
Korean: 
이 동영상에서 제가 하고 싶은 것은
지수함수의 도함수에 대한 탐구입니다
우리는 이미 x에 대한
e^x의 미분값이
e^x임을 살펴보았습니다
이는 놀라운 것입니다
e가 특별한 이유들 중 하나입니다
만약 밑이 e인 지수함수를
가지고 있다면 그것의
미분값은 원함수에서의
기울기와 같습니다
하지만 다른 밑을 가질 때를 생각해봅시다
a^x의 x에 대한
미분값을 만약 a가 임의의 상수라면
알 수 있을까요?
알아낼 방법이 있을까요?
혹시 e^x의 미분에 대한 지식을
사용할까요?
만약 약간의 대수와
e의 성질로 다시 쓸 수 있다면

English: 
- [Voiceover] What I
want to do in this video
is explore taking the derivatives
of exponential functions.
So we've already seen that the derivative
with respect to x of e to the x
is equal to e to x,
which is a pretty amazing thing.
One of the many things that
makes e somewhat special.
Though when you have an exponential
with your base right over here as e,
the derivative of it,
the slope at any point,
is equal to the value
of that actual function.
But now let's start exploring
when we have other bases.
Can we somehow figure out
what is the derivative,
what is the derivative with respect to x
when we have a to the x,
where a could be any number?
Is there some way to figure this out?
And maybe using our
knowledge that the derivative
of e to the x, is e to the x.
Well can we somehow use
a little bit of algebra
and exponent properties to rewrite this

Bulgarian: 
В настоящия урок искам
да изследвам намирането на
 производни на функции със степени.
Вече видяхме, че производната
спрямо x на e^x
е равна на e^x,
което е удивително нещо.
Едно от многото неща, които правят 
'e' специално число по някакъв начин.
Въпреки че е дадена функция 
със степен,
и с числото e за основа ето тук,
производната на функцията, т.е. наклонът
в коя да е точка от функцията,
ще бъде равен на стойността 
на самата функция.
Нека сега да изследваме случаи,
в които имаме други числа за основа.
Можем ли по някакъв начин 
да намерим на какво
е равна производната спрямо x,
когато имаме a^x, където 
a може да е всяко число?
Има ли начин да отговорим 
на този въпрос?
Може да е като използваме 
знанието, че производната
на e^x е равна на e^x.
А може ли да използваме 
малко алгебра
и свойства на степените, 
за да запишем функцията,

Czech: 
V tomto videu se chci zabývat
derivacemi exponenciálních funkcí.
Už jsme se setkali s derivacemi
podle x z (e na x), což je (e na x),
to je docela zajímavá věc.
Jedna z mnoha věcí,
která činí e neobvyklým.
Když zde máte exponenciální
funkci o základu e,
její derivace, sklon v jakémkoli bodě,
je roven hodnotě aktuální funkce.
Pojďme teď prozkoumat
funkce o jiném základu.
Můžeme nějak přijít na to,
co je derivací podle x,
pokud máme (a na x),
kde 'a' může být jakékoli číslo?
Lze to nějak vyřešit?
A možná s využitím znalosti,
že derivace (e na x) je (e na x)?
Můžeme nějak použít trochu
algebry a vlastností exponentu

