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f ′(0, 0) = lim

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, ( * ( . . , " u = x2 + y2

(0,0) , ( (. (, "

lim	f (0 + x, 0)	− f (0, 0)	= lim	(0 + x)2	+ 02	− 0	= lim	\| x \|

	x			x				x
x→ 0	x		x→ 0	x			x→ 0	x
x→ 0			x→ 0				x→ 0

,, , f x′(0, 0) . *

,, " , ( ' (0,0) .

2 " + ,

( * ( ( ( ,

). $", , '

	2xy		, x2	+ y2	> 0,


f ( x, y) = x2		+ y2
		0, x = y = 0.

% +, ' * f x′(0, 0), f y′(0, 0) . (

f ′(0, 0) = lim		f (0,		y) − f (0, 0)				= lim	0	= 0 .

y	y→ 0			x				x→ 0	y

$ " " , * ' (0,0) ( . . 6,
2,4), + , ( ( '.
. * ! " u = f ( x1 , x2 ,..., xn ) M ( x1 , x2 ,..., xn )
			∂u	( j =			), M . +
						1, n
			∂x j

" u = f ( x1 , x2 ,..., xn )					" ! M .
. " " ! + * 2-

u = f ( x, y) (* ( " , * " ). $" (

u = f ( x + x, y + y) − f ( x, y) . &

f ( x, y + y) , :

u = [ f ( x + x, y + y) − f ( x, y + y)] + [ f ( x, y + y) − f ( x, y)] .

% 6 + " *				f ( x, y ) ' x
	y + y		.	2	' 7 +
:
f (x + x, y + y ) − f ( x, y + y ) = f x′ (ξ , y + y )					x ,
ξ * " + x x +		x .
:
f ( x, y + y ) − f ( x, y ) = f y′ ( x,η )			y ,
η * " +	y y +	y .
		22

" ,

y ". & f x′ , f y′

lim

f x′ (ξ , y +

y ) =

f x′ ( x, y ),

lim

f y′ ( x,η ) = f y′ ( x, y ) .

x→0

y →0

) +:

f x′ (ξ , y + y ) = f x′ ( x, y ) + α ,

f y′ ( x,η ) = f y′ ( x, y ) + β ,

α , β –

x → 0 ,

y → 0 .

z = ( f x′ ( x, y )

x + f y′ ( x, y )

y ) + α x + β y ,

! "

z = f ( x, y )

( ( x, y ) , ( ! !

9. .

$" ' u = f ( x1 , x2 ,..., xn ) , (

x = ( x1 , x2 ,..., xn ) . # ( , ":

∂u

u = ∑

x j + ∑α j

x j ,

∂x j

j =1

α j ( j =

)

" ' * "

x1 ,..., xn . # + :

1, n

∑α j

x j = o (

x12 + ... +

xn2 ).

j =1

5 |

|< 1, | x

|k < |

( j =

; k > 1). ) +

1, n

x ,...,

∂u

∑α j

x j

" ∑

x j , (

∂x

j 1

∂u

( ', * " ∑

x j – (

∂x

j 1

x1 ,..., xn . # ' * " ' '

! .

. " ( x = ( x1 , x2 ,..., xn )

u = f ( x1 , x2 ,..., xn )

, * "

∂u

du = df ( x) = ∑

x j .

∂x

j 1

% ( u = x1 . #

∂	0,	j ≠ 1,
u	=
∂x j	=	j = 1,
∂x j	1,	j = 1,

, dx1 = x1 . dx2 = x2 ,..., dxn = xn , !

+ ' * . # ,:

du = df ( x) = ∑n ∂u dx j .

j =1 ∂x j

# , (! *6 6 '. 2, " 2- u = f ( x, y) ,:

du = ∂u dx + ∂u dy .

∂x ∂y

" 3- u = f ( x, y, z) ,:

du = ∂u dx + ∂u dy + ∂u dz .

∂x ∂y ∂z

1. 2 ( (

z = ln (y + x2 + y2 ).

/ ,:

f x′ =

y + x2 + y2

x2 + y 2

y x2 + y 2 + x2 + y2

x2 + y2

+ y

f y′ =

1 +

(y +

)

y +

+ y

x2 + y2

# :

dz =

x dx

y x2 + y2 + x2 + y2

x2 + y2

2 ( ( u = ( xy )z .

/ ,:

u′

= z ( xy )z −1

y = x z −1

y z z ,

u′

= z ( xy )z −1

x = x z y z −1 z , u′

= ( xy )z ln ( xy ),

du = x z −1 y z z dx + x z y z −1 z dy + ( xy )z ln ( xy )dz .

u = ln

r =

+ x

+ ... + x2 .

