
Czech: 
Máme funkci y rovná se logaritmus
o základu 4 z (x na druhou plus x).
Čemu se rovná
derivace y podle x?
Možná už teď vidíte,
že toto je složená funkce.
Nemáme logaritmus
o základu 4 jenom z x,
ale z jiného výrazu
obsahujícího proměnnou x.
Mohli bychom si tedy říct, že tento
modrý výraz označíme jako u(x).
Napíšu to modře.
Modrý výraz
označíme jako u(x),
takže u(x) se bude rovnat
x na druhou plus x.
Později se bude hodit vědět,
čemu se rovná ‚u‘ s čárkou v bodě x.
To se rovná, když použiji pravidlo
pro derivaci mocniny, 2 krát x plus 1.
2 jsem napsal dopředu
a zmenšil jsem exponent
a derivace x podle x je 1.
Logaritmus o základu 4 z tohoto
výrazu můžeme označit jako funkci ‚v‘.

Bulgarian: 
Нека да кажем, че функцията y e равна
на логаритъм от (x^2 + x) с основа 4.
На какво ще бъде равна 
производната на функцията y
спрямо x?
Може би веднага ще познаеш,
че това е съставна функция.
Вземаме основата на логаритъма 4, 
а не само x.
Вземаме това като друг израз,
който включва x.
Бихме могли да кажем,
можем да кажем, 
че това нещо в синьо е u(x).
Нека да направя това в синьо.
И така, това нещо в синьо е u(x).
Функцията u(x) е равна на (x^2 + x).
А по-късно ще бъде полезно
 да разберем,
на какво е равно u'(x).
Това ще бъде равно...
Просто ще използвам правилото 
за намиране производна на степен.
2x + 1
Изнесох тези два члена отпред
и намалих степента.
Производната на x спрямо x e 1.
Можем да кажем, че
логаритъм от тези неща с основа 4
бихме могли 
да го наречем функция V...
Бихме могли да кажем,
ако кажем, че това е v(x),

Thai: 
สมมุติว่า y เท่ากับ
ล็อกฐาน 4 ของ x กำลังสองบวก x
อนุพันธ์ของ y
เทียบกับ x จะเท่ากับอะไร?
ตอนนี้ คุณอาจสังเกตได้ทันทีว่า
นี่คือฟังก์ชันประกอบ
เรากำลังหาล็อกฐาน 4 ไม่ใช่แค่ x
แต่เรากำลังหาของอีกพจน์
ที่เกี่ยวกับ x
เราบอกได้ว่า
เราบอกได้ว่าตัวสีฟ้านี้
มันคือ u ของ x
ขอผมใช้สีฟ้านะ
อันนี้สีฟ้า
มันคือ u ของ x
u ของ x เท่ากับ x กำลังสอง
บวก x
และมันจะมีประโยชน์ต่อไปเพื่อหาว่า
u ไพรม์ของ x คืออะไร
มันจะเท่ากับ
ผมจะใช้กฎยกกำลังตรงนี้นะ
2x บวก 1
ผมนำ 2 นั่นมาไว้ข้างหน้า
แล้วลดเลขชี้กำลังลง
อนุพันธ์เทียบกับ x ของ x ได้ 1
และเราบอกได้
ล็อกฐาน 4 ของตัวนี้
เราเรียกมันว่าฟังก์ชัน v ได้
เราบอกได้ว่า v ของ
ถ้าเราบอกว่า v ของ x

Korean: 
y가 log4(x²+y)일 때
y가 log4(x²+y)일 때
x에 대한 y의 도함수는
무엇이 될까요
x에 대한 y의 도함수는
무엇이 될까요
이는 합성함수이기 때문에
이는 합성함수이기 때문에
log4x의 도함수만 구하는 것이 아닌
x를 포함하는 이 함수의
도함수를 구해야 합니다
우선 파란 부분을
U(x)라는 함수라고 해봅시다
우선 파란 부분을
U(x)라는 함수라고 해봅시다
우선 파란 부분을
U(x)라는 함수라고 해봅시다
우선 파란 부분을
U(x)라는 함수라고 해봅시다
그렇다면 U(x)=x²+x로
나타낼 수 있습니다
그렇다면 U(x)=x²+x로
나타낼 수 있습니다
그렇다면 U(x)=x²+x로
나타낼 수 있습니다
그렇다면 U(x)=x²+x로
나타낼 수 있습니다
y의 도함수를 구할 때 필요한
U'(x)의 값을 먼저 구해줍시다
U'(x)를 멱의 법칙을
이용해 구해주면
U'(x)를 멱의 법칙을 이용해
지수의 2를 계수에 곱하고
지수를 1 감소시켰습니다
x의 x에 관한 도함수는 1이므로
U'(x)=2x+1로 나타낼 수
있게 됩니다
그리고 log4로 시작되는 식을
그리고 log4로 시작되는 식을
함수 V라고 두었을 때
V(x)=log4(x)라고
표현 가능합니다
V(x)=log4(x)라고
표현 가능합니다

