 8х.
Мога да разчистя малко.

English: 
So I could just throw the
u of x in here if I like,
actually let me just do that.
So, this is going to be x to the third
plus four x squared plus seven.
And then I wanna multiply that,
and this is the chain rule.
I took the derivative of the outside
with respect to the inside and
then I'm gonna multiply that
times the derivative of the inside.
So what's the derivative of u of x?
U prime of x, let's see we just gonna use
the power rule a bunch of times,
it's gonna be three x squared
plus two times four is eight x
to the two minus one is
just one power, first power,
so I can just write that as eight x,
and then the derivative
with respect to x of seven,
well, the derivative with
respect to x of a constant
is just gonna be zero.
So that's u prime of x.
So then I'm just gonna
multiply by u prime of x
which is three x squared plus,
three x squared plus eight x.
And so, well, I can clean
this up a little bit,

Thai: 
อันนี้จะเท่ากับ
อันนี้จะเท่ากับ
ที่จริง ขอผมเขียนเลขชี้กำลังนั่นใหม่นะ
1/4 ลบ 1 นี่ ผมเขียนมันใหม่ได้
1/4 ลบ 1 คือลบ 3/4
ลบ 3/4
ยกกำลังลบ 3/4
แล้วคุณจัดการมันได้หลายวิธี ถ้าต้องการ
แต่ประเด็นคือแค่สังเกตว่านี่คือ
การใช้กฎลูกโซ่
อนุพันธ์ของตัวนอก ที่จริงแล้ว
สิ่งแรกที่ต้องสังเกตคือว่า รากที่สี่
ก็เหมือนกับการยกกำลังด้วย 1/4
สมบัติเลขยกกำลังพื้นฐาน แล้วสังเกตว่า
โอเค ฉันมีฟังก์ชันประกอบตรงนี้
ฉันหาอนุพันธ์ของตัวนอก
เทียบกับตัวใน นั่นคือสิ่งที่เราทำตรงนี้
คูณอนุพันธ์ของตัวในเทียบกับ x
แล้วถ้ามีคนถามคุณ
ถ้ามีคนถามคุณว่า เอาล่ะ f ของ x
f ของ x เท่ากับรากที่สี่ของ

Czech: 
Tohle bych možná mohl
trochu zjednodušit.
Bude se
to rovnat...
Vlastně jenom
přepíšu exponent.
(1 lomeno 4) minus 1 se rovná 
minus (3 lomeno 4).
Tohle ještě můžete
různě upravovat.
Důležité je ale poznat, že je třeba použít
pravidlo o derivaci složené funkce.
Derivace vnější
funkce podle...
První důležitou věcí
vlastně bylo to,
že čtvrtá odmocnina je totéž
jako mocnění na (1 lomeno 4).
To je základní
vlastnost mocnin.
Poté bylo důležité poznat,
že jde o složenou funkci,
takže je třeba zderivovat vnější
funkci podle vnitřní, což máme tady,
a to vynásobit derivací
vnitřní funkce podle x.
Kdyby vám teď někdo zadal, že f(x)
se rovná čtvrtá odmocnina z výrazu:

English: 
so this would be equal to,
this would be equal to.
Actually let me just
rewrite that exponent there.
So this 1/4 minus one, I can rewrite it,
1/4 minus one is negative 3/4,
negative 3/4,
negative 3/4 power.
And you can manipulate this in
different ways, if you like,
but the key is to just
recognize that this is
an application of the chain rule.
Derivative of the outside, well, actually,
the first thing to
realize is the fourth root
is the same thing as taking
something to the 1/4 power,
basic exponent property, and then realize,
okay, I have a composite function here.
So I can take the
derivative of the outside
with respect to the inside,
that's what we did here,
times the derivative of the
inside with respect to x.
And so if someone were to tell you,
if someone were to say,
"All right, f of x,
"f of x is equal to the fourth root of

Korean: 
식을 다시 깔끔하게
적어보겠습니다
식을 다시 깔끔하게
적어보겠습니다
지수를 -3/4로 다시 적겠습니다
지수를 -3/4로 다시 적겠습니다
지수를 -3/4로 다시 적겠습니다
지수를 -3/4로 다시 적겠습니다
지수를 -3/4로 다시 적겠습니다
식을 다르게 바꿔도 되지만 핵심은
이 식이 연쇄법칙을 응용했단 겁니다
식을 다르게 바꿔도 되지만 핵심은
이 식이 연쇄법칙을 응용했단 겁니다
식을 다르게 바꿔도 되지만 핵심은
이 식이 연쇄법칙을 응용했단 겁니다
네제곱근이 ¼승과 같다는
기본적인 지수의 성질을 염두에 두고
네제곱근이 ¼승과 같다는
기본적인 지수의 성질을 염두에 두고
네제곱근이 ¼승과 같다는
기본적인 지수의 성질을 염두에 두고
이 식이 합성함수라는 점을
보시면 됩니다
이 식이 합성함수라는 점을
보시면 됩니다
여기 보이시는 것처럼
안쪽에 대한 밖의 식을 미분하고
여기 보이시는 것처럼
안쪽에 대한 밖의 식을 미분하고
x에 대한 안쪽 식의 미분을
곱하시면 됩니다
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는

