
English: 
- [Voiceover] In a previous
video we used the quotient rule
in order to find the derivatives
of tangent of x and cotangnet of x.
And what I what to do in
this video is to keep going
and find the derivatives of
secant of x and cosecant of x.
So let's start with secant of x.
The derivative with respect
to x of secant of x.
Well, secant of x is the same thing as
so we're going to find the
derivative with respect to x
of secant of x is the same thing as
one over,
one over the cosine of x.
And that's just the definition of secant.
And there's multiple
ways you could do this.
When you learn the chain rule,
that actually might be a
more natural thing to use
to evaluate the derivative here.
But we know the quotient rule,
so we will apply the quotient rule here.
And it's no coincidence that
you get to the same answer.
The quotient rule actually can be derived
based on the chain rule
and the product rule.
But I won't keep going into that.
Let's just apply the quotient
rule right over here.

Bulgarian: 
В предишно видео използвахме
правилото за производна на частно,
за да намерим производните на
tgx и cotgx.
В това видео ще продължим
и ще намерим производните 
на secx и cosecx.
Да започнем със secx.
Производната спрямо х на secx.
secx е същото нещо като...
Ще намираме производната 
спрямо х
на secx, което е същото
 нещо като
1 върху cosx.
Това е дефиницията на секанс.
Тук имаме няколко варианта.
Когато научиш верижното правило
за диференциране,
то може би ще е по-добър начин
да сметнеш производната тук.
Но ние знаем правилото за 
производна на частно,
затова тук ще приложим него.
Не е случайно, че ще получим
същия отговор.
Правилото за производна на частно
всъщност може да бъде изведено
от верижното правило и 
правилото за произведение.
Но няма да навлизам в това.
Нека просто приложим 
правилото за частно.

Thai: 
ในวิดีโอที่แล้ว เราใช้กฎผลหาร
เพื่อหาอนุพันธ์
ของแทนเจนต์ของ x กับโคแทนเจนต์ของ x
และสิ่งที่ผมอยากทำในวิดีโอนี้คือทำต่อไป
หาอนุพันธ์ของเซคแคนต์ของ x 
กับโคเซคแคนต์ของ x
ลองเริ่มด้วยเซคแคนต์ของ x กัน
อนุพันธ์เทียบกับ x ของเซคแคนต์ของ x
เซคแคนต์ของ x เท่ากับ
เราจะหาอนุพันธ์เทียบกับ x
ของเซคแคนต์ของ x เท่ากับ
1 ส่วน
1 ส่วนโคไซน์ของ x
และนั่นก็แค่นิยามของเซคแคนต์
และมันมีวิธีทำได้หลายวิธี
เวลาคุณเรียนกฎลูกโซ่
มันเป็นวิธีที่ธรรมชาติกว่า เมื่อใช้
หาค่าอนุพันธ์ตรงนี้
แต่เรารู้กฎผลหาร
เราจึงจะใช้กฎผลหารตรงนี้
และมันไม่บังเอิญที่คุณได้คำตอบเหมือนกัน
กฎผลหารจริงๆ แล้วก็พิสูจน์
มาจากกฎลูกโซ่กับกฎผลคูณ
แต่ผมจะไม่ลงรายละเอียดในที่นี้
ลองใช้กฎผลหารตรงนี้กัน

Czech: 
Minule jsme použili pravidlo o derivaci
podílu k nalezení derivací tan(x), cot(x).
V tomto videu odvodíme
derivace funkcí sekans a kosekans.
Začněme
s derivací sec(x).
sec(x) je dle definice to stejné
jako 1 lomeno cos(x).
Derivaci lze najít pomocí
několika různých způsobů.
Přirozeným způsobem by se dalo
použít pravidlo o derivaci složené funkce.
Avšak my použijeme pravidlo 
o derivaci podílu, které, jak víte,
může být odvozeno pomocí pravidel
o derivaci součinu a složené funkce.