English: 
- [Voiceover] Let's say that Y is equal to
log base four of X squared plus X.
What is the derivative of Y
with respect to X going to be equal to?
Now you might recognize immediately that
this is a composite function.
We're taking the log
base four, not just of X,
but we're taking that
of another expression
that involves X.
So we could say
we could say this thing in blue
that's U of X.
Let me do that in blue.
So this thing in blue
that is U of X.
U of X is equal to X squared
plus X.
And it's gonna be useful later on to know
what U prime of X is.
So that's gonna be
I'm just gonna use the power rule here
so two X plus one
I brought that two out front
and decremented the exponent.
Derivative with respect to X of X is one.
And we can say the
log base four of this stuff
well we could call that a function V.
We can say V of
well if we said V of X

Czech: 
Funkce v(x) tedy bude
logaritmus o základu 4 z ‚x‘.
V jiných videích
jsme si ukázali,
že ‚v‘ s čárkou v bodě x
bude velmi podobná tomu,
kdyby to byl logaritmus o základu e,
neboli přirozený logaritmus,
ale ještě to
musíme přenásobit.
Bude to 1 lomeno
logaritmus o základu 4...
Pardon, 1 lomeno
(přirozený logaritmus ze 4 vynásobený x).
Kdyby v(x) bylo přirozený logaritmus x,
tak by derivace byla 1 lomeno x,
ale jde o logaritmus
o základu 4...
Tohle plyne ze vzorce pro
změnu základu logaritmu,
o kterém máme
samostatné video.
Jmenovatel tedy musíme
přenásobit přirozeným logaritmem ze 4,
neboli celý výraz přenásobíme výrazem
1 lomeno přirozený logaritmus ze 4.
Tohle teď využijeme, protože na y se nyní
můžeme dívat jako na ‚v‘ v bodě...

English: 
this would be log base four
of X.
And then we've shown in other videos
that V prime of X
is, we're gonna be very
similar that if this
was log base E, or natural log,
except we're going to scale it.
So it's going to be
one over
one over log base four.
Sorry, one over
the natural log.
The natural log of four
times X.
If this was V of X, if V of X was just
natural log of X, our
derivative would be one over X.
But since it's log base four
and this comes straight
out of the change of base
formulas that you might have seen.
And we have a video where we show this.
But we just scale it in the denominator
with this natural log of four.
You think of scaling the whole expression
by one over the natural log of four.
But we can now use this
information because Y
this Y can be viewed as
V of
V of.

Bulgarian: 
то това ще бъде логаритъм от
x^2 + x с основа 4.
Показахме и в други уроци,
че v'(x) ще бъде 
твърде подобна на израза, когато
основата на логаритъма е 'e' 
или натурален логаритъм,
освен че ще го умножим.
Ще бъде равно на 1 върху...
1 върху логаритъм с основа 4.
Извинявам се, 1 върху 
натурален логаритъм.
Натурален логаритъм от 4, 
умножен по x.
Ако тази функция беше v(x), 
ако v(x) беше просто
натурален логаритъм от x, 
производната щеше да бъде 1 върху x.
Но умножаваме поради това, че логаритъмът има основа 4,
а това се получава директно
 от формулата
за смяна на основата, която 
вече сме показвали.
Има урок, където го показваме.
Просто увеличихме знаменателя
с този натурален логаритъм от 4.
Мислиш за увеличаване 
на целия израз чрез
1 върху натурален логаритъм от 4.
Сега можем да използваме 
тази информация.
Тази функция y може да бъде
 разглеждана като v от...
V  от....

