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Национальный минерально-сырьевой университет «Горный»

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y = f (x) x &		x0	" y8, y8x
f '(x0 ) . F,				f ( x0 + 6x) f ( x0 )
y8 = f 8(x0 ) = lim	6y	= lim			.	(4.1)
	6x
6x 0		6x	0	6x
.	f '(x0 )		& &-

', " (4.1).

= ", "

! / 0 y = f (x) +

+ X , + X / 0, -

+ x 7 ,

. > , / 0 / 0 f										" f 8 .
* 1. > / 0							f (x) = x
x .
:					(x + 6x) x				6x
		f '(x) = x ' = lim			(x + 6x) x	=	lim		6x	= 1.
		f '(x) = x ' = lim		0		=	lim	0 6x		= 1.
			6x	0	6x		6x	0 6x
= ", x ' = 1. .+ + ,
			(xn ) ' = nxn 1,n N.
(" [3]).
M 0			& + .
		-			y
'					y					l
'
Oxy			l , -
Oxy			l , -	y0 + 6y						M
	' /		/ 0							T
y = f (x)	( .4.13).		= "

, -

M0 (x0 , y0 ) .

> l - " '

M (x0 + 6x, y0 + 6y)

M0M , "

Ox '

H -

l M0

M "-

	6y
	O
	dy
y0	M0
y0	P

	0
	0
O		x0	x0 + 6x	x
			M 4.13.

156

+ M0 , + ((M0, M ) 0 . !

, M0M " " +c +-

M 0T , ' + M0M M 0T ,

M 0T "

l M0 .

, ' M0

M 0T ,

0 ,

0 = lim

tg 0

= lim tg .

(4.2)

> .4.13 ,

f (x0 + 6x) f (x0 )

tg =

= 6y =

M0 P

', ' ' ,// 0 k0

M 0T -

f (x0 + 6x) f (x0 )

= tg 0 = lim

= f 8(x0 ).

(4.3)

6x 0

C ' ,// 0 k0 =

f 8(x0 ) M 0T M0 (x0 , y0 ),

, - ,

y y0 = f 8(x0 )(x x0 ).

(4.4)

/ 0

y = f (x) x0 '

f (x)

" ' ,// 0 ' / , / 0

" 0 x0 .		f 8(x)	x ,-
D	&. T

y = f (x)	, ( " 0 x ). , -

, Oy , "

– , Oy .

H // 0 / 0

/ 0 y = f (x) X x0

. C , ' 6x 0 , "

x0 + 6x X .

157

. E 0 y = f (x) &-

x0 , / 0 , ,

6x ', + /
6y = A6x + +(6x) 6x ,	(4.5)
' A - , +(6x) - / 0 6x , " 6x	0 ,
+(6x) 0 6x 0 .

C, A / x0

6x . x0 A , " ', -

T // 0 / 0 -

.
@ . H ', " / 0 y = f (x)			" // 0
x0 , " ,	" , / 0 " ,
.
. + , [3], A , (4.5), -
f 8(x0 )	f 8(x) x0 . ,
+ /
6y = f 8( x ) 6x + +(6x) 6x,	'	+(6x) 0		6x	0	.	(4.6)
0	'					.	(4.6)

D &. " // 0-

/ 0 -

+ + // 0 . @ . ! / 0 y = f (x) // 0 x0 ,

, .

158

D &. @" + , -

/ 0 , " ', // 0-

/ 0 , . M, , / 0, ' / -

. 4.14. @" , / 0 x0,

" // 0 , . O ' /

) x0

/ 0 M0 (x0 , y0 ) Oy , , / 0 "-

x0 " . < / / 0 x0

" , , / 0 x0 + .

. =, " x0 . 4.14 , &

/ 0, " x0 . 4.14 " – 0 & .

. E 0 y = f (x) -

(a,b) ( "), , / 0

// 0 + , ' +.

@ ,, + X ' +

// 0 ( -

). ,				X = [a,b]	- , ' '-
a b			(a < b)	+ , , -
							f (b + 6x) f (b)
lim		f (a + 6x) f (a)			lim			.

6x	0		6x		6x	0	6x
6x	>0				6x	<0

< / / 0, // 0 +, +

- " ' . = " -

M 4.14

159

H // 0 / 0

/ 0 y = f (x) // 0 x0 . = ' x0

" ' 6x 0 - (4.6):

6y = f 8( x	) 6x + + (6x) 6x	, '	+ (6x) 0		6x	0	,
0		, '					,

" '. , '

f 8( x0 ) 6x / 0 6x , '

+(6x)6x - / 0 6x .

. f 8(x0 )6x , "

6x / 0 x0 , -

y = f (x) , "

dy df (x0 ). F,

dy = f 8( x0 ) 6x.

(4.7)

C, 6x ' x , -

, " " dx -

. J -

// 0, + f (x) = x ([3]). = ",	/
(4.7) // 0 / 0 -
dy = f 8(x0 )dx,	(4.8)

// 0 / 0 -

/ 0 , // 0 ( ) -

F / (4.8) ,	f 8(x0 ) = dy	. = ",
	dx

/ 0 + - // 0 / 0 /-

/ 0 . P dydx "-

/ 0 y x .

' / / 0 y = f (x) , .4.13. O ' , // 0 / 0 y = f (x) x0 '

" ' / , / 0

M0 (x0 , y0 ) [x0, x0 + 6x][3].

@ 0 + // 0 / 0 -

, / 0. @" " 0

"b . / (4.8) // 0 / 0 +

6x 9 dx.

* 2. > // 0 / 0 y = 3x2 + x

x = 1 , 6x = 0,1 .

