Directional derivative Khan Academy English (auto-generated) Korean

Directional derivativeYouTube↗ IA↗(long shot)
Education 387.4K views 2.9K likes 167 dislikes 11 May 2016

Khan Academy 9.4M subscribers United States YouTube↗

Click to Play Video

hello everyone so here I'm going to talk

안녕하세요 여러분

about the directional derivative and

오늘은 방향도함수에 대해서 알아보도록 하겠습니다

that's a way to extend the idea of a

이는 편도함수의 개념을

partial derivative and partial

확장시킨 것입니다

derivatives if you'll remember have to

편도함수는

do with functions with some kind of

여러 개의 변수로 이루어진

multi variable input and I'll just use

다변수함수와 관련된 내용입니다

two inputs because that's the easiest to

저는 간단하게 그냥 2개의 변수를 입력받아

think about and it could be some single

하나의 값을 출력하는 함수를

variable output it could also deal with

생각해 보도록 하겠습니다

vector variable outputs we haven't

사실 이는 벡터값을 출력하는 함수에 대해서도 적용할 수 있지만

gotten to that yet so we'll just think

아직 다루지 않은 내용이니까 넘어가겠습니다

about a single variable ordinary real

그러니까 그냥

number output that's you know an

하나의 실수값을 출력하는

expression of x and y and the partial

x와 y로 이루어진 함수의

derivative one of the ways that I said

편도함수를 생각해 보겠습니다

you could think about it is to take a

이것을 보다 시각적으로 받아들이기 위해

look at the input space your x and y

값을 입력받을 공간인

plane so this would be the x axis this

xy평면 위에서 생각해 보겠습니다

is y and you know vaguely in your mind

이것이 x축이고요, 이것이 y축입니다

you're thinking that somehow this

그리고 여러분에 머리 속에서 모호하게나마나 이 함수의 반환값은

outputs to a line this outputs to just

실수직선 위에만 있다는 것을 알고 있습니다

the real numbers and maybe you're

그리고 여러분은 이 입력 공간으로부터

thinking about a transformation that takes it there or maybe you're just

저 출력 직선 사이의 관계를

thinking okay this is the input space

생각해 보고 있을지도 모르죠

that's the output and when you take the

그리고 여러분이 특정 점에서

partial derivative at some kind of point

편미분을 하게 되면

so I'll write it out like partial

derivative of F with respect to X at a

f의 x에 대한 편도함수에

point like 1/2 you think about that

point you know 1 Y is equal to 2 and if

대입하게 되면 여기 x =1, y = 2인 점에서

you're taking with respect to X you

x축 방향으로

think about just nudging it a little bit

in that X direction and you see what the

이동했을 때

resulting nudge is in the output space

출력값이 얼마나 이동해하는지를 확인할 수 있습니다

and the ratio between the size of that

그리고 입력값이 이동한 거리와

resulting nudge and the original one the

출력값이 이동한 거리의 비가

ratio between you know partial F and

f의 x에 대한 편도함수에

partial X is the value that you want and

입력값을 대입한 값입니다

when you did it with respect to Y you

그리고 y에 대해서 편미분하면

know you were thinking about traveling

in a different direction maybe you you

y 방향으로 아주 조금 이동시켰을 때

nudge it straight up and you're

wondering okay how does that influence

출력값에 어떤 영향을 끼치는지를 알 수 있습니다

the output and the question here with

그런데 방향도함수는

directional derivatives what if you have some vector V I'll give a little vector hat on top of it that you know I don't

예를 들어 어떤 벡터 v가 있을 때, 이 v 위에 벡터 표시를 쓰겠습니다

know let's say it's negative 1/2 is the vector so you'd be thinking about that

이 벡터 v가 [-1, 2]라고 합시다

as a step of negative 1 in the X

direction and then 2 more in the Y

x축 방향으로 -1만큼 가고

direction so it's going to be something

y축 방향으로 2만큼 간 방향입니다

that ends up there this is your vector V

즉 이런 모습이겠죠

at least if you're thinking of V is

이것이 벡터 v입니다

stemming from the original point and

물론 기존 점 (1, 2)를

you're wondering what does it nudge in

시점으로 하는 벡터입니다 그런데 만약 이 방향으로 점을 움직였을 때

that direction due to the function

출력값에는 어떤 영향이 있을까요?