Korean: 
이전 영상에서 tan(x)와 cot(x)의
도함수를 찾기 위해
몫의 미분법을 사용했죠
이번 영상에서도 계속 해보려고 합니다
sec(x)와 csc(x)의 
도함수를 찾아봅시다
먼저 sec(x)부터 시작해봅시다
sec(x)의 x에 대한 도함수도
마찬가지입니다
sec(x)의 x에 대한 도함수를
같은 방법으로 구할 거예요
sec(x)는
1 / cos(x)입니다
이것이 sec(x)의 정의입니다
sec(x)의 도함수를 구하는 방법은 
여러 가지가 있죠
사실 연쇄법칙으로
도함수를 구하는 것이
더 자연스럽습니다
그러나 우리는
몫의 미분법을 알고 있기 때문에
여기에 몫의 미분법을
적용시킬 거예요
어느 방법을 써도
같은 답이 나올 거예요
몫의 미분법은 사실
연쇄법칙과 곱의 미분을 이용해서
유도해낼 수 있어요
하지만 여기서 
유도는 하진 않을 거예요
그냥 몫의 미분법만
적용해봅시다

English: 
So this derivative is
going to be equal to,
it's going to be equal to
the derivative of the top.
Well, what's the derivative
of one with respect to x?
Well, that's just zero.
Times the function on the bottom.
So, times cosine of x.
Cosine of x.
Minus,
minus the function on the top.
Well, that's just one.
Times the derivative on the bottom.
Well, the derivative on the bottom is,
the derivative of cosine
of x is negative sine of x.
So we could put the sine of x there.
But it's negative sine of x,
so you have a minus and
it'll be a negative,
so we can just make that a positive.
And then all of that over the function
on the bottom squared.
So, cosine of x, squared.
And so zero times cosine of x,
that is just zero.
And so all we are left with is sine of x
over cosine of x squared.

Czech: 
Takže derivace bude
rovna následujícímu.
Derivace funkce v čitateli, tedy 0,
krát funkce ve jmenovateli, tedy cos(x).
Od toho odečtěme součin
funkce v čitateli, tedy 1,
a derivace funkce 
ve jmenovateli, tedy −sin(x).
Napišme tedy sin(x), jelikož
minus a minus je plus.
To celé vydělme čtvercem funkce
ve jmenovateli, tedy cos(x) na druhou.
Jelikož 0 krát cos(x) je 0, tak celkem
máme sin(x) lomeno cos(x) na druhou.

Korean: 
1 / cos(x)에서
분자인 1의 x에 대한 도함수를
먼저 구해봅시다
1의 x에 대한 도함수는 뭐죠?
그냥 0이죠
여기에 분모 함수인
cos(x)를 곱하고
0 × cos(x)에서
빼기
분자 함수인
1을 뺍니다
여기에 분모의 도함수를 곱합니다
분모인 cos(x)의 도함수는
-sin(x)이므로
여기에 sin(x)를 넣어줍니다
-sin(x)이기 때문에
앞의 마이너스 부호와 음수가 만나
양수를 만듭니다
그리고 이 모두를
분모를 제곱한 함수인
cos²x로
나눕니다
0과 cos(x)를 곱하면
그냥 0이 됩니다
그래서 최종적으로
sin(x)/cos²x가 되겠네요

Bulgarian: 
Производната ще бъде
равна на производната
на горната функция...
На колко ще е равна 
производната на 1 спрямо х?
Ами, просто 0.
По долната функция,
т.е. по cosx,
минус
горната функция,
която е просто 1,
по производната на долната.
Производната на долната е...
Производната на cosx е –sinx.
Можем да сложим sinx там.
Обаче това е –sinx,
и тъй като имаме минус,
ще бъде отрицателно.
Затова можем просто 
да направим това положително.
Тогава цялото това върху
долната функция на квадрат.
cosx на квадрат.
0 по cosx
е просто 0.
Остава ни само sinx
върху cosx на квадрат.

