Directional derivative, formal definition Khan Academy English Korean

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- [Voiceover] So I have written here the formal definition

여기에 다변수 함수 f의 x에 대한 편미분계수의

for the partial derivative of a two-variable function with respect to X, and what

수학적 정의를 적어 놓았습니다

I wanna do is build up to the formal definition

저는 이를 토대로 하여

of the directional derivative of that same function

벡터 V 방향으로의

in the direction of some vector V, and you know,

방향도함수의 수학적 정의를 이끌어 내고 싶습니다

V with the little thing on top,

V 위에 벡터 모자가 달렸으니

this will be some vector in the input space, and I have

V는 입력 공간 위의 어떤 벡터가 됩니다

another video on the formal definition

편도함수의 수학적 정의에 대한 내용은

of the partial derivative if you want to

다른 영상에 다루어져 있으니

check that out, and just to really quickly go through here,

한번 확인해 보시길 바랍니다만, 빠르게 훑고 넘어가기 위해

I've drawn this diagram before,

그림을 하나 그려 보겠습니다

but it's worth drawing again, if you

저번에 그린 그림이랑 똑같은 그림입니다

think of your input space, which is the X Y plane,

여기에 입력 공간인 xy 평면이 있습니다

and you think of it somehow mapping over to

그리고 함수 f는 이 xy 평면으로부터 값을 입력받아

the real number line, which is where

실수직선 위의 실수를 출력하게 됩니다

your output F lives, and when you're taking

즉 xy 평면과 실수직선이 어떻게든 연결되어 있는 것입니다

the partial derivative at a point A B, you're looking

그리고 여러분이 점 (a, b)에서의 편미분계수를 생각해보면

over here and you say, maybe that's your point, some point A B, and you imagine nudging it

점 (a, b)가 여기에 있다고 할게요

slightly in the X direction, and saying,

이 점 (a, b)로부터 x축 방향으로 살짝 이동했을 때

hey, how does that influence the function?

이 이동이 함숫값에 어떤 영향을 주게 될까요?

So, maybe this is where A B lands, and maybe

f(a, b)가 이 점이라고 하면

the result is a nudge that's a little bit negative.

이동 후에는 이 값이 조금 음의 방향으로 이동했을 지도 모르죠

That would be a negative partial derivative, and you

그러면 음의 편미분계수가 나오게 되겠죠

think of the size of that nudge as partial X, and the size

이 때 입력값이 이동한 거리가 ∂x 이고,

of the resulting nudge in the output space as partial F.

이에 따라 함숫값이 이동한 거리가 ∂f 입니다

So, the way that you read this formal definition

그래서 여러분이 이 편도함수의 정의를 받아들이는 방법은

is you think of this variable H, you know, people,

이 변수 h를

you could say delta X, but H seems to be

h라 하지 않고 Δx라고 할 수도 있습니다만 그래도

the common variable people use, you think of it

h가 가장 흔히 쓰이는 것 같더라고요

as that change in your input space,

이 h를 입력값의 x축으로의

that slight nudge, and you look at how that influences

매우 작은 이동으로 생각하고,

the function when you only change the X component here,

이 이동으로 인해 f의 함숫값에

you know, you're only changing the X component

어떤 일이 일어나는지를

with that nudge, and you say what's the change in F?

f의 함숫값이 얼마만큼 이동하는지를

What's that partial F?

∂f의 값이 얼마인지를 구해내는 것이죠

So, I'm gonna write this in a slightly different way,

그러면 이 정의를

using vector notation.

벡터를 이용해 다르게 표현해 보겠습니다

Instead I'm gonna say, you know, partial F,

∂f/∂x라고 표기하는 대신

partial X, and instead of saying the input is A B, I'm gonna say it's a, you know,

입력값이 (a, b)라고 하는 대신

just A, and then make it clear that that's

그냥 a 벡터라고 하겠습니다

a vector, and this will be a two-dimensional vector,

이차원 벡터입니다

so I'll put that little arrow on top

벡터임을 표현해주기 위해

to indicate that it's a vector, and if we rewrite

작은 화살표를 달아주고요

this definition, we'd be thinking

이를 토대로 위의 정의를 다시 써보자면

the limit, as H goes to zero,

리미트 h가 0으로 갈 때

of

something divided by H, but that thing,

h 분의 무언가가 될 것입니다만

now that we're writing in terms of

그 무언가를

vector notation, is gonna be F of,

벡터로 표기해 줘야 합니다

so it's gonna be our original starting point A,

그런데 f에 우리의 시작점이 될 a에

무엇을 더해줘야 할까요?