Thai: 
สิ่งที่ผมอยากทำในวิดีโอนี้
คือสำรวจเรื่องการหาอนุพันธ์ของ
ฟังก์ชันเอกซ์โพเนนเชียล
เราเห็นมาแล้วว่าอนุพันธ์
เทียบกับ x ของ e กำลัง x
เท่ากับ e กำลัง x
ซึ่งเป็นสิ่งที่น่าอัศจรรย์ทีเดียว
เป็นสิ่งหนึ่งที่ทำให้ e นั้นพิเศษ
เวลาคุณมีเอกซ์โพเนนเชียล
ที่ฐานของคุณตรงนี้เป็น e
อนุพันธ์ของมัน ความชันที่จุดใดๆ
เท่ากับค่าของฟังก์ชันตรงนั้น
แต่ตอนนี้ เราเริ่มสำรวจกรณีที่เรามีฐานอื่นบ้าง
เราหาได้ไหมว่าอนุพันธ์คืออะไร
อนุพันธ์เทียบกับ x
เมื่อเรามี a กำลัง x เมื่อ a เป็นจำนวนใดๆ
คืออะไร?
มันมีวิธีหาไหม?
อาจจะใช้ความรู้ของเราที่ว่าอนุพันธ์
ของ e กำลัง x เท่ากับ e กำลัง x
เราใช้พีชคณิต
และสมบัติเลขยกกำลัง เขียนอันนี้

Korean: 
e가 밑인 것으로 볼 수 있을까요?
여러분은 a를 볼 수 있습니다
a가 다시 쓰자면
이렇게 다시 쓰겠습니다
a가 e^(ln a)임을
볼 수 있습니다
만약에 이것이 직관적이지 않다면
여러분이 이것에 대해 
다시 생각해보기를 바랍니다
ln a 가 무엇일까요?
a의 자연 로그는 e에 지수로 취해
a를 얻기 위한 것입니다
만약에 e의 지수 형태로 얻고
싶을 때 여러분은
a를 얻기 위해 e의 지수형태로
올릴 수 있습니다
그럼 a를 얻을 수 있을 것입니다
이것에 대해 생각해 보겠습니다
이것을 맹신하지 마십시오
이것이 말이 되는 것처럼 보일 수 있습니다
이것은 진짜 로그에 대해 보여줍니다
따라서 우리는 a를 이 표현으로 대체할 수 있습니다
만약 a가 e^(ln a)와 같다면

Bulgarian: 
така че да изглежда като нещо,
 което има числото e за основа?
Може да разглеждаш числото a,
може да го разглеждаш като нещо,
 което е свързано с числото e.
Нека да го запиша по следния начин.
Добре. Разглеждаме a като равно на
е на степен натурален 
логаритъм от а.
Ако сега не е очевидно за теб,
то наистина искам
да помислиш върху това.
Какво представлява 
натуралният логаритъм от a?
Натуралният логаритъм от a
е степента, на която следва
да повдигнеш числото е, 
за да получиш числото а.
Ако действително повдигнеш 
e на степен...
ако повдигнеш e на степента,
 която е необходима,
за да се получи числото а.
Тогава просто ще получиш 
числото а.
Така че наистина помисли 
върху това.
Недей наготово
да приемаш това за истина.
Наистина трябва да го разбереш.
И това ще дойде от знанието 
какво всъщност е логаритъм.
Може да заместим a с целия 
този израз ето тук.
Ако a е същото нещо като 
e на степен натурален логаритъм от a,

English: 
so it does look like
something with e as a base?
Well, you could view a,
you could view a as being equal to e.
Let me write it this way.
Well all right, a as being equal to e
to the natural log of a.
Now if this isn't obvious to you,
I really want you to think about it.
What is the natural log of a?
The natural log of a is the power you need
to raise e to, to get to a.
So if you actually raise e to that power,
if you raise e to the power you need
to raise e too to get to a.
Well then you're just going to get to a.
So really think about this.
Don't just accept this as a leap of faith.
It should make sense to you.
And it just comes out of
really what a logarithm is.
And so we can replace a with
this whole expression here.
If a is the same thing as
e to the natural log of a,