Thai: 
อันนี้จะเป็นล็อกฐาน 4
ของ x
แล้วเราแสดงไปในวิดีโออื่น
ว่า v ไพรม์ของ x
เรารู้ว่า มันจะคล้ายกับกรณีที่เป็น
ล็อกฐาน e หรือล็อกธรรมชาติมาก
ยกเว้นแต่ว่าเราจะย่อขยายมัน
มันจะเท่ากับ
1 ส่วน
1 ส่วนล็อกฐาน 4
โทษที 1 ส่วน
ล็อกธรรมชาติ
ล็อกธรรมชาติของ 4
คูณ x
ถ้านี่คือ v ของ x ถ้า v ของ x คือแค่
ล็อกธรรมชาติของ x อนุพันธ์ของเรา
จะเป็น 1 ส่วน x
แต่เนื่องจากมันเป็นล็อกฐาน 4
และอันนี้ตรงมาจากสูตรการเปลี่ยนฐาน
ที่คุณอาจเคยเห็นแล้ว
และเรามีวิดีโอที่เราแสดงอันนี้
แต่เราย่อขยายมันตรงตัวส่วน
ด้วยล็อกธรรมชาติของ 4 นี้
คุณคิดถึงการย่อขยายทั้งพจน์
ด้วย 1 ส่วนล็อกธรรมชาติของ 4
แต่ตอนนี้เราใช้ข้อมูลนี้ได้เพราะ y
y นี้มองเป็น
v ของ
v ของ

Korean: 
V(x)=log4(x)라고
표현 가능합니다
V(x)=log4(x)라고
표현 가능합니다
그리고 지난 비디오와 같은
방법을 이용해
V'(x)를 구해줄 수 있습니다
log의 밑이 e인 자연로그의
미분형과 똑같이 써준 뒤
log의 밑이 e인 자연로그의
미분형과 똑같이 써준 뒤
약간 보정해주면 됩니다
분자는 1이 되고
분자는 1이 되고
분모는 (ln4)x가 됩니다
분모는 (ln4)x가 됩니다
분모는 (ln4)x가 됩니다
분모는 (ln4)x가 됩니다
분모는 (ln4)x가 됩니다
만약 V(x)가 자연로그를
포함한 함수였다면
x를 분모로 갖고 1을 분자로 갖는
평범한 분수꼴이었을 것입니다
하지만 로그의 밑이 4이기 때문에
기존의 형태와는 다른
함수가 나오게 됩니다
기존의 형태와는 다른
함수가 나오게 됩니다
기존의 형태와는 다른
함수가 나오게 됩니다
우리는 최종식을 분모에 ln4를
곱해줌으로써 얻을 수 있습니다
우리는 최종식을 분모에 ln4를
곱해줌으로써 얻을 수 있습니다
우리는 최종식을 분모에 ln4를
곱해줌으로써 얻을 수 있습니다
우리는 최종식을 분모에 ln4를
곱해줌으로써 얻을 수 있습니다
이제 이 정보들을  이용해
Y를 V에 관한 함수로
표현할 수 있게 됩니다
Y를 V에 관한 함수로
표현할 수 있게 됩니다
Y를 V에 관한 함수로
표현할 수 있게 됩니다

Czech: 
‚v‘ jsme definovali jako
logaritmus o základu 4 z něčeho.
Nebude to ovšem v(x),
protože tady není jenom x,
ale máme tu výraz, 
který jsme označili jako u(x),
takže sem musíme
napsat u(x).
Nakreslím zde čáru, aby se nám věci
na jednotlivých stranách nepletly.
Podle pravidla o derivaci složené funkce
víme, že derivace y podle x se rovná:
derivace ‚v‘ podle ‚u‘,
neboli ‚v‘ s čárkou v bodě u(x)...
u(x) napíšu modře.
...‚v‘ s čárkou v bodě u(x)
krát ‚u‘ s čárkou v bodě x.
Čemu se rovná
‚v‘ s čárkou v bodě u(x)?
Víme, čemu se rovná
v(x) s čárkou.
Když chceme znát ‚v‘ s čárkou v bodě u(x),
musíme všude místo x napsat u(x),
takže toto se rovná
‚v‘ s čárkou v bodě u(x)...

Thai: 
นึกดู v คือล็อกฐาน 4 ของอะไรสักอย่าง
แต่มันไม่ใช่ v ของ x
เราไม่ได้มีแค่ x ตรงนี้
เรามีทั้งพจน์
ที่กำหนด u ของ x ขึ้นมา
เรามี u ของ x ตรงนี้
ขอผมลากเส้นเล็กๆ ตรงนี้นะ
เราจะได้ไม่เอาสองด้านมาปนกัน
และเรารู้จากกฎลูกโซ่
อนุพันธ์ y เทียบกับ x
อันนี้จะ
อันนี้จะเป็นอนุพันธ์ของ v
เทียบกับ u
หรือเราเรียกมันว่า v ไพรม์ก็ได้
v ไพรม์ของ u ของ x
v ไพรม์ของ u ของ x
ขอผมเขียน u ของ x ด้วยสีฟ้านะ
v ไพรม์ของ u ของ x
คูณ u ไพรม์ของ x
แล้ว v ไพรม์ของ u ของ x คืออะไร?
เรารู้ว่า v ไพรม์ของ x คืออะไร
ถ้าเราอยากรู้ว่า v ไพรม์ของ u ของ x คืออะไร
เราก็แค่แทนทุกที่ที่เราเห็น x
ด้วย u ของ x
อันนี้จึงเท่ากับ
v ไพรม์ของ u ของ x, u ของ x
และที่คุณทำ คือคุณหาอนุพันธ์

