Formal definition of partial derivatives Khan Academy English Korean

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- [Voiceover] So, I've talked about

지난 시간에는

the partial derivative and how you compute it,

편미분을 어떻게 계산하는지

how you interpret in terms of graphs,

또 그래프에서 어떻게 해석할 수 있는지를 배웠습니다

but what I'd like to do here is give its formal definition.

오늘은 편미분의 형식적 정의를 소개해보겠습니다

So, it's the kind of thing, just to remind you,

복습해보자면 편미분은

that applies to a function that has a multi-variable input.

변수가 두 개 이상인 함수를 다룰 때 사용합니다

So, X, Y, and you know, I'll emphasize that

f(x,y,…) x와 y뿐만아니라 더 많은 변수도

it could actually be a number of other inputs.

가능하다는 것을 …으로 표현했습니다

You could have 100 inputs, or something

100개의 변수도 있을 수 있습니다

like that, and as with a lot of things here,

여기서 일어나는 여러가지를 보기 전에

I think it's helpful to take a look at

먼저 우리가 예전에 변수가 하나밖에 없을 때

the one dimensional analogy and think about

도함수를 어떻게 정의했는지로부터

how we define the derivative, just the ordinary derivative,

오늘 배울 편미분의 정의를

when you have a function that's just one variable.

유추해보는 것이 좋을 것 같습니다

You know, this would be just something simple,

변수가 하나밖에 없는 간단한

F of X, and you know, if you're thinking

f(x)=x^2 같은

in the back of your mind that it's a function like F of X

함수를 떠올려봅시다

equals X squared, and the way to think about

the definition of this is to just actually spell out

이 함수의 도함수를 정의하기 위해서는

how we interpret this D F and D X, and then

어떻게 우리가 df/dx를 이해했는지를 떠올려보면 됩니다

slowly start to tighten it up into a formalization.

그리고 그것을 수학적으로 갖춰 표현해보면 되겠죠

So, you might be thinking of the graph of this function.

이 함수의 그래프를 그려봅시다

You know, maybe it's some kind of curve, and when you

곡선(여기서는 포물선)의 형태가 그려질 것입니다

think of evaluating it at some point, you know,

특정 점에서의 df/dx를 구한다고 생각해보세요

let's say you're evaluating it at a point A,

노란점 a에서 df/dx를 구해보겠습니다

you're imagining D X here as representing a slight nudge,

dx는 x의 증가량(증분)을 의미합니다

just a slight nudge in the input value.

So, this is in the X direction.

x축의 방향으로 dx증가한 것이죠

Got your X coordinate, your output,

지금 우리는 x로 나타낸 f 함수를 보고 있습니다

F of X is what the Y axis represents here, and then

그러므로 세로축이 f(x), 즉 함수값을 의미하죠

you're thinking of D F as being the resulting nudge here,

df는 f의 증가량(변화량)이라고 생각할 수 있습니다

the resulting change to the function.

x의 증가량으로 인해 생기는 함수값의 변화죠

So, when we formalize this, we're gonna be thinking of

그래서 이것을 식으로 표현할 때에는

a fraction that's gonna represent D F over D X, and I'll

df/dx를 대표할 수 있는 분수를 생각할 수 있습니다

leave myself some room.

등호 옆에 약간의 공간을 남겨두겠습니다

You can probably anticipate why if you know where

지금 무엇을 하고 있는지 짐작이 가시나요?

this is going, and instead of saying D X, I'll say H.

dx 대신에 h를 써보겠습니다

So instead of thinking D X is that tiny nudge,

dx 대신에 h를 아주 작은 증가량이라고 생각하는 겁니다

you'll think H, and I'm not sure why H is used necessarily,

왜 h라고 쓰는지는 모르겠지만

but just having some kind of variable that you

단순히 어떤 변수의 증가량이

think of as getting small, maybe all the other

굉장히 작아지는 것을 생각하는 것이므로

letters in the alphabet were taken.

h 대신에 어떤 문자를 사용하든 상관 없을겁니다

Now, when you actually say, what do we mean by

f의 변화량이 의미하는게 무엇일까요?

the resulting change in F, we should be writing,

h만큼 조금 갔을 때

as well, where does it go after you nudge?

f는 얼마만큼 변하나요?