Thai: 
ให้มันดูเหมือนว่า e เป็นฐานได้ไหม?
คุณมอง a
คุณมอง a ว่าเท่ากับ e ได้
ขอผมเขียนแบบนี้นะ
เอาล่ะ a เท่ากับ e
กำลังล็อกธรรมชาติของ a
ถ้าคุณไม่เห็นชัดว่าทำไม
ผมอยากให้คุณคิดดู
ล็อกธรรมชาติของ a คืออะไร?
ล็อกธรรมชาติของ a คือกำลังที่คุณต้อง
ยก e จึงจะได้ a
ถ้าคุณยกกำลัง e ด้วยกำลังนั้น
ถ้าคุณยก e กำลังค่าที่คุณต้อง
ยก e เพื่อให้ได้ a
คุณก็จะได้ค่า a
คิดดู
อย่ายอมเชื่อง่ายๆ
แล้วคุณจะเข้าใจ
มันตรงมาจากนิยามว่าลอการิทึมคืออะไร
แล้วเราแทน a ด้วยพจน์ทั้งหมดนี่ตรงนี้ได้
ถ้า a เท่ากับ e ยกกำลังล็อกธรรมชาติของ a

Czech: 
a přepsat to tak,
že 'e' bude základ?
Můžeme uvažovat,
že 'a' rovná se 'e'…
Napíšu to takto.
'a' se rovná 'e' na
přirozený logaritmus 'a'.
Pokud vám to není jasné,
chci, abyste o tom přemýšleli.
Čemu je roven přirozený
logaritmus z 'a'?
Přirozeným logaritmem 'a' je mocnina,
kterou umocníte 'e', abyste dostali 'a'.
Takže pokud umocníte e
exponentem, který potřebujete,
abyste po umocnění dostali 'a',
potom dostanete hodnotu 'a'.
Tak o tom popřemýšlejte.
Nepřijměte to jako slepou pravdu.
Mělo by vám to dávat smysl.
Vychází to z toho,
co je logaritmus.
Takže můžeme nahradit 'a'
tímto celým výrazem.
Pokud 'a' je shodné
s 'e' na přirozený logaritmus,

Thai: 
แล้วอันนี้จะเท่ากับ
อันนี้จะเท่ากับอนุพันธ์
เทียบกับ x
ของ e กำลังล็อกธรรมชาติ
ผมเขียน la อยู่นั่นแหละ [หัวเราะ]
ล็อกธรรมชาติของ a
แล้วเราจะยกกำลังมันด้วย x
เราจะยกกำลังมันด้วย x
และตอนนี้ เมื่อใช้สมบัติเลขยกกำลัง
อันนี้จะเท่ากับอนุพันธ์
เทียบกับ x ของ
ผมจะใช้สีแทนความหมายไปเรื่อยๆ
ถ้าผมยกกำลังอะไรสักอย่าง
ด้วยเลขชี้กำลังค่าหนึ่ง
แล้วยกกำลังมันด้วยเลขชี้กำลังอีกตัว
มันจะเท่ากับ การยกกำลังฐานเดิม
ด้วยผลคูณของเลขชี้กำลังเหล่านั้น
นั่นก็แค่สมบัติเลขยกกำลังพื้นฐาน
มันจะเท่ากับ e
กำลังล็อกธรรมชาติของ a
ล็อกธรรมชาติของ a คูณ x
คูณ x
แล้วตอนนี้เราใช้กฎลูกโซ่

Korean: 
이것은
x에 대해서
e ^(ln a)의 미분이
될 것입니다
계속 웃는 것으로 표기하였군요
a의 자연로그입니다
이것을 x 제곱할 것입니다
이것을 x만큼 제곱하면
e의 성질으로부터
x에 대한 미분값과
동일할 것입니다
계속하여 색을 칠하겠습니다
만약 제가 e에
지수를 취하면
원래 밑에
지수의 곱을 지수로 취하는 것과 같습니다
이것은 지수의 기본적인 성질입니다
결국 저것은 e에
a의 자연로그 제곱에
x제곱을 한 것과
같습니다
이제 우리는 연쇄 법칙을 사용해서