# " , * " . / ,:

∂u

= −

∂r

= −

∂xi

r ∂xi

+ x2

+ ... + x2

2xi = −

x12

+ x22

+ ... + xn2

2 x2

+ x2

+ ... + x2

# :

dxi

du = − ∑

= −

∑ xi dxi .

+ x2

+ ... + x2

r 2

i =1 x2

i =1

10. ! " .

5 ,, ( x = ( x1 , x2 ,..., xn )

u = f ( x1 , x2 ,..., xn ) , ":

n	∂u
u = ∑	∂u		x j + o ( x12 + ... + xn2 ).
u = ∑			x j + o ( x12 + ... + xn2 ).
=	∂x	j
j 1		j

5 o ( x12 + ... + xn2 ), , ! +

∑n ∂u

u ≈ x j .

j =1 ∂x j

		n
f (x1 + x1,..., xn	+ xn ) ≈ f (x1	,..., xn ) + ∑ fx′ ( x1 ,..., xn ) x j .	(10.1)
		j

j=1

% , ! ! + ! "

u= f ( x1, x2 ,..., xn ) x1 = ( x11,..., x1n ) . 5 , * " ( 6

x0 = ( x01 ,..., x0 n ) , " * ! * x1 , " ( "

u= f ( x1, x2 ,..., xn ) 1- " ,

(10.1) ,:

x1 = x01,..., xn = x0 n , x1 + x1 = x11,..., xn + xn = x1n .

# , ! + :

		n
f (x11	,..., x1n ) ≈ f ( x01	,..., x0 n ) + ∑ fx′ ( x01,..., x0 n )( x1 j − x0 j ) .	(10.2)
		j
		j =1
2, " 2- u = f ( x, y) ,:
f ( x1, y1 ) ≈ f ( x0 , y0 ) + f x′( x0 , y0 )( x1 − x0 ) + f y′( x0 , y0 )( y1 − y0 ) .			(10.3)

1. )! ! + (4, 05)2 + (3, 07)2 .

; " " " u = x2 + y2

( x	= 4, 05; y	= 3, 07)	. %: x	= 4, y	0	= 3	. # f ( x , y		) = 42 + 32	= 5 .
1	1		0		0		0	0

2 (:

f ′ ( x , y

) =

= 0, 8 ,

x2 + y2

x =4

y =3

f ′ (x , y

) =

= 0, 6 .

x2 + y2

x =4

y =3

% "' ( " (10.3), ,:

(4, 05)2 + (3, 07)2 ≈ 5 + 0,8 0, 05 + 0, 6 0, 07 = 5, 08 .

# * " ": 5,082.

2. 8 , 6 :

R = 2, 5 , H = 4 l = 1 . 2 ( ! + !’,

, " .

$" ' V = F ( x, y ) = π x2 y . )!’, + , * "

V= V1 −V0 ,

V1 = F (R + l, H + l ) , V0 = F (R, H ). 2 ' (10.3) ,:

V ≈ ∂F ( R, H ) l + ∂F (R, H ) l ,

	∂R	∂H
∂F ( x, y )	= 2π xy ,	∂F ( x, y )	= π x2 .

∂x		∂y

V ≈ (2π RH + π R2 )l = (2π 2, 5 4 + π 6, 25) 0,1 = 2, 625π ≈ 8, 2 ( 3 ).

11. ) *.

$" ' n u = f ( x) = f ( x1 , x2 ,..., xn ) . %,

x1 , x2 ,..., xn ' , " t :

x1 = x1 (t ), x2 = x2 (t ),..., xn = xn (t ) .

# " u ! ' ,' t : u = f ( x(t )) = f ( x1 (t ), x2 (t ),..., xn (t )) .

. " & x1 (t ), x2 (t ),..., xn (t ) " ! t0 ,

" u = f ( x) = f ( x1 , x2 ,..., xn ) " ! ! x0 =

=( x1 (t0 ), x2 (t0 ),..., xn (t0 )) . + " F (t) = f ( x(t)) =

=f ( x1 (t ), x2 (t ),..., xn (t )) " ! t0 , ":

F′(t

) =

f ′ (x0 )x′(t

) + f ′ ( x0 ) x′

) + ... + f

′ ( x0 )x′

) .

. . ( t

t ,

x1 , x2 ,..., xn

' *

x1 , x2 ,...,

xn . ' "

u . ) * u ( x0 ,

(11.1)

u =

∂f ( x0 )

x +

∂f (x0 )

+ ... +

∂f ( x0 )

+ ε

x + ε

+ ... + ε

∂x1

∂x2

∂xn

ε1 , ε2 ,..., εn

–

x1 → 0,

x2 → 0,...,

xn → 0 . %

! ,

t :

∂f (x0 )

∂f ( x0 )

∂f (x0 )

+ ... +

+ ε

+ ... + ε

∂x1

∂x2

∂xn

) * x1 (t ), x2 (t ),..., xn (t )

( t0 ,

lim

= x′(t

), lim

= x′

),..., lim

= x′

) .

t →0

2 ( ( x1 (t ), x2 (t ),..., xn (t )

* ( , +

lim

x1 = 0, lim

x2 = 0,..., lim

xn = 0 ,

t →0

ε1 → 0, ε2 → 0,..., εn → 0 .