Bulgarian: 
Това е равно на...
Всъщност нека запиша 
тази степен тук.
Мога да запиша това 
1/4 минус 1...
1/4 минус 1 е –3/4.
На степен –3/4.
Можем да преработим това по
 различни начини,
но по-важното е да разберем, 
че това
е приложението на 
верижното правило.
Производната на външната...
 Всъщност
първото трябва да осъзнаем,
че корен четвърти
е същото нещо като
 нещо на степен 1/4.
Това е основно свойство на 
степените. После да осъзнаем:
"Добре, тук имам 
сложна функция.
Следователно мога да намеря 
производната на външната функция
спрямо вътрешната. 
Това, което направихме тук,
по производната на 
вътрешната спрямо х.
И ако някой ти каже:
"Добре, f(x)
е равно на корен четвърти от

Bulgarian: 
х^3 + 4х^2 + 7."
и после каже: "А колко е f прим от...
например –3?"
Ще сметнем това при –3.
Нека го направя.
1/4 по...
Получаваме –27...
Надявам се, че ще се получи
 сравнително добре.
плюс 36
плюс 7 на степен –3/4.
На какво е равно това?
Това ще е равно на:
Това тук е 16, нали?
–27 плюс 7 е –20,
плюс 36, следователно това е 16.
Мисля, че ще се получи добре.
После по
3 по... 3 по 9,
което е 27, минус 24.
Това тук ще бъде...
Това ще бъде 3.
Колко е 16 на степен –3/4?

Thai: 
x กำลังสามบวก 4x กำลังสองบวก 7
แล้วเขาบอกว่า อืม f ไพรม์ของ
ไม่รู้สิ ลบ 3 เป็นเท่าใด?
คุณก็หาค่านี้ที่ลบ 3
ขอผมทำนะ
มันคือ 1/4 คูณ
ดูนะ คุณมีลบ 27
ผมหวังว่ามันจะออกมาใช้ได้
บวก 36, บวก 36
บวก 7 กำลังลบ 3/4
อันนี้ให้อะไร?
อันนี้จะเท่ากับ
ค่านี่ตรงนี้คือ 16 จริงไหม?
ลบ 27 บวก 7 ได้ลบ 20
บวก 36 นี่ก็คือ 16
ผมว่ามันจะออกมาสวยงาม
แล้วคูณ
3 คูณลบ, 3 คูณ 9
ซึ่งก็คือ 27 ลบ 24
อันนี้จะอยู่ตรงนี้
มันจะเท่ากับ 3
ทีนี้ 16 กำลังลบ 3/4 คืออะไร?

Czech: 
x na třetí plus 4 krát
x na druhou plus 7,
a ptal se, čemu se rovná
f s čárkou v bodě −3,
tak byste tohle museli
vyčíslit pro x rovno −3.
Tak to udělejme.
Bude to (1 lomeno 4)
krát (−27...
Doufám, že to vyjde
nějak hezky.
...plus 36 plus 7) umocněno na
minus (3 lomeno 4).
Čemu se
tohle rovná?
Bude se
to rovnat...
Tohle je 16, protože −27 plus 7 je −20,
k čemuž ještě přičteme 36.
Tady tedy
bude 16.
Myslím, že to
vyjde hezky.
Tohle musíme
vynásobit...
3 krát 9 je 27 a ještě
musíme odečíst 24,
takže to musíme
vynásobit 3.
Čemu se rovná
16 na minus (3 lomeno 4)?

Korean: 
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는
누가 f(x)=∜(x³+4x²+7)일 때
f(-3)이 무엇이냐 물었을 때는
이 식에 -3을 대입하면 됩니다
한번 해봅시다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-3을 대입하면
¼(-27+36+7)-3/4이 됩니다
-27+36+7=16이므로
꽤 간단히 풀릴 겁니다
-27+36+7=16이므로
꽤 간단히 풀릴 겁니다
-27+36+7=16이므로
꽤 간단히 풀릴 겁니다
-27+36+7=16이므로
꽤 간단히 풀릴 겁니다
-27+36+7=16이므로
꽤 간단히 풀릴 겁니다
이제 여기에 (27-24)를 곱하는데
그냥 3이라고 하겠습니다
이제 여기에 (27-24)를 곱하는데
그냥 3이라고 하겠습니다
이제 여기에 (27-24)를 곱하는데
그냥 3이라고 하겠습니다
이제 여기에 (27-24)를 곱하는데
그냥 3이라고 하겠습니다
이제 여기에 (27-24)를 곱하는데
그냥 3이라고 하겠습니다
16의 3/4승이 뭘까요?