Bulgarian: 
то тогава производната ще бъде...
Производната ще бъде 
равна на производната
спрямо x
на е на степен натурален логаритъм.
О, продължавам да пиша la!
На степен натурален логаритъм от а.
След това ще повдигнем
този израз на степен x.
ще повдигнем този израз на степен x.
И сега, просто като използваме 
свойствата на степените,
това ще бъде равно на производната
спрямо x на...
Продължавам да използвам
 различни цветове.
Ако повдигна нещо на степен,
а след това повдигна 
резултата на степен,
то това е нещо, което представлява 
повдигането на нашата основа
на произведението 
от тези две степени.
Това е просто основно 
свойство на степените.
Следователно този израз ще бъде
 същото нещо като
е на степен 
натурален логаритъм от a...
На степен натурален логаритъм от a,
 умножено по x.
Умножено по x 
в степенния показател.
Сега може да използваме 
верижното правило,

English: 
well then this is going to be,
then this is going to be
equal to the derivative
with respect to x
of e to the natural log,
I keep writing la (laughs),
to the natural log of a
and then we're going to
raise that to the xth power.
We're going to raise that to the x power.
And now this, just using
our exponent properties,
this is going to be
equal to the derivative
with respect to x of,
and I'll just keep color-coding it.
If I raise something to an exponent
and then raise that to an exponent,
that's the same thing as
raising our original base
to the product of those exponents.
That's just a basic exponent property.
So that's going to be the same thing as e
to the natural log of a,
natural log of a times x power.
Times x power.
And now we can use the chain rule

Czech: 
potom to bude rovno
derivaci podle x…
Derivaci e na ln(a) a potom to
umocníme na x-tou mocninu.
Teď s užitím vlastnosti exponentu
to bude rovno derivaci podle x…
Tady to zvýrazním.
Pokud něco umocním 
a ještě znovu to umocním,
je to stejné jako umocnit
původní základ na součin exponentů.
To je základní
vlastnost exponentů.
Tedy to bude stejné jako 'e' na přirozený
logaritmus 'a' krát 'x'.

Czech: 
A teď to můžeme použít pravidlo
o složené funkci a vyčíslit derivaci.
Takže teď uděláme to,
že první vezmeme
derivaci vnější funkce.
'e' na přirozený logaritmus
'a' krát 'x' podle vnitřní funkce,
tedy podle přirozeného
logaritmu 'a' krát x.
Bude se to rovnat 'e' na
přirozený logaritmus 'a' krát 'x'.
A potom vezmeme derivaci
vnitřní funkce podle x.
Přirozený logaritmus 'a'…
Nemusí to být hned
patrné, ale to je číslo.
Bude to vynásobené derivací.
Kdyby to byla derivace
(3x), byly by to 3.
Když to je derivace
přirozeného logaritmu 'a' krát 'x',
bude to přirozený
logaritmus 'a'.
A to nám dá přirozený logaritmus 'a'
krát 'e' na přirozený logaritmus 'a'.

English: 
to evaluate this derivative.
So what we will do is
we will first take the derivative
of the outside function.
So e to the natural log of a times x
with respect to the inside function,
with respect to natural log of a times x.
And so, this is going to be equal to e
to the natural log of a times x.
And then we take the derivative
of that inside function
with respect to x.
Well natural log of a,
it might not immediately jump out to you,
but that's just going to be a number.
So that's just going to be,
so times the derivative.
If it was the derivative of
three x, it would just be three.
If it's the derivative
of natural log a times x,
it's just going to be natural log of a.
And so this is going to give us
the natural log of a times e
to the natural log of a.

Korean: 
이 미분을 할 수 있습니다
우리가 할 것은
바깥의 함수의 미분을 취하고
따라서 e^(ln a)x
의 안쪽 함수 즉,
(ln a)x에 대한 미분을 구할 것입니다
따라서 이것은
e^(ln a)x와 같을 것입니다
그리고 안쪽의 함수의 x에 관한
미분을 할 것입니다
ln a는 미분이 바로 생각나지는
않을지는 몰라도
상수입니다
따라서 이것은
미분에 영향을 주지 않습니다
3x 였다면 미분값은 x가 될것입니다
만약 이것이 (ln a)x였다면
ln a가 될 것입니다
따라서 이것은
(ln a)e^(lna)^x와
같다는 것을 알 수 있습니다