itself and remember with these original

물론, 우리가

you know nudges in the X Direction

x축으로나 y축으로나 이동했을 때,

nudges in the Y you're not really

우리는 이것을

thinking of it is you know this is kind

가시적인 이동으로 생각하기 보다는

of a large step you're really thinking

of it is something eating any bitty bit

매우 매우 매우 작은

of a tea you know it's not that but it's

really something very very small and

매우 매우 매우 작은 이동입니다

formally you'd be thinking about limit as this gets really really really

실제로는 0으로의 극한에 대해

small approaching zero and this gets

생각하는 것이죠

really really small approaching zero

물론 출력값의 이동도 매우 작은 값입니다

what does the ratio of the two approach

이 때 두 매우 작은 이동의 비를 따지는 것이죠

and similarly with the Y you're not

똑같이 y의 경우에도

thinking of it is something this is this

가시적인 이동이 아닌

is pretty sizable but it would be something really really small and the

매우 매우 작은 이동에 대해 생각합니다

directional derivative is similar you're

방향도함수도 마찬가지입니다

not thinking of the actual vector

실제로 이 벡터 크기만큼

actually taking a step along that but be

이동하는 것이 아닌

thinking of taking a step along say H

이 벡터 방향으로

multiplied by that vector and H might

매우 작은, 예를 들어 h만큼 이동했을 때를 생각하는 것입니다

represent some really really small

이 때 h는 매우 매우 작은 값입니다

numbers I know maybe this here is like 0.001 and when you're doing this formula

예를 들자면 0.001 같은 값이죠

you just be thinking the limit as H goes

물론 실제 계산에서는 h를 0으로의 극한으로 생각하게 됩니다

to zero so the directional derivative is

즉 방향도함수가 의미하는 것은,

saying when you take a slight nudge in

주어진 벡터 방향으로 매우 조금 이동했을 때,

the direction of that vector what is the resulting change to the output and one

출력값이 어떻게 변화할지를 보는 것입니다

way to think about this is you say well

이를 받아들이는 한가지 방법은,

that's slight nudge of the vector if we

이 벡터 방향으로의 이동은

actually expand things out and we look

전개해 보자면

at the definition itself it'll be

벡터 v의 정의에 따라

negative H negative one times that

-h 만큼 이동한 후

component and then to H here so it's

2h 만큼 이동한 것입니다

kind of like you took negative one nudge

즉 x축 방향으로 음의 방향으로 한번 이동한 뒤

in the X direction and then two nudges

y축 방향으로 2번 이동하는 것이죠

in the Y direction you know so for whatever your whatever your nudge in the

v 방향으로 이동은

V Direction there you take a negative

뒤로 한 걸음

one step by X and then 2 of them up by Y

그리고 위로 두 걸음이죠

so when we actually write this out the

이것을 실제로 써 보자면

notation by the way is you take that

이를 표기하는 방법은

same nabla from the gradient but then

그라디언트에서 쓰는 똑같은 기호 나블라(∇)를 이용하는데

you put the vector down here so this is

하지만 벡터를 밑에 적습니다

the directional derivative in the

즉 이것이 v 방향으로의

direction of V and there's a whole bunch

방향도함수입니다

of other notations too you know I think

물론 다른 표기법도 많죠

there's like derivative of F with

어떤 사람들은

respect to that vector is one way people

f를 v에 대해서

think about it some people will just

편미분하는 이런 표기법도 쓰고요

write like partial with a little

어떤 사람들은

subscript vector there's a whole bunch

파셜 디(∂)에 벡터 기호를 첨자로 쓰기도 하고요

of different notations but this is the

표기법은 엄청 다양합니다

one I like you think that nabla with a

하지만 이 표기법이 제가 제일 좋아하는 표기법입니다

little little F down there with a little

나블라 밑에 달린 작게

V for your vector of F and it's still a

벡터 v가 달려있고

function of x and y and the reason I

x, y로 이루어진 f를 옆에 써주는데,

like this is it's indicative of how you

제가 이 표기법을 좋아하는 이유는

end up calculating it which I'll talk

이 표기법이 방향도함수의 계산법을 내포한다는 점입니다

about at the end of the video and for

이에 대해서는 영상의 끝부분에 말하도록 하겠습니다

this particular example a good guess

다시 이 예로 돌아가자면

that you might have is to say well we

여러분이 생각해낼 만한 생각 중 하나는,

take a negative step in the X direction

x축 방향으로 음의 방향으로 한 번 이동하니까

so you think of it as whatever the

원래 x축 방향으로 이동했을 때

change that's caused by such a step in

일어나는 출력값의 변화에

the X Direction you do the negative of

마이너스를 취하고

that and then it's two steps in the Y

y축 방향으로 2만큼 이동하니까

direction so whatever the change caused by a tiny

원래 y축 방향으로 이동했을 때

step in the Y direction let's just take

일어나는 출력값의 변화에