# ,

du(t

)

= lim

∂f ( x0 )

dx (t

)

∂f (x

0 )

dx (t

)

+ ... +

∂f ( x0 )

dx (t

)

∂x1

∂x2

∂xn

→0

# .

" 2- u = f ( x, y) , x = x(t),

y = y(t) , ,:

∂f

(11.2)

∂x dt

∂y dt

2, " t = x ,

y = y(t) = y( x) ,

∂f

(11.3)

∂x

∂y dx

6 ', * " ! *6 * ( . .( ,

x1 = x1 (t1 ,..., tm ) ,…, xn = xn (t1 ,..., tm ) (

t 0 = (t

,...,t

)

m . .( x0 = ( x

,..., x

) , x

= x

(t 0 ) =

0 n

0 j

= x

( j =

) – n .

,...,t

0 m

)

1, n

. " & x1 = x1 (t ), x2 = x2 (t ),..., xn = xn (t ) , t = (t1 ,...,tm ) ,

" ! t 0 , " u = f ( x) = f ( x1 , x2 ,..., xn )

" ! ! x0 , " F (t) = f ( x(t)) =

= f ( x1 (t1 ,..., tm ),..., xn = xn (t1 ,..., tm )) " ! t 0 ,

∂F (t

0 )

∂f (x

0 )

∂x (t

)

∂f (x0 )

∂x (t

)

... +

∂f (x0 )

∂x (t

)

2 0

n 0

∂t1

∂x1

∂t1

∂x2

∂t1

∂xn

∂t1

…

∂F (t

0 )

∂f (x

0 )

∂x (t

)

∂f (x0 )

∂x (t

)

... +

∂f (x0 )

∂x (t

)

2 0

n 0

∂tm

∂x1

∂tm

∂x2

∂tm

∂xn

∂tm

" 2- u = f ( x, y) , x = x(t, s), y = y(t, s) , ,:

∂u

∂x

∂u

∂y

∂u

∂x

∂u

∂y

(11.4)

∂t

∂x ∂t

∂y ∂s

∂y ∂t

∂s ∂x ∂s

1. 2 (

, " z = e2 x−3 y , x = tg t , y = t 2 − t .

2 ' (11.2) ,:

2 x−3 y

− 3e

2 x−3 y

(2t −1)

= e

2 tg t −3(t 2 −t )

−

3(2t −1)

cos2 t

2. 2 ( dy , " y = (ln x)sin x . dx

«* " , »

( 6 " "

' ". # , * ' "

( * . %: u = ln x , v = sin x . # y = uv . 2 ' (11.2) ,:

∂y du

∂y dv

= vu

v−1 1

+ u

ln u cos x =

∂u dx ∂v dx

(sin x)(ln x)sin x−1

+ (ln x)sin x ln ln x cos x .

3. 2 ( du , " u = x2 y − xy2 , x = t sin s, y = s cos t .

/ ,: u = F (t, s) , du = ∂u dt + ∂u ds . 2 ' (11.4):

							∂t	∂s
∂u	=	∂u	∂x	+	∂u	∂y	= (2xy − y2 ) sin s + (x2 − 2xy)(−s sin t) ,
∂t	=	∂x ∂t		+			= (2xy − y2 ) sin s + (x2 − 2xy)(−s sin t) ,
∂t		∂x ∂t			∂y ∂t
∂u	=	∂u	∂x	+	∂u	∂y	= (2xy − y2 )t cos s + ( x2 − 2xy) cos t .
∂s	=	∂x ∂s		+			= (2xy − y2 )t cos s + ( x2 − 2xy) cos t .
∂s		∂x ∂s			∂y ∂s

) +:

du = ((2xy − y2 ) sin s − ( x2 − 2 xy)s sin t )dt + ((2xy − y2 )t cos s + ( x2 − 2xy) cos t )ds .

12. $ *.

5 ,, " " " ( 6 ' * "

" " + ! y = f ( x) ,

" 6 " ! *6 *, " ", "

’" , x y :

	F ( x, y) = 0 .				(12.1)
., " " :
	x2	+	y2	= 1.
	a2		b2

2 " " * * " ! *

. ’" , ": "

" " (12.1) y " ' x ? 0 +

y′ , ? 2 +, ’" " "

(12.1) y (+, " x ) , * " +. 5, , " " +:

	b
y = ±	b	a2 − x2 ,
y = ±	a	a2 − x2 ,
	a

, , " ": y − x − εsin y = 0 (0 < ε < 1)

+ .