English: 
"x to the third plus four
x squared plus seven,"
and then they said,
"Well, what is f prime of
"I don't know, negative three?"
Well, you would evaluate
this at negative three.
So let me just do that.
So it's 1/4 times,
see, you have negative 27,
I hope this works out reasonably well,
plus 36, plus 36,
plus 7 to the negative 3/4,
what does this result to?
This is going to be equal to,
this right over here is 16, right?
Negative 27 plus seven is negative 20
plus 36, so this is 16.
I think this is going to work out nicely.
And then times
three times negative, so three times nine,
which is 27 minus 24.
So this is going to be right over here,
that is going to be three.
Now, what is 16 to the negative 3/4?

English: 
So this is 1/4 times,
so 16 times to the 1/4
is two and then you raise that to the,
let me, actually I don't
want to skip steps here.
At this point we're dealing with algebra,
or maybe even pre-algebra.
So this is going to be times, times,
16 to the 1/4,
and then we're gonna raise
that to the negative three
times that three out front.
So we could put that three there.
16 to the 1/4 is two,
two to the third is eight,
so two to the negative third power is 1/8,
so that is 1/8.
So we have 3/4 times 1/8
which is equal to three over 32, 3/32.
So that would be the
slope of the tangent line
of the graph y is equal to f of x
when x is equal to negative three.

Bulgarian: 
Това е 1/4 по...
16 на степен 1/4 е 2 
и после го повдигаме на...
Не искам да пропуснем 
някоя стъпка.
В този момент се занимаваме 
с алгебра
или може би въведение в алгебрата.
Това ще бъде по
16 на степен 1/4,
и после ще го повдигнем
 на степен –3,
по това 3 отпред.
Можем да сложим това 3 там.
16 на степен 1/4 е 2.
2 на трета е 8.
Следователно 
2 на степен  –3 е 1/8.
Това е 1/8.
Получаваме 3/4 по 1/8,
което е равно на 3 върху 32.
Това ще е наклонът на допирателната
към графиката у = f(x),
когато х е равно на –3.

Thai: 
นี่ก็คือ 1/4 คูณ, 16 กำลัง 1/4
ได้ 2 แล้วคุณยกกำลังมัน
ขอผม ผมไม่อยากข้ามขั้นตรงนี้
ถึงจุดนี้ เราแค่คิดเลขด้วยวิชาพีชคณิตแล้ว
อาจจะก่อนพีชคณิตด้วยซ้ำ
อันนี้จะคูณ คูณ
16 กำลัง 1/4
แล้วเรายกกำลังมันด้วยลบ 3
คูณ 3 นั่นข้างหน้า
แล้วเราใส่ 3 นั่นตรงนั้น
16 กำลัง 1/4 ได้ 2
2 กำลัง 3 ได้ 8
2 กำลังลบ 3 ได้ 1/8
นั่นก็คือ 1/8
เราจึงได้ 3/4 คูณ 1/8
ซึ่งเท่ากับ 3 ส่วน 32, 3/32
มันก็คือความชันของเส้นสัมผัส
ของกราฟ y เท่ากับ f ของ x
เมื่อ x เท่ากับลบ 3

Korean: 
지금의 과정은 대수학입니다
이 과정을 생략하고 싶진 않습니다
지금의 과정은 대수학입니다
이 과정을 생략하고 싶진 않습니다
지금의 과정은 대수학입니다
이 과정을 생략하고 싶진 않습니다
지금의 과정은 대수학입니다
이 과정을 생략하고 싶진 않습니다
지금의 과정은 대수학입니다
이 과정을 생략하고 싶진 않습니다
뒤의 3을 앞으로 빼서
3/4(16¼)-3으로 만듭시다
뒤의 3을 앞으로 빼서
3/4(16¼)-3으로 만듭시다
뒤의 3을 앞으로 빼서
3/4(16¼)-3으로 만듭시다
뒤의 3을 앞으로 빼서
3/4(16¼)-3으로 만듭시다
뒤의 3을 앞으로 빼서
3/4(16¼)-3으로 만듭시다
16¼=2고 2³=8이므로
2-3=1/8입니다
16¼=2고 2³=8이므로
2-3=1/8입니다
16¼=2고 2³=8이므로
2-3=1/8입니다
16¼=2고 2³=8이므로
2-3=1/8입니다
결과적으로 값은
3/4(1/8)=3/32가 됩니다
결과적으로 값은
3/4(1/8)=3/32가 됩니다
이 값이 바로 x=-3일 때
y=f(x) 그래프의 접선의
기울기가 됩니다

Czech: 
Toto se tedy rovná
(1 lomeno 4) krát...
16 na (1 lomeno 4) je 2,
což následně mocníme na...
Raději nebudu
přeskakovat kroky.
V tuhle chvíli už je to algebra
nebo možná pokročilejší aritmetika.
Bude to tedy krát 16 na (1 lomeno 4),
což následně umocníme na minus třetí,
krát tato 3, kterou
mohu napsat sem.
16 na (1 lomeno 4) je 2.
2 na třetí je 8, takže
2 na minus třetí je 1 lomeno 8.
Máme tedy (3 lomeno 4) krát (1 lomeno 8),
což se rovná 3 lomeno 32.
Taková je směrnice tečny ke grafu
funkce y rovná se f(x) v bodě x rovno −3.