English: 
Remember, V is the log
base four of something.
But it's not V of X.
We don't have just an X here.
We have the whole expression
that defines U of X.
We have U of X right there.
And let me draw a little line here
so that we don't get
those two sides confused.
And so we know from the chain rule
the derivative Y with respect to X.
This is going to be
this is going to be the derivative of V
with respect to U.
Or we could call that V prime.
V prime of U of X.
V prime of U of X.
Let me do the U of X in blue.
V prime of U of X
times U prime of X.
Well, what is V prime of U of X?
We know what V prime of X is.
If we want to know what V prime of U of X
we would just replace wherever we see an X
with a U of X.
So, this is going to be equal to
V prime of U X, U of X.
And you just do is you take the derivative

Bulgarian: 
Не забравяй, че v е логаритъм с основа 4 от нещо.
Но това не е v(x).
Тук нямаме просто x.
Разполагаме с цял израз, 
който дефинира u(x).
Ето тук ни е дадена функцията u(x).
Нека да начертая една 
малка линия ето тук,
така че да не объркаме 
тези две части.
От верижното правило знаем,
че намираме производната 
на функцията y спрямо x.
Това ще бъде
производната на v спрямо u.
Може да наречем това v'.
v'(u(x)).
Нека да направя u(x) в син цвят.
v'(u(x))
умножено по u'(x).
На какво е равно v'(u(x))?
Знаем на какво е равно v'(x)
Ако искаме да знаем 
на какво е равно v'(u(x)),
то просто заместваме там, където
 виждаме x с функцията u(x).
Производната ще бъде равна на
v'(u(x)), u(x).
И това, което правиш, е просто 
да намериш производната

Korean: 
V(x)는 log4(x)이기 때문에
V(x)는 log4(x)이기 때문에
V(x)는 log4(x)이기 때문에
준식을 V(U(x))로
표현 가능합니다
준식을 V(U(x))로
표현 가능합니다
준식을 V(U(x))로
표현 가능합니다
준식을 V(U(x))로
표현 가능합니다
준식을 V(U(x))로
표현 가능합니다
그리고 이는 미분의 연쇄법칙에 의해
그리고 이는 미분의 연쇄법칙에 의해
그리고 이는 미분의 연쇄법칙에 의해
U에 대한 V의
도함수로 표현 가능하고
U에 대한 V의
도함수로 표현 가능하고
이는 곧 V'이 됩니다
그래서 y의 도함수는
U'(x)V'(U(x))
가 됩니다
U'(x)V'(U(x))
가 됩니다
U'(x)V'(U(x))
가 됩니다
U'(x)V'(U(x))
가 됩니다
V'(U(x))의 값은 무엇이 될까요
우리는 V'(x)의 결과에서
이를 유도해 낼 수 있습니다
V'(U(x))를 얻어내려면
x가 들어갈 곳에
U(x)를 넣어주면 됩니다
x가 들어갈 곳에
U(x)를 넣어주면 됩니다
그래서 이 식은
그래서 이 식은
파란색 식에 대한 초록색 식의
도함수를 계산해주면 됩니다

Czech: 
Jde vlastně o derivaci zelené
funkce podle modré funkce.
Bude to tedy 1 lomeno
(přirozený logaritmus ze 4 vynásobený...
Místo toho, abychom napsali x,
tam musíme napsat u(x).
...vynásobený u(x)), tohle celé krát
‚u‘ s čárkou v bodě x.
Dělám to ve více krocích, aby bylo
lépe vidět, co tu dělám.
Toto se rovná 1 lomeno
(přirozený logaritmus ze 4...
u(x) se rovná 
x na druhou plus x,
...krát (x na druhou plus x)),
tohle celé krát ‚u‘ s čárkou v bodě x, 
tedy krát (2 krát x plus 1).
Tohle ještě můžeme
přepsat jako:
(2 krát x plus 1) lomeno přirozený
logaritmus ze 4 krát (x na druhou plus x).