Thai: 
อนุพันธ์นี้จะเท่ากับ
มันจะเท่ากับอนุพันธ์ของตัวบน
อนุพันธ์ของ 1 เทียบกับ x คืออะไร?
มันก็แค่ 0
คูณฟังก์ชันข้างล่าง
คูณโคไซน์ของ x
โคไซน์ของ x
ลบ
ลบฟังก์ชันตัวบน
มันก็แค่ 1
คูณอนุพันธ์ข้างล่าง
อนุพันธ์ของตัวล่างคือ
อนุพันธ์ของโคไซน์ของ x คือลบไซน์ของ x
เราก็ใส่ไซน์ของ x ตรงนั้นได้
มันคือลบไซน์ของ x
คุณจึงได้ลบ มันจะเป็นลบ
เราทำให้มันเป็นบวกได้
แล้วทั้งหมดนั้นส่วนฟังก์ชัน
ข้างล่างกำลังสอง
โคไซน์ของ x กำลังสอง
แล้ว 0 คูณโคไซน์ของ x
มันก็แค่ 0
แล้วที่เราเหลือคือไซน์ของ x
ส่วนโคไซน์ของ x กำลังสอง

Thai: 
และมันมีวิธีเขียนอันนี้ใหม่ได้หลายวิธี
ถ้าต้องการ
คุณบอกได้ว่า อันนี้เท่ากับไซน์ของ x
ส่วนโคไซน์ของ x คูณ 1 ส่วนโคไซน์ของ x
และแน่นอน นี่คือแทนเจนต์ของ x
คูณเซคแคนต์ของ x
เซคแคนต์ของ x
คุณจึงบอกได้ว่า อนุพันธ์ของเซคแคนต์ของ x
เท่ากับไซน์ของ x ส่วนโคไซน์กำลังสองของ x
หรือมันก็คือแทนเจนต์ของ x คูณ
เซคแคนต์ของ x
ทีนี้ลองทำโคเซคแคนต์บ้าง
อนุพันธ์เทียบกับ x ของโคเซคแคนต์ของ x
มันจะเท่ากับอนุพันธ์
เทียบกับ x ของ 1 ส่วนไซน์ของ x
โคเซคแคนต์คือ 1 ส่วนไซน์ของ x
ผมจำได้ เพราะคุณคิดว่ามันคือโคเซคแคนต์
บางทีมันน่าจะเป็นส่วนกลับของโคไซน์
แต่มันไม่ใช่
มันตรงข้ามกับสิ่งที่คุณคิด
ส่วนกลับของโคไซน์ไม่ใช่โคเซคแคนต์
แต่มันคือเซคแคนต์

Korean: 
당신만 괜찮다면
이를 다양한 방법으로
표기할 수 있습니다
이것은 sin(x) / cos(x)곱하기
1 / cos(x)와 같습니다
그리고 이것은 tan(x)
곱하기
sec(x)와도 같습니다
그래서 sec(x)의 도함수는
sin(x) / cos²x 또는
tan(x) sec(x)입니다
이제 csc(x)를 해봅시다
csc(x)의 x에 대한 도함수는
1/sin(x)의 x에 대한 도함수와
같습니다
csc(x)는 1/sin(x)입니다
이것이 csc(x)인걸 기억하세요
csc(x)는 cos(x)의 역수가 아닙니다
보통 생각하는 것과는 반대죠
cos(x)의 역수는 csc(x)가 아니라
sec(x)입니다

English: 
And there's multiple ways
that you could rewrite this
if you like.
You could say that this
is same thing as sine of x
over cosine of x times
one over cosine of x.
And of course this is tangent of x,
times secant of x.
Secant of x.
So you could say derivative
of secant of x is
sine of x over cosine-squared of x.
Or it is tangent of x
times the secant of x.
So now let's do cosecant.
So the derivative with
respect to x of cosecant of x.
Well, that's the same
thing as the derivative
with respect to x of one over sine of x.
Cosecant is one over sine of x.
I remember that because
you think it's cosecant.
Maybe it's the reciprocal
of cosine, but it's not.
It's the opposite of
what you would expect.
Cosine's reciprocal isn't
cosecant, it is secant.