Thai: 
เพื่อหาค่าอนุพันธ์นี้ได้
สิ่งที่เราจะทำ
คือเราจะหาอนุพันธ์ของฟังก์ชันนอก
e กำลังล็อกธรรมชาติของ a คูณ x
เทียบกับฟังก์ชันใน
เทียบกับล็อกธรรมชาติของ a คูณ x
แล้ว อันนี้จะเท่ากับ e
กำลังล็อกธรรมชาติของ a คูณ x
แล้วเราหาอนุพันธ์ของฟังก์ชันในนั้น
เทียบกับ x
ล็อกธรรมชาติของ a
มันอาจไม่สะดุดตาคุณทันที
แต่มันจะเท่ากับจำนวนนั้น
มันจะเท่ากับ
คูณอนุพันธ์
ถ้ามันเป็นอนุพันธ์ของ 3x มันจะได้ 3
ถ้ามันเป็นอนุพันธ์ของล็อกธรรมชาติ a คูณ x
มันจะได้แค่ล็อกธรรมชาติของ a
แล้วอันนี้จะให้
ล็อกธรรมชาติของ a คูณ e
กำลังล็อกธรรมชาติของ a

Bulgarian: 
за да намерим производната.
Това, което ще направим,
е първо да намерим
 производната на външната функция.
Тоест на числото e на степен 
натурален логаритъм от a по x,
спрямо вътрешната функция,
спрямо натурален логаритъм
 от а по x.
И така, това ще бъде равно
на e на степен натурален
 логаритъм от a по x.
След това намираме 
производната на вътрешната функция
спрямо x.
E, натурален логаритъм от a,
може да не се досетиш веднага,
но това просто ще бъде числото ln(a).
Следователно това число просто ще бъде
умножено по производната.
Ако беше производната на 3 по x, 
щеше да бъде просто 3.
Ако е производната на 
натурален логаритъм от a по x,
то ще бъде просто 
натурален логаритъм от a.
И това ще ни даде като резултат
натурален логаритъм от a
по e на степен 
натурален логаритъм от a.

English: 
And I'm going to write it like this.
Natural log of a to the x power.
Well we've already seen this.
This right over here is just a.
So it all simplifies.
It all simplifies to the natural log of a
times a to the x,
which is a pretty neat result.
So if you're taking the
derivative of e to the x,
it's just going to be e to the x.
If you're taking the
derivative of a to the x,
it's just going to be the natural
log of a times a to the x.
And so we can now use this result
to actually take the
derivatives of these types
of expressions with bases other than e.
So if I want to find the derivative
with respect to x
of eight times three to the x power,
well what's that going to be?
Well that's just going to be eight times

Bulgarian: 
Ще го запиша по следния начин.
Повдигнато на степен x.
Вече сме виждали това.
Това ето тук е просто числото а.
Следователно резултатът 
се опростява.
Опростява се до 
натурален логаритъм от a
умножено по а на степен x,
което изглежда сравнително
 добре като резултат.
Следователно ако търсиш
 производната на e на степен x,
то просто ще е равна на 
е на степен x.
Ако търсиш производната на
 а на степен x,
тя е равна на натурален логаритъм от а, умножено по а на степен x.
Сега вече може да използваме
 този резултат,
за да намираме производните 
на такива видове
изрази, които имат основа,
 различна от числото e.
Ако искам да намеря 
производната спрямо x
на 8 по 3 на степен x,
то на какво ще е равна тя?
Това просто ще бъде равно 
на 8 умножено