English: 
of the green function with
respect to the blue function.
So it's going to be one over
the natural log of four.
The natural log of four.
Times, instead of putting an X there
it would be times U of X.
Times U of X.
And of course, that whole thing
times U prime of X.
And so, and I'm doing
more steps just hopefully
so it's clearer what I'm doing here.
So this is one over
the natural log of four.
U of X is X squared plus X.
So
X squared plus X.
And then we're gonna multiply that
times U prime of X.
So times two X plus one.
And so we can just rewrite this as
two X plus one over
over
over
the natural log of four.
The natural log of four
times X squared plus X.
Times X squared
plus X.

Korean: 
파란색 식에 대한 초록색 식의
도함수를 계산해주면 됩니다
그래서 이 식은 1을 분자로 하고
(ln4)x를 분모로
가지는 식에서
(ln4)x를 분모로
가지는 식에서
x 대신 U(x)를 대입해준 뒤
x 대신 U(x)를 대입해준 뒤
x 대신 U(x)를 대입해준 뒤
전체에 U'(x)를 곱해주면 됩니다
전체에 U'(x)를 곱해주면 됩니다
U(x)와 U'(x)를 x에 관한 식으로
다시 바꿔주는 과정을 거친다면
분자는 그대로 1이고
(ln4)U(x)에서 U(x)를
x²+x로 바뀌며
x²+x로 바뀌며
x²+x로 바뀌며
곱해준 U'(x)는
곱해준 U'(x)는
2x+1이 됩니다
이를 다시 정리해보면
2x+1을 분자로 갖고
2x+1을 분자로 갖고
2x+1을 분자로 갖고
(ln4)(x²+x)를
분모로 갖는
(ln4)(x²+x)를
분모로 갖는
(ln4)(x²+x)를
분모로 갖는
(ln4)(x²+x)를
분모로 갖는
(ln4)(x²+x)를
분모로 갖는

Bulgarian: 
на зелената функция 
спрямо синята функция.
Ще бъде равно на 1 върху
натурален логаритъм от 4.
Умножено по...На мястото на x
ще бъде умножено по u(x).
По u(x).
И цялото това нещо разбира се
е умножено по u'(x).
Правя повече стъпки, защото
 се надявам, че така
ще бъде по-ясно какво правя тук.
И така, това е 1 върху 
натурален логаритъм от 4.
u(x) е равно на (x^2 + x).
x^2 + x
След това умножаваме 
този резултат по u'(x).
Следователно по (2x + 1).
Може просто да запишем
 израза отново
като 2x + 1 върху...
върху...
натурален логаритъм от 4.
Натурален логаритъм от 4,
умножено по (x^2 + x).
По x^2
плюс x.

Thai: 
ของฟังก์ชันสีเขียวเทียบกับฟังก์ชันสีฟ้า
มันจะเท่ากับ 1 ส่วน
ล็อกธรรมชาติของ 4
ล็อกธรรมชาติของ 4
คูณ แทนที่จะใส่ x ตรงนั้น
มันจะเป็น คูณ u ของ x
คูณ u ของ x
แน่นอน ทั้งหมดนั้น
คูณ u ไพรม์ของ x
แล้ว ผมจะทำอีกขั้นเพื่อหวังว่า
คุณจะเห็นชัดขึ้นว่าผมทำอะไรอยู่ตรงนี้
นี่ก็คือ 1 ส่วน
ล็อกธรรมชาติของ 4
u ของ x คือ x กำลังสองบวก x
แล้ว
x กำลังสองบวก x
แล้วเราจะคูณมัน
ด้วย u ไพรม์ของ x
คูณ 2x บวก 1
แล้วเราจะเขียนอันนี้ใหม่ได้
เป็น 2x บวก 1 ส่วน
ส่วน
ส่วน
ล็อกธรรมชาติของ 4
ล็อกธรรมชาติของ 4
คูณ x กำลังสองบวก x
คูณ x กำลังสอง
บวก x

English: 
And we're done, and we could distribute
this natural log of four
if we found that interesting.
But, we have just found
the derivative of Y
with respect to X.

Czech: 
A máme hotovo, i když ještě můžeme
roznásobit přirozeným logaritmem ze 4,
ale právě jsme spočítali
derivaci y podle x.

Korean: 
식이 구해지게 됩니다
이러한 방법으로
이러한 방법으로
y의 x에 관한 도함수를
구할 수 있습니다

Bulgarian: 
Готови сме и можем 
да разкрием скобите за
този натурален логаритъм от 4,
ако това ни се стори интересно.
Но просто намерихме 
производната на функцията y
спрямо x.

Thai: 
และเราก็เสร็จแล้ว เราแจกแจง
ล็อกธรรมชาติของ 4 นี้ได้
ถ้าเราว่ามันน่าสนใจ
แต่เราได้หาอนุพันธ์ของ y
เทียบกับ x แล้ว