Czech: 
Výraz můžeme přepsat například jako
sin(x) lomeno cos(x) krát 1 lomeno cos(x).
Což je
tan(x) krát sec(x).
Můžeme říct, že derivace sec(x)
je sin(x) lomeno cos(x) na druhou,
jinak řečeno
tan(x) krát sec(x).
Počítejme nyní
derivaci csc(x).
To je opět dle definice stejná věc
jako derivace 1 lomeno sin(x).
Pomůckou na zapamatování je,
že csc je to opačné, než bychom očekávali.

Bulgarian: 
Това можем да го запишем 
по няколко начина.
Можем да кажем, че това е
 същото нещо като sinx
върху cosx по 1 върху cosx.
Разбира се, че това е tgх
по secx.
Можем да кажем, че 
производната на secх е
sinx върху cos квадрат х
или tgx по secx.
Нека сега направим
 cosec.
Производната спрямо х на cosecх.
Това е същото нещо 
като производната
спрямо х на 1 върху sinx.
Косеканс е едно върху синус х.
Помня това, защото може би
 си мислиш,
че косеканс е реципрочното
 на косинус, но не е така.
Обратното е на това, 
което очакваш.
Реципрочното на косинус не е 
косеканс, а е секанс.

Thai: 
ย้ำอีกครั้ง มันตรงข้ามกับที่คุณคิด
อันนั้นเริ่มต้นด้วย s อันนี้เริ่มต้นด้วย c
อันนั้นเริ่มต้นด้วย c อันนั้นเริ่มต้นด้วย s
มันก็แค่วิธีที่เขานิยามขึ้นมา
เอาล่ะ ลองหาค่านี้กัน
ย้ำอีกครั้ง เราจะใช้กฎผลหาร
แต่คุณทำได้ด้วยกฎลูกโซ่เช่นกัน
มันจะเท่ากับ
อนุพันธ์ของพจน์ข้างบน ซึ่งก็คือ 0
คูณพจน์ข้างล่าง ซึ่งก็คือไซน์ของ x
ไซน์ของ x
ลบพจน์ข้างบน ซึ่งก็คือ 1
คูณอนุพันธ์ของพจน์ข้างล่าง
ซึ่งก็คือโคไซน์ของ x
ทั้งหมดนั้นส่วนพจน์ข้างล่างกำลังสอง
ไซน์กำลังสองของ x
นั่นคือ 0
เราจึงได้ลบโคไซน์ของ x
ส่วนไซน์กำลังสองของ x
นั่นคือวิธีคิดอย่างหนึ่ง
หรือถ้าต้องการ คุณทำแบบนี้ได้
อย่างที่เราทำไปตรงนี้

Bulgarian: 
Отново обратното на това, 
което би очаквал.
Това започва с "s", а това
започва с "c".
Това започва с "c", а това
 започва с "s".
Просто така е дефинирано.
Нека просто го пресметнем.
Отново ще използвам формулата за 
намиране на производна на частно,
но може и да се направи
с верижното правило.
Ще бъде
производната на израза отгоре, 
който е 0,
по израза отдолу, 
който е sinx,
минус израза отгоре,
 който е 1,
по производната на 
израза отдолу,
която е cosx,
всичко това върху
израза отдолу на квадрат,
т.е. sin квадрат х.
Това е 0.
Получаваме –cosx
върху sin квадрат х.
Можем да го разгледаме 
по този начин
или сякаш това
е същото нещо като
това тук.