So, when you take, you know, from that input point,

우리가 시작했던 점에서

plus that nudge, plus that little H,

조금(h만큼) 더 간 점의 함수값으로 변할 것입니다 f(a+h)

what's the difference between that and the original

그 값이 원래 함수값과 얼마나 차이가 나는지를 알기위해

function, or the original value of

원래 함수값을 빼줍니다

the function, at that point?

h만큼 가기 전에 있던 원래 함수값 말입니다

So, this top part is really what's representing D F.

이 분수에서 분자는 df를 의미합니다

You know, this is what's representing D F over here,

그래프에서의 y값의 증가량말이죠

but you don't do this for any actual value of H.

그러나 여러분은 h가 특정한 실제 값을 가질 때

You don't do it for any specific nudge.

이런 식을 세우지 않습니다

Largely, the whole point of calculus is that you're

미적분에서는 우리가 세운 이 식을

considering the limit as H goes to zero of this,

h가 0으로 가까이 가는 극한으로 보냅니다 lim h→0 f(a+h)-f(h)/h

and this is what makes concrete the idea of, you know,

이 식이 의미하는 것이 무엇일까요?

a tiny little nudge or a tiny little resulting change.

x값의 작은 증가량과 그에 따른 f값의 증가가

It's not that it's any specific one.

단순히 특정한 경우만 의미하는 것이 아니라는 겁니다

You're taking the limit, and you know, you could

극한(lim)을 취하면서

get into the formal definition of a limit,

but it gives you room for rigor as soon as

you start writing something like this.

Now, over in the multi-variable world, very similar story.

다변수함수일 경우에도 굉장히 비슷합니다

We can pretty much do the same thing, and we're gonna

거의 같은 작업을 할 겁니다

look at our original fraction, and just start

원래 분수 ∂f/∂x를 보고

to formalize what we think of each of

각 변수가 무엇을 의미하는지를

these variables as representing.

식으로 나타내봅시다

That partial X, still it's common to use the letter H,

∂x 도 아까처럼 h로 나타내보겠습니다

just to represent a tiny nudge in

x축방향의 아주 작은 증가량을

the X direction, and now if we think about

h로 나타냅시다

what is that nudge, and here, let me

여기서 저 증가량이 무엇일까요?

draw it out, actually.

그려보겠습니다

The way that I kinda like to draw this out is

제가 이것을 그리는 방법은

you think of your entire input space as,

전체 변수를(여기서는 x, y)

you know, the X Y plane.

xy 평면 위에 얹는 것입니다

If it was more variables, this would be

변수가 더 많다면

a high dimensional space, and you're thinking

더 높은 차원이 되겠죠 (z라는 변수가 더 있으면 xyz 좌표공간)

at some point, you know, maybe you're thinking of it

as A B, or maybe I should specify that,

(a,b)라는 점을 생각해봅시다.

actually, where we're doing this

여러분이 특정 점에서

at a specific point how you define it.

어떻게 편미분을 하는지 알아보는 겁니다

We're doing this at a very specific point,

특정 점인 (a,b)에서의

A B, and when you're thinking of your tiny little

∂f/∂x 값을 구해보죠

change in X, you'd be thinking, you know,

x값의 아주 작은 증가량을 생각해보면

a tiny little nudge in the X direction,

다시말해 x축 방향으로 조금 증가했을 때

a tiny little shift there, and the entire function

함수 f의 모든 정의역을

maps that input space, whatever it is, to

이 수직선에 대응시킵니다

the real number line.

This is your output space, and you're saying,

그러므로 이 수직선이 함수값의 집합(치역)이 됩니다

hey, how does that tiny nudge influence the output?