I mean, up here, it was clear we could just add it

제 말은 여기 위에서는 벡터가 아니었기에

to the first component, but if I'm not writing in terms

그냥 첫번째 구성 성분에 h를 더해줄 수 있었지만

of components, and I have to think in terms of

이제는 각각의 구성 성분으로 분리해서 생각하지 않고

vector addition, really what I'm adding

벡터의 덧셈으로 생각해 줘야 하니 제가 더해야 할 것은

is that H times the vector, the unit vector

h 곱하기 x축 방향으로의

in the X direction, and it's common

단위 벡터입니다

to use, you know, this little I with a hat

그리고 x축 방향의 단위 벡터는

to represent the unit vector in the X direction.

i위에 작은 모자가 달린 것으로 표기하는 것이 일반적입니다

So when I'm adding these, it's really the same.

그래서 보면

You know, this H is only gonna go to

x축 성분은 1만 곱해져서 그대로고

that first component, and the second component is multiplied

y축 성분은 0이랑 곱해져서 사라지니까

by zero, and what we subtract off

원래 정의와 정확히 똑같은 의미입니다

is the value of the function at

그리고 빼주는 값은

that original input, that original

마찬가지로 원래의 함숫값인

two-dimensional input that I'm just thinking of

f에 a 벡터를 빼주면 되겠죠

as a vector here, and when I write it like this,

그리고 이렇게 쓰면 x, y축으로만의 이동이 아닌

it's actually much clearer how we might extend

다양한 방향으로의 이동으로

this idea to moving in different directions.

확장하기가 더 쉬워집니다

'Cause now, all of the information

왜냐하면 움직일 방향에 대한

about what direction you're moving

모든 정보가

is captured with this vector here,

이 벡터에 담겨있기 때문이죠

what you multiply your nudge by as you're adding the input.

기존 입력값에 아주 조금만큼을 벡터 방향으로 이동합니다

So let's just rewrite that over here

그래서 여기에 방향도함수의 개념을

in the context of directional derivative.

다시 한번 써보도록 하죠

What you would say is that the directional derivative

함수 f의, 어떤 점에서의

in the direction of some vector, any vector,

어떤 벡터 방향 v로의 방향미분계수에

of F, evaluated at a point, and we'll think about

대해 생각해 봅시다

that input point as being a vector itself, A.

함수에 입력할 어떤 점을 벡터 a로 표기할게요

Here, I'll get rid of this guy.

이것 좀 지우고요

It's also gonna be a limit, and as always,

항상 그래왔듯이 극한을 써야겠죠

with these things, we think of some,

리미트를 쓰고

not, I mean, always, but with derivatives,

미분을 할 때는

you think of some variable as going to zero, and then

어떤 변수가 0에 가까워지는 상태에 대해 생각합니다

that's gonna be on the denominator, and the change

이 변수가 분모에 들어가고요

in the function that we're looking for

함숫값의 변화가 분자에 들어가는데

is gonna be F, evaluated at that initial input vector

함수 f에 원래 벡터에다가

plus H, that scaling value, that little nudge of a value,

아주 작은 값은 h에

multiplied by the vector whose direction

벡터 v를 곱한 만큼을

we care about, and then you subtract off

더해준 값의 함숫값에서

the value of F at that original input.

원래 함숫값을 빼주게 됩니다

So, this right here is the formal definition

그래서 이것이 바로 벡터 a에서의

for the directional derivative, and you see how

방향미분계수의 수학적 정의입니다

it's much easier to write in vector notation,

그리고 벡터를 이용해 나타내니

because you're thinking of your input

입력값과 이동 방향을 모두 벡터로 생각할 수 있게 되어

as a vector and your output as just some nudge by something.

훨씬 더 수월하게 이해할 수 있습니다

So, let's take a look at what that

그럼 이것이 좌표평면 위에서

would feel like over here.