Czech: 
1 lomeno cos(x), není csc(x),
nýbrž sec(x).
Toto začíná na ‚s‘ 
a toto začíná na ‚c‘.
Toto začíná na ‚c‘
a toto začíná na ‚s‘.
Vraťme se však
k odvozování.
Užijme pravidla
o derivaci podílu,
jde užít i pravidlo o
derivaci složené funkce.
Obdržíme derivaci výrazu nahoře, což
je 0, krát výraz dole, což je sin(x).
Odečteme výraz nahoře, což je 1,
krát derivace výrazu dole, což je cos(x).
To celé vydělme čtvercem výrazu
dole, tedy sin(x) na druhou.
Toto je 0.
Tedy dostaneme −cos(x)
lomeno sin(x) na druhou.
Výraz můžeme upravit
stejně jako minule.

English: 
Once again, opposite of
what you would expect.
That starts with an s,
this starts with a c.
That starts with a c,
that starts with an s.
It's just way it happened to be defined.
But anyway, let's just evaluate this.
Once again, we'll do the quotient rule,
but you could also do
this using the chain rule.
So it's going to be
the derivative of the expression
on top, which is zero,
times the expression on the
bottom, which is sine of x.
Sine of x.
Minus the expression on
top, which is just one.
Times the derivative of the
expression on the bottom,
which is cosine of x.
All of that over the expression
on the bottom squared.
Sine-squared of x.
That's zero.
So we get negative cosine of x
over sine-squared of x.
So that's one way to think about it.
Or if you like, you could do this,
the same thing we did over here,

Korean: 
다시 말하지만
생각한 것과는 반대죠
저것은 s로 시작하고
이것은 c로 시작하고
저것은 c로 시작하고
이것은 s로 시작합니다
이게 정의일 뿐이에요
어쨌든 일단 이걸 풀어봅시다
몫의 미분법을 이용할 겁니다
연쇄법칙도 이용해서 풀어볼 거예요
이를 위해
분자의 도함수인 0에
분모함수인 sin(x)를 곱합니다
0 × sin(x)에서
분자의 함수인 1에다가
분모 함수의 도함수인
cos(x)를 곱해서 빼는 거예요
그리고 분모를 제곱한
sin²x로 나눕니다
여기는 0이기 때문에
분자는 -cos(x)가 되고
분모는 sin²x이 되겠죠
이것은 표기하는 방법 중 하나입니다
아니면,
다르게도 표현해 봅시다

Korean: 
마찬가지로,  -cos(x)/sin(x)곱하기
1 / sin(x)로 나타낼 수 있습니다
-cos(x) / sin(x)는 -cot(x)가 됩니다
-cot(x)가 됩니다
이렇게 쓰는게 편하겠군요
1/sin(x)는 csc(x)이므로
csc(x)를 곱하면 -cot(x)csc(x)가 됩니다
둘 중 더 편한 걸로 쓰시면 됩니다

Thai: 
อันนี้เท่ากับลบโคไซน์ของ x
ส่วนไซน์ของ x คูณ 1 ส่วนไซน์ของ x
และนี่คือลบโคแทนเจนต์ของ x
ลบโคแทนเจนต์ของ x คูณ
ผมจะเขียนแบบนี้นะ
คูณ 1 ส่วนไซน์ของ x คือโคเซคแคนต์ของ x
โคเซคแคนต์ของ x
อันไหนก็ได้ที่คุณคิดว่ามีประโยชน์กว่า

Bulgarian: 
Това е същото нещо
 като –cosx
върху sinx по 1 / sinx.
Това е –cotgx.
–cotgх по...
Може би ще го запиша така.
По 1 /sinx е равно на cosecх.
Косеканс от х.
Което на теб ти е по-полезно.

Czech: 
Tedy máme −cos(x) lomeno sin(x)
krát 1 lomeno sin(x).
Což je
−cot(x) krát csc(x).
Tím jsme hotovi.

English: 
this is the same thing
as negative cosine of x
over sine of x, times one over sine of x.
And this is negative cotangent of x.
Negative cotangent of x, times,
maybe I'll write it this way,
times one over sine of x is cosecant of x.
Cosecant of x.
So, which ever one you find more useful.