어떻게 저 조그마한 증분이 함수값에 영향을 끼칠까요?

I've drawn this diagram a lot, this loose sketch.

저는 이 증분와 수직선 그림을 자주 그립니다

I think it's actually a pretty good model,

이 그림이 꽤 좋다고 생각합니다

because once we start thinking of

왜냐하면 우리가 더 높은 차원의

higher dimensional outputs or things like that,

함수값 등을 생각할 때

it's pretty flexible, and you're thinking of this

꽤 유연하게 설명이 가능하거든요

as your partial X, your changing

아까 x축 방향의 증분을 ∂x라고 생각하면

the X direction, and this is that resulting

수직선위의 화살표는 그 증분의 결과인 ∂f입니다

change for the function.

But, we go back up here, and we say,

다시 식으로 돌아가서

well what does that mean, right?

방금 본 기하학적 의미를 되새겨봅시다

If H represents that tiny change to your X value,

h가 ∂x라면

well then you have to evaluate the function at the point A,

a+h에서의 함수값을

but plus that H, and you're adding it

구해야할 것입니다

to the X value, that first component,

x좌표인 a에 h를 더하는 이유는

just because this is the partial derivative

이것이 ∂x, 즉 x값의 증분이기 때문입니다

with respect to X, and the point B

그러므로 y좌표인 b는

just remains unchanged, right?

아무 변화가 없겠죠?

So, this is you evaluating it, kind of, at

그러므로 새로운 점 (a+h,b)에서의

the new point, and you have to say,

함수값을 계산하는 것입니다

what's the difference between that and the old

이 새로운 점 (a+h,b)에서의 함수값과

evaluation, where it was just at A and B.

원래 (a,b)에서의 차이를 알기 위해

둘을 서로 빼주면 끝입니다

That's the formal definition of your partial derivative,

이것이 편미분의 정의입니다

except, oh, the most important part, right?

가장 중요한 부분을 빼먹었네요

The most important part, given that

이것이 미적분이고

this is calculus is that we're not doing this

아무 값을 가지는 h를 생각하는게 아니죠

for any specific value of H, but we're actually,

일단 방금 적은 식을

let me just move this guy.

조금 옮겨봅시다

Give a little bit of room here.

But, we're actually taking the limit

h에 아무 값을 넣는 것이 아니니까

here, limit, as H goes to zero,

여기 lim를 달고 h를 0으로 보냅시다

and what this means is you're not considering any specific

다시말해 ∂x를 0으로 보내므로

size of D X, any specific size of this.

∂x의 크기를 무시하는 것이죠

Really, this is H, considering the notation up here,

위의 식을 고려하면 ∂x는 h니까

but any size for that partial X.

You're imagining that nudge shrinking

이 증분이 점점 작아지는 것을 상상하면

more and more and more, and the resulting change

그 증분으로 인한 함수값의 변화량도

shrinks more and more and more, and you're

점점 작아질 것입니다

wondering what the ratio between them approaches.

그렇게 될 때의 ∂f/∂x 값을 구하고 싶은거니까

So, that would be the partial derivative

그것이 x의 편미분입니다

with respect to X, and just for practice,

이제 연습삼아

let's actually write out what the partial derivative

y의 편미분도

with respect to Y would be.

같이 적어봅시다

So, we'll get rid of some of this

아까 했던 변수가 1개인

one dimensional analogy stuff here.

상미분을 좀 지우도록 할게요

Don't need that anymore, and let's just think about

이제 다른 변수를 기준으로 한

what the partial derivative with respect to

편미분에 대해서 생각해봅시다

a different variable would be.

So, if we were doing it as partial derivative

∂f/∂y를 알아봅시다

of F with respect to Y, now we're nudging slightly

아까와는 반대방향으로

in the other direction, right?

아주 작은 증가량이 생기겠죠?