Thai: 
และผมจะเขียนมันแบบนี้
ล็อกธรรมชาติของ a กำลัง x
เราเห็นอันนี้มาแล้ว
อันนี้ตรงนี้ก็แค่ a
มันก็ลดรูปได้
มันลดรูปเหลือ ล็อกธรรมชาติของ a
คูณ a กำลัง x
ซึ่งเป็นผลที่สวยงามทีเดียว
ถ้าคุณหาอนุพันธ์ของ e กำลัง x
มันก็แค่ e กำลัง x
ถ้าคุณหาอนุพันธ์ของ a กำลัง x
มันจะเท่ากับล็อกธรรมชาติของ a
คูณ a กำลัง x
แล้วเราก็ใช้ผลนี้
หาอนุพันธ์ของพจน์ประเภทนี้
ที่มีฐานอื่นนอกจาก e ได้
ถ้าผมอยากหาอนุพันธ์
เทียบกับ x
ของ 8 คูณ 3 ยกกำลัง x
มันจะเท่ากับอะไร?
มันจะเท่ากับ 8 คูณ

Korean: 
이렇게 쓰겠습니다
(ln a)e^(lna)^x입니다
이것을 이미 보였으며
이부분이 a이기에
단순화할 수 있습니다
따라서 ln a에
a^x를 곱한 것과 같습니다
매우 간단합니다
따라서 여러분이 e^x의 미분을 구하면
그냥 e^x가 됩니다
만약 여러분이 a^x을 미분하면
단순히 (ln a)a^x가 됩니다
따라서 우리는 이 결과를
이러한 경우에 대한 표현을
e가 아닌 밑에 대해 적용할 수 있습니다
만약 제가
x에 관한 미분을
8*3^x에 대해 하고 싶다면
어떻게 될까요?
이것은 8을

Czech: 
A zapíšu to takto:
přirozený logaritmus 'a' na 'x'.
Už jsme to viděli.
Toto zde vpravo je jen 'a'.
Vše se zjednoduší na přirozený
logaritmus 'a' krát (a na x),
což je pěkný výsledek.
Takže derivace
e na x je e na x.
Pokud derivujete (a na x),
bude to přirozený logaritmus
'a' krát (a na x).
Takže můžeme
použít výsledek,
abychom dostali derivaci
výrazů o jiném základu než 'e'.
Takže když chceme najít derivaci
podle x výrazu 8 krát (3 na x),
kolik to bude?
Jednoduše: 8 krát…

Korean: 
오른쪽의 미분값에 곱한 형태가 될것이며
우리가 방금 본 결과에 의해
밑의 자연로그인
ln3에 곱하는 값이
3^x가 되어
(8ln 3)*3^x의
형태로 표현이 됨을
알 수 있습니다

Thai: 
แล้วอนุพันธ์ของตัวนี้ตรงนี้
จะเท่ากับ จากสิ่งที่เราเพิ่งเห็นไป
จะเท่ากับล็อกธรรมชาติของฐานเรา
ล็อกธรรมชาติของ 3 คูณ 3 กำลัง x
คูณ 3 กำลัง x
มันจึงเท่ากับ 8 ล็อกธรรมชาติ
ของ 3 คูณ 3 กำลัง x
คูณ 3 ยกกำลัง x

Bulgarian: 
по производната на този израз тук.
Като се основаваме на това, 
което току-що намерихме,
това  ще бъде натурален 
логаритъм от основата,
т.е. натурален логаритъм 
от 3, по 3 на степен x.
Умножено по 3 на степен x.
Следователно производната 
е равна на 8
по натурален логаритъм 
от 3, по 3 на степен x.
Умножено по 3 на степен x.

Czech: 
A teď derivace tady toho bude…
Na základě toho,
co už jsem říkal,
to bude přirozený logaritmus
našeho základu,
přirozený logaritmus 3 krát (3 na x).
Takže je to rovno 8 krát přirozený 
logaritmus 3 krát (3 na x).

English: 
and then the derivative
of this right over here
is going to be, based on what we just saw,
it's going to be the
natural log of our base,
natural log of three times three to the x.
Times three to the x.
So it's equal to eight natural log
of three times three to the x.
Times three to the x power.