2배를 해주면 되지 않을까 하고 생각해 볼 수 있겠네요

two times partial F partial Y and this

파셜 디 에프 파셜 디 와이(∂f/∂y) 곱하기 2

is actually this is Iraq is actually how

그리고 실제로 이게 맞습니다

you calculate it and if I was going to

조금 더 일반적으로 쓰자면,

be more general you know let's say we've got a vector W

벡터 w에 대해

I'm going to keep it abstract and just

일반적으로 쓰겠습니다

call it a and B as its components other

w의 구성 성분을 a와 b라고 하겠습니다

than the specific numbers you would say

일반적으로 쓰기 위해

that the directional derivative in the

이 때 벡터 w가 뭐든 간에

direction of W whatever that is of F is

함수 f에 대한 벡터 w의

equal to a times the partial derivative of F with respect to X plus B times the

방향도함수는 f의 x에 대한 편도함수에 a를 곱한 값에 f의 y에 대한 편도함수에 b를 곱한 값을

partial derivative of F with respect to Y and this is it this is the formula

더한 값입니다

that you would use for the directional

이게 여러분들이 방향도함수를 구할 때

derivative and again the way that you're

쓸 공식입니다

thinking about this is you're really

다시 말하지만 이것을 받아들이는 방법은

saying you know you take a little nudge that's a in the X direction and B in the

x축 방향으로 a만큼 이동하고

Y direction so this should kind of make

y축 방향으로 b만큼 이동하니까

sense and sometimes you see this written

일리있는 공식입니다

not with respect to the partial

가끔씩은 이가 편도함수와

derivatives themselves and the actual

a와 b로 표현되기 보다는

components a and B but with the with

그라디언트로 표현되기도 합니다

respect to the gradient and this is

이는 이것이 다른 차원에서

because it makes it much more compact more general if you're dealing with

훨씬 함축적이고 일반적이기 때문이죠

other dimensions so I'll just write it

여기에 한번 적어 볼까요

over here if you look at this expression it looks like a dot product if you take

이 식을 보시면 일종의 내적처럼 보이기도 합니다

the dot product of the vectors a B and

벡터 [a, b]와

the one that has the partial derivatives

첫 번째 값으로는

in it so what's lined up with a is the partial derivative with respect to X partial F partial X and what's lined up

f의 x에 대한 편도함수가

with B is the partial derivative with

두 번째 값으로는 f의 y에 대한 편도함수가 들어가서

respect to Y and you look at this and

벡터 내적을 하게 된 값이죠

you say hey a B I mean that's that's

그리고 이걸 보면

just the original vector right that's W

이 식의 첫번째 벡터 [a, b]는

that's the vector W and then your

벡터 w입니다

dotting this with well partial

그리고 이 벡터 w를

derivative with respect to X and one

∂f/∂x를 첫번째 성분으로

component the other partial derivative

∂f/∂y를 두번째 성분으로 가지는 벡터와

and the other component that's the dirt

내적을 하게 되는데

that's just the gradient that is the

이건 그라디언트죠

gradient of F and here you know it's

f의 그라디언트입니다

nabla without that little that little W

즉 첨자 w가 없는 나블라를 써서

at the bottom and this is why we use

다음과 같이 표기할 수 있습니다

this notation because it's so suggestive

그리고 이 표기법을 쓰면

of the way that you ultimately calculate

어떻게 계산할 지가 명확해집니다

it so this is this is really what you'll

그래서 교과서에서는 이렇게 표기가 되어있고

see in a textbook or C as the compact

방향도함수를 표기하는 함축적인 방법입니다

way of writing it and you can see how this is more flexible for dimensions so

또한 이 표기법은 다양한 차원에 대해서도 적용가능한 유연한 표기법입니다

if we were talking about something that

예를 들어서

has like a five dimensional input

입력값이 5차원인 함수라면

the vector the direction you move has

이동 방향의 벡터도

five different components this is

5개의 성분으로 구성되어 있겠죠

flexible when you expand that the

하지만 이 표기법을 사용하면

gradient would have five components and

전개해 보면, 그라디언트가

the vector itself would have five

5개의 구성 성분을 가질 테고,

components so this is the directional

벡터 역시 5개의 구성 성분을 가질 테니 방향도함수를 계산하는데

derivative and how you calculate it and

아무런 문제가 없습니다

the way you interpret you're thinking of

정리하자면, 방향도함수는

moving moving along that vector by a

주어진 벡터 방향으로

tiny nudge by a tiny you know little

매우 작은 값만큼을

value multiplied by that vector and

이동했을 때

saying how does that change the output

이 이동이 출력값에 어떤 영향을 미치는지를

and what's the ratio of the resulting

입력값의 이동과 출력값의 이동 비율로써 알려줍니다

change and in the next video I'll

다음 동영상에서는

clarify that with the formal definition

방향도함수의 수학적 정의에 대해

of the directional derivative itself

알아보도록 하겠습니다

Similar videos:

Gradient and graphs
Khan Academy
189.1K views 1K likes 17 dislikes 11 May 2016100% match

Partial derivatives, introduction
Khan Academy
836.1K views 6.5K likes 294 dislikes 11 May 2016100% match

Second derivative test | Using derivatives to analyze functions | AP Calculus AB | Khan Academy
Khan Academy
136.7K views 365 likes 20 dislikes 26 Jul 2016100% match

Multiplying & dividing powers (integer exponents) | Mathematics I | High School Math | Khan Academy
Khan Academy
326.6K views 423 likes 39 dislikes 29 Mar 2017100% match

Meet the preposition | The parts of speech | Grammar | Khan Academy
Khan Academy
385.8K views 2K likes 204 dislikes 16 May 2016100% match

Introduction to average rate of change | Functions | Algebra I | Khan Academy
Khan Academy
241.8K views 254 likes 16 dislikes 10 Jun 2015100% match

Units of the rate constant | Kinetics | AP Chemistry | Khan Academy
Khan Academy
242.2K views 448 likes 52 dislikes 16 May 2016100% match

Evaluating composite functions: using tables | Mathematics III | High School Math | Khan Academy
Khan Academy
172.9K views 83 likes 7 dislikes 12 Feb 2016100% match

Calculating internal energy and work example | Chemistry | Khan Academy
Khan Academy
146.2K views 137 likes 23 dislikes 24 May 2016100% match

Introduction to Islam | World History | Khan Academy
Khan Academy
125.3K views 1.1K likes 141 dislikes 03 Mar 2017100% match

Jesus Christ and Christianity | World History | Khan Academy
Khan Academy
148.4K views 944 likes 83 dislikes 09 Jan 2017100% match

Breaking apart 3-digit addition problems | 2nd grade | Khan Academy
Khan Academy
337.9K views 102 likes 45 dislikes 14 Aug 2017100% match

How to evaluate a piecewise function (example) | Functions | Algebra I | Khan Academy
Khan Academy
411.4K views 359 likes 13 dislikes 28 May 2015100% match

Formal definition of partial derivatives
Khan Academy
148.9K views 552 likes 19 dislikes 11 May 2016100% match

Evaluating quotient of fractional exponents | Mathematics I | High School Math | Khan Academy
Khan Academy
131.2K views 98 likes 6 dislikes 11 Feb 2016100% match

Lac operon
Khan Academy
599.9K views 6.1K likes 92 dislikes 23 Jun 2016100% match

Vector components from magnitude and direction
Khan Academy
129.8K views 302 likes 13 dislikes 13 Jan 2016100% match

Operons and gene regulation in bacteria
Khan Academy
186.5K views 877 likes 17 dislikes 20 Jun 2016100% match

Addition using groups of 10 and 100 | 2nd grade | Khan Academy
Khan Academy
330K views 87 likes 17 dislikes 14 Aug 2017100% match

Commas and introductory elements | Punctuation | Grammar | Khan Academy
Khan Academy
139.1K views 277 likes 20 dislikes 06 Jun 2016100% match

Basic derivative rules (Part 1) | Derivative rules | AP Calculus AB | Khan Academy
Khan Academy
175.4K views 44 likes 2 dislikes 20 Jul 2016100% match

Introduction to the coordinate plane
Khan Academy
128.3K views 160 likes 47 dislikes 03 May 2017100% match

Graphing a shifted and stretched absolute value function
Khan Academy
173.6K views 109 likes 5 dislikes 29 Mar 2017100% match

Multiplying binomials intro | Mathematics II | High School Math | Khan Academy
Khan Academy
131K views 34 likes 6 dislikes 22 Sep 2015100% match

Radius & diameter from circumference | High School Geometry | High School Math | Khan Academy
Khan Academy
167.4K views 141 likes 33 dislikes 22 Dec 2015100% match

Biology overview
Khan Academy
569.4K views 875 likes 33 dislikes 14 Mar 201638% match

What is an operational amplifier?
Khan Academy
705.4K views 6.5K likes 190 dislikes 25 Jul 201638% match

The scientific method
Khan Academy
1M views 4.3K likes 375 dislikes 15 Mar 201638% match

Surface area of a box
Khan Academy
464.9K views 547 likes 75 dislikes 04 Sep 201533% match

Overview of cell signaling
Khan Academy
431.1K views 2.4K likes 55 dislikes 05 Nov 201533% match

Settings Privacy About Contact Donate