. 9 " y = f ( x) , * " , " + !

’" y " " " (12.1). &, ! * " " " " (12.1) , y " "

' x . . , " " "

x2 + y2 + 1 = 0

+ .

', , " ', , " " " (12.1)

, y " " ' x , " ! "

, .

( " " ). * !

1) " F ( x, y) ( x0 , y0 )

Fx′( x, y), Fy′( x, y) ;

2)F ( x0 , y0 ) = 0 ;

3)Fy′( x0 , y0 ) ≠ 0 .

+ K = {( x, y) : | x − x0 | ≤ a, | y − y0 | ≤ b} ,

(12.1) y " & x . , y = f ( x)

" ! ( x0 − a, x0 + a) ,

f ′( x) = −	Fx′( x, y)	.	(12.2)
	Fy′( x, y)
		y = f ( x )

" , . 0 , * "

" * ":

F1 ( x1 ,..., xn , y1 ,..., ym ) = 0 ,

…

Fm ( x1 ,..., xn , y1 ,..., ym ) = 0 .

. , " "

y − εsin y = x (0 < ε < 1)

(12.3)

, y " " ' x ( .

, * ' " " .

F ( x, y) = y − εsin y − x . ), F (0, 0) = 0 . 2 (:

Fy′(0, 0) = (1 − ε cos y) y=0 = 1 − ε ≠ 0 .

# " ( x0 , y0 ) + ! (0,0). #

' , " K0 = {( x, y) : | x | ≤ a, | y | ≤ b} , "

" " (12.3) , y " " ' x . 2 ( ,

. / ,: Fx′ = −1, Fy′ = 1 − ε cos y . 2 ' (12.2):

y′	= −	Fx′	=	1	.

x		Fy′		1 − ε cos y

13. .

$" ' 3- u = f ( x, y, z) , " " (

+ G 3 . .( M ( x0 , y0 , z0 ) – 6 " + G . & M " ( " l , " ( , * " "

cos α, cosβ, cos γ ( cos2 α + cos2 β + cos2 γ = 1). ) * M –

6 " + G , M " l !’" ( 6 M ′( x0 + x, y0 + y, z0 + z) , " + ,

6 * ' ' + G . % M M ′

" u = f ( x, y, z) , :

l u = f ( x0 + x, y0 + y, z0 + z) − f ( x0 , y0 , z0 ) .

. (	l = \| MM ′ \| . #
x =	l cos α, y = l cos β, z = l cos γ .
	30

) +:

l u = f ( x0 + l cos α, y0 + l cos β, z0 + l cos γ) − f ( x0 , y0 , z0 ) .

. " & u = f ( x, y, z) M ( x0 , y0 , z0 )

l , * " " 6 " ,

" ", , " , ":

∂u	= lim	l u
			.
∂l
	l → 0	l

, * " " 2-

u = f ( x, y) . & " " + " *

u = f ( x1 , x2 ,..., xn ) .

∂u

+ " " u = f ( x, y, z)

∂x ∂y ∂z

" ( Ox,Oy,Oz .

∂u

+ , 6 * u = f ( x, y, z) +

∂l

" l . 5 + "

∂u

> 0 , "

∂l

" u = f ( x, y, z) ,, "

∂u

< 0 , " u = f ( x, y, z)

∂l

. " u = f ( x, y, z)

" !

M ( x0 , y0 , z0 ) , l

! # !

∂u

cos α +

∂u

cosβ +

∂u

cos γ .

(13.1)

∂l

∂x

∂y

∂z

. ) * " u = f ( x, y, z) (

M ( x0 , y0 , z0 ) ,

l u = f ( x0 +

l cos α, y0 +

l cos β, z0 +

l cos γ) − f ( x0 , y0 , z0 ) =

= f x′( x0 , y0 , z0 )

l cos α + f y′( x0 , y0 , z0 ) l cosβ + f z′( x0 , y0 , z0 ) l cos γ +

+ε1 l cos α + ε2

l cos β + ε3

l cos γ ,

ε1 → 0, ε2 → 0, ε3 → 0

l → 0 . 2:

l u	= fx′( x0	, y0 , z0 ) cos α + f y′( x0	, y0 , z0 ) cos β + f z′( x0 , y0 , z0 ) cos γ +
l

+ε1 cos α + ε2 cos β + ε3 cos γ .

% " l → 0 , , (13.1).

1. 2 ( z = x2 − xy + y2 M (1;1) "

l = 6i + 8 j .

2 ( " M :

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