We're nudging in the Y direction, and you'd be thinking,

y축방향의 증분이 되겠죠

okay, so we're still gonna divide something

그럼 ∂f라는 것을 그 증분(∂y)로 나누는 것이라고

by that nudge, and again I'm just using the same variable.

생각할 수 있지만 저는 아까처럼 h를 사용할 것입니다

Maybe it would be clearer to write something like

아마 여기에서는 y값의 증가량(Δy)로 쓰고

the change in Y, or to go up here and write

아까 했던 x의 편미분에서는

something like, you know, the change

Δx라고 적으면 훨씬 분명할 것 같은데

in X, and people will do that, but it's less common.

가끔 그렇게도 쓸 수 있지만 주로 h로 표기합니다

I think people just kinda want

제 생각에는 사람들이

the standard go-to limiting variable.

극한을 취할 수 있는 대표 변수가 필요했나봐요

But, this time when you're considering

그렇지만 이번에는

what is the resulting change, oh, and again,

아 다시 까먹었네요

I always forget to write in we're evaluating this

우리는 이 ∂f/∂y를

at a specific point, at a specific point A B,

특정한 점 (a,b)에서 계산할 것입니다

and as a result, maybe I'll give myself

그러므로 자리를 조금 더

a little bit more room here.

넓혀보겠습니다

So, we're taking this whole thing,

어쨌든 어떤 것을 h로 나눌 것인데

dividing by H, but what is the resulting change in F?

이 ∂y로 인해서 f는 어떻게 변할까요?

This time you say F, the new value is still gonna be at A,

여기서 새로운 점의 x좌표는 그대로 a일 것입니다

but the change happens for that second variable.

이번에는 y좌표(두번째 변수)가 변하겠죠

It's gonna be that B, B plus H.

b+h가 될 것입니다

So, you're adding that nudge to

h를 이번에는

the Y value, and as before, you subtract off.

y좌표에 더하는 것이죠

You see the difference between that and how you

그럼 아까처럼 원래 점과의 함수값의 차이를

evaluate it at the original point, and again,

알아보기 위해 뺄셈을 해봅시다

the whole reason I move this over and give myself

아까 이 전체 식을 지우고 조금 오른쪽으로 옮긴 이유는

some room is because we're taking the limit,

여기에 lim를 취하기 위해서입니다

as this H goes to zero, and the way that

h가 0으로 가면서

you're thinking about this is very similar.

아까와 유사한 방식으로 이해하면 됩니다

It's just that when you change the input

h를 y좌표에 더함으로써

by adding H to the Y value, you're shifting it upwards.

아까와는 달리 위로 이동하게 됩니다

So, again, this is the partial derivative,

그러므로 이 식이

the formal definition of the partial derivative.

편미분의 정의가 되는 것이죠

Looks very similar to the formal definition of

상미분과 아주 유사해보이지만

the derivative, but I just always think about this

저는 항상 이것에 의미를 조금 부여해봅니다

as spelling out what we mean by

partial Y and partial F, and kinda spelling out

∂y와 ∂f가 무엇을 뜻하는지,

why it is that the Leibniz's came up

그리고 왜 라이프니츠가 이런 형식의

with this notation in the first place.

표기법을 생각해냈는지도 말해보는 거죠

Well, I don't know if Leibniz came up with the partials,

라이프니츠가 ∂ 형태를 생각했었는지는 잘 모르겠지만

but the D F, D X portion, and this is good

df/dx 표기법은 항상 머릿속에

to keep in the back of your mind,

잘 넣어주는 것이 좋아요

especially as we introduce new notions,

특히 오늘은 우리가 새로운 표기법과

new types of multi-variable derivatives,

방향도함수와 같은 새로운 종류의

like the directional derivative.

다변수미분을 해보았기 때문에

I think it helps clarify what's really go on

그것을 머릿속에 넣어두면 지금 무엇을 하고 있는지를

in certain contexts.

확실하게 하는 것에 도움을 줄 것 같습니다

See ya next video.

다음 영상에서 뵙겠습니다

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