어떻게 보일지 한번 보도록 합시다

You know, instead of thinking of D X and a nudge

순수하게 x축 방향으로만 이동하는

purely in the X direction, and I'll erase these guys,

∂x에 대해서 생각하는 대신

you would think of this point as being A,

이 점을 벡터 A로써

as being a vector valued A, so

생각하고 나서

just to make clear how it's a vector,

벡터임을 확실하게 표현해주기 위해

you'd be thinking of it starting at

원점을 시점으로 하는

the origin, and the tip represents

화살표를 그려주겠습니다

that point, and then H times V, you know,

그리고 순수 x축 또는 y축 방향이 아닌

maybe V is some vector, often,

다른 방향으로 뻗어 있는 벡터 v에

you know, a direction that's neither

매우 작은 값인 h를

purely X nor purely Y, but when you

곱해주게 되면

scale it down, it'll just be a tiny little nudge

방향은 같지만 매우 작은

that's gonna be H, that tiny little value,

벡터가 됩니다

scaling your vector V, so that tiny little nudge, and what

그리고 이 매우 작은 벡터를 따라

you wonder is, hey, what's the resulting nudge

입력값을 아주 조금 이동하게 되면

함숫값은 어떻게 변하게 될까요?

And the ratio between the size of that

여기서 입력값의 변화량과

resulting nudge to the output and the original guy there

함숫값의 변화량의 비가

is your directional derivative, and more importantly,

바로 방향미분계수의 값입니다

as you take the limit for that original nudge

더욱더 중요하게는, 벡터 hv를

getting really really small, that's gonna be

0으로의 극한으로 생각해 주게 되면

your directional derivative, and you can probably anticipate

방향미분계수를 구할 수 있게 되는데, 아마 여러분은

there's a way to interpret this as the slope of a graph.

이를 그래프에서의 기울기로 해석할 수 있음을 짐작할 수 있을 것입니다

That's what I'm gonna talk about next video,

이에 대해서는 다음 영상에서 짚도록 하겠습니다

but you actually have to be a little bit careful,

하지만 한 가지 주의할 점은,

because we call this the directional derivative,

우리는 방향미분계수를 구하고 있습니다만,

but notice, if you scale the value V by two,

만약 이 벡터 v를 2배 하면,

you know, if you go over here and you start plugging in

여기로 다시 돌아가서 v가 아닌

two times V and seeing how that influences things,

2v를 대입해 보면

it'll be twice the change, because here,

방향미분계수의 값도 2배가 됩니다

even if you're scaling by the same value H,

왜냐하면 분자의 입력값의 차이가 2hv만큼 남에 따라

it's gonna double the initial nudge that you had, and it's

분모도 이를 맞춰주기 위해 2h가 되어야 하므로

gonna double the resulting nudge out here,

방향미분계수의 값이

even though the denominator H doesn't stay changed.

2배가 됩니다

So when you're taking the ratio, what you're considering

그래서 실제로는 이동 벡터의 방향 뿐만 아니라

is the size of your initial nudge

크기도 방향미분계수의 값에

actually might be influenced.

영향을 끼치게 됩니다

So, some authors, they'll actually change

그래서 어떤 사람들은

this definition, and they'll throw a little

원래 정의에다가 분모에 벡터 v의 크기를 추가함으로써

absolute value of the original vector, just to make sure

벡터 v의 방향에만 상관있고 크기에는 상관없이

that when you scale it by something else, it doesn't

방향미분계수의 값이 나오도록

influence things, and you only care about the direction.

정의를 바꾸는 사람들도 있습니다만

But, I actually don't like that.

저는 그다지 선호하지 않습니다

I think there's some usefulness in the definition

왜냐하면 원래 정의에도

as it is right here, and that there's kind of

나름의 유용성이 있기 때문인데

a good interpretation to be had,

직관적으로 받아들이자면

for when, if you double the size of your vector,

이동 방향의 벡터의 크기가 2배가 되면

why that should double the size of your derivative,

마찬가지로 변화량도 2배가 되어야겠죠

but I'll get to that in following videos.

하지만 이에 대해서는 나중에 더 자세히 다루도록 하고요

This right here is the formal definition

아무튼 이것이 방향미분계수의 수학적 정의입니다

to be thinking about, and I'll see you next video.

다음 동영상에서 봐요 :)

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