Диференціальне числення ФОЗ

y = A( x ) x + α ( x, x ) x ,

A( x ) x + *, α + * " , "

x , ! lim α ( x, x ) = 0 .

x→ 0

6 ,,

". (, " 3 * (

x :

" ( ! * " ( ".

6’" , * ", . % " " ( x

’" " ( , ', * " '

'.

!. , % " y = f ( x ) "

x , " , % " f ′( x ) .

# " , " , ',

( . «& ») + :

y = f ′( x ) + α (x, x ) , x

α " , " x . 6 , ,: y = f ′(x ) x + α (x, x ) x ,

( ! ! . # A( x) = f ′( x ) . # ! *. .(:

y = A( x ) x + α ( x, x ) x ,

α →0 x →0 . % ! , x . / ,:

y = A(x ) + α ( x, x) . x

% ( x →0 :

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( * f ′(x ) . # ! !, -

( , A(x ) ', f ′(x ) .

# ( * , +

' ( . # * , ! «( », + * «' - '». , " ! «* " ( !-

* », " (, " + * * + -

, ( * " (

. # ( .

% " " ( ’" + " "

( .

!. & % y = f (x ) x , -

,, lim y = 0 , ( , * x .

x→0

". )! + " , , !

x , ( * ( -

. $" .

# 1. $" ' y =| x | ( . 7).

$ . 7.

5" " + ( ", -

x = 0 . # 3, " " , ( ' ( . (-, % 1 .3 ! , " x = 0 , -

, + , ( ( '. 6 - ,, x = 0 , .

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# 2. $" ' y = 3 x . 5" " +

( ( " (, x = 0 ( . 5). % 2 . 3 ! , x = 0 " " ,. ) +

, ( ' x = 0 .

8 " ( , ( ,

* ; , ' ,'. # (-

' ' * '. 8 ( * " -

", + * ,

( . 8).

$ . 8.

. * . & (, 3 (1815 – 1897) ! !-

, " , ( ' -

. 5 ! * + ": * ", " ( ,

. " " " , " +, + ( . $" ( + (

. . + ( ( 3 * ! ! ,

( ( . 9):

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$ . 9.

+ , 3 3

+ ( 3 * ! ! , ( . ) , ( . 10):

$ . 10.

. + ! +

' .

. . 10 ! + , " , * " " 5 :

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$ . 11.

0 ' * ’' , " "' * ! " ! * " , " , + ( 0.

5 ( + + . , * "

", " ' * ' ’" * (1870–1924), " 1904 3 ! . 0 " " , . 2 " * " .

+ " * " +, ! *

3. ) * + ( ,. & -

" * " ! * " *1.

0 ' * ! *3 , " -

, , ( ( '. $" ':

), lim y = 0 , * x ≠ 0 " y , ! -

x = 0 . % +, " " , * !

x = 0 . $":

1 ., .: 7 ;. 7 <. – /., /, 1991. – 254 .

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5 , , " , , ( . «& »), + , -

x = 0 . , ( . # "

y, ( ' x = 0 .

, x = 0 * + ' -

. . . 12 , , ! + ! (

' Maple V.

$ . 12.

5. # .

!. & % y = u ( x), y = v ( x) x ,

y = u ( x) + v (x ), y = u (x ) − v (x ), y = u (x )v (x )

x , :

(u + v)′ = u′ + v′ ,

(u − v )′ = u′ − v′ ,

(uv )′ = u′v + uv′ .

u ( x)

& % v ( x ) ≠ 0 , y = v ( x )

x , :

. " ( ,:

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x→0

x ( ( .

6 , , ( ( *: " v ( x ) =

= c = const ,

(uv )′ = (cu )′ = u′c + uc′ = cu′ ,

! $ $ .

0 3 " :

6 " , lim v ( x + x ) = v (x ), * " v (x )

x→0

x ( ( .

18

$ . & % uk (x) (k = 1, n) x , ck

(k = 1, n) – ,

# ! ( ! ( ( ', - ( ( ( ( ! (.

6. # .

$" " z = F ( x ) = f (ϕ( x)).

!. ( y = ϕ( x ) x ,

. ) * " y = ϕ( x ) ( x ,

( , ": y = ϕ′( x) x + α (x, x ) x ,

α →0 x →0 . ) * " z = f ( y ) ( y ,

( , ":

z = f ′( y ) y + β( y, y ) y ,

β → 0 y → 0 . #:

x →0 . # y →0 y = ϕ( x ) ,

' , (, + β → 0 . # ,:

lim z = f ′( y )ϕ′( x ) ,

x→0 x

!:

F ′( x) = f ′( y )ϕ′( x ).

# .

#.

1. 6 ( z = sin x2 . 5" " , ,' - ( y = ϕ( x) = x2 z = f ( y ) = sin y . # ' (6.1) ,:

19

(sin x2 )′ = (sin y )′ (x2 )′ = cos y 2x = 2x cos x2 . 2. 6 ( y = ln sin x .

5 + " ( y = ϕ( x ) = sin x z = f ( y ) = ln y . #:

(ln sin x )′ = (ln y )′ (sin x)′ = 1 cos x = cos x = ctg . y sin x

$" " ! .

!. ( y = f (x ) $ :

1) Df = (a,b), E f = (c, d ),

2)y = f (x ) (a,b),

3)y = f (x ) (a,b),

4)x (a,b ): f ′( x) ≠ 0 .

. 0 " x = ϕ( y ) , ! " y = f (x ),

, y = f (x ). ) * "

y = f (x ) (, + (a,b), " x = ϕ( y )

(c, d ) ( . «& »). .(

y (c, d ) . . * ' y , ! y + y (c, d ) . 7-

# .

5" , ( ( . $"

y = f (x ) ( . 13). # + " ! x = ϕ( y ) . 6 -

,, : 20

f ′( x ) = tg α, ϕ′( y ) = tg β .

7.# .

1.% .

2. % .

( 2– * ). 6 a = e ,: 21