160

> G // 0 / 0 :

6y = 3( x + 6x)2 + ( x + 6x) 3x2 x = 6x6x + 3(6x)2 + 6x = (6x + 1)6x + 3(6x)2 .

= ' dy = (6x +1) 6x . 6y dy x = 1 , 6x = 0,1

6y = 7 , 0,1+ 3, 0,01 = 0,73; dy = 7 , 0,1 = 0,7.

, ' / 0

> - // 0-

, / 0,

" ' . + [3]. @ . ! / 0 u = u(x) v = v(x) // 0 x ,

				(u ± v)8 = u8 ± v8 ,
					8	8	8
				(uv)		= u v + v u ,
8			8		8
! u "		=	u v v u			v = v(x) 0 .
$	%
$	%			v	2
& v '				v

(4.9)

(4.10)

(4.11)

T + , dx , + ,

// 0


d (u ± v) = du ± dv ,								(4.12)
d (uv) = vdu + udv ,								(4.13)
	! u "			vdu udv
d	$		% =				.	(4.14)
				v	2
	& v '
// 0 / 0 y = C ( - -
x X ) .
= ", x X
C	8	= 0;				8		(4.15)
				dC = C dx = 0 .
H, " + 7 / 0
+ , ' 6y 9 0						"		x 6x ,
x, x + 6x X. @,								// 0,

/ (4.15).

E (4.9) " " " ' ' '

/ 0.

u = C , ' C const , / (4.10) (4.13), (4.15), -

: (Cv)8 = Cv8, d (Cv) = Cdv. = , + +

// 0.

161

		* 1
F / ( + [3]).
1.	(sin x)8 = cos x,																2.	(cos x)8						= sin x,
3.	(tgx)8 =				1				,								4.	(ctgx)8					=							1					,
3.	(tgx)8 =		cos2 x						,								4.	(ctgx)8					=					sin			2		x		,
			cos2 x																									sin			2		x
5.	(loga x)8		=			1	loga				e =	1	,				6.	(ln x)8					=		1		.
5.	(loga x)8		=				loga				e =	x ln a	,				6.	(ln x)8					=				.
						x						x ln a													x
											+ , &.
D & 1. / 0																			f (x) =					1		x		4				1		x		2	+ x + 2
D & 1. / 0																			f (x) =					4		x					2			x			+ x + 2
x = 3 .																								4							2
x = 3 .
> / 0 f (x) ,														// 0-

	8	! 1		4				1		2		"8	! 1		4	"8	!		1		2	"8				8					8
f	8		x	4					x	2	+ x + 2%		= $	x	4	%	+ $			x	2	%				8					8
f	(x) = $		x					2	x		+ x + 2%		= $	x		%	+ $		2	x		%	+ x						+ 2 .
		& 4						2				'	& 4			'	&		2			'

+ :

! 1	"8	! 1		"8		1	8	1	8
$	x4 %	+ $		x2 %	+ x8 + 28 =		(x4 )		(x2 ) + x8 + 28,
			2			4		2
& 4	'	&		'

/ // 0 / 0 -

/ 0,

14 (x4 )8 12 (x2 )8 + x8 + 28 = 14 , 4x3 12 , 2 , x + 1 + 0 = x3 x +1.

F, f 8(x) = x3 x + 1.

> x = 3 , '

+ x 3:

f 8(3) = 33 3 + 1 = 25.

D & 2. > / 0 f (x) = (2x2 + 1), cos x .

, / 0 , // 0-

/ 0:

f 8(x) = -1/(2x2 +1)cos x.208 = (2x2 + 1)8 cos x + (2x2 +1)(cos x)8.

' + +, 1,

' + , (cos x)8 = sin x :

162

(2x2 +1)8 cos x + (2x2 +1)(cos x)8 = (2, 2x + 0)cos x + (2x2 +1),( sin x) = 4xcos x (2x2 +1)sin x.

= ",

(

+1)sin x .

f (x) = 4x cos x

D & 3. > // 0 df (x)

/ 0

f (x) = tgx x = .

> // 0 / 0, / df (x) = f

(x)dx .

= (tgx)8 =

cos2

df (x) =

, dx

cos2 x

, , cos =

= 2dx

df $

4 '

D & 4. P f ( x) = sin x

x0 = .

T ' / / 0 ( x0 , f (x0 ))

y f (x0 ) = f 8(x0 )(x x0 ) .

> ' :

f (x0 ) = sin = 0

f 8(x) = cos x,

f 8(x0 ) = cos = 1.

= ' y = ( 1)(x ) ,

y = x - -

; 4.5

1.D / 0 ?

2.' ?

3.O ' ' / / 0 ?

4.O / 0 // 0: ) ; ") +-?

5.D // 0 / 0?

6.O ' // 0?

7.D ) ; ") ; ) '

/ 0?

8.H + / + / 0.

163

4.6. * , & -

. * , -

•C .

•* " .

•C . *

0 & .

•@ .

•* , .

•C , &.

- . ! "

, " [3], ' 3, . 67-73 ' –

( &-& & & , " - & " F 2 F

71-80, 81-90, 91-100.

H // 0 + / 0.

/ 0

+ / 0 y ' x / y = f (u), ' u = (x) .

@ ( ). ! / 0 y = f (u), u = (x) // 0 ' ' u

x , + / 0 f [ (x)]	+ // 0	x ,
y8 = f 8(u)u8	y8x = y8u , u8x.	(4.16)

T + (4.16) dx , + //- 0 + / 0 dy = f 8(u)du.

D &. H // 0 / 0 y = f (u) " +

, " ' u " / 0, -

. , ( -

) / // 0 - '. P -, u - , du = 6u -

, + u - + ' ( / 0), du - /- / 0 , / 0, ,

6u.

P ' / // 0-

/ 0 [3]:

164

1) (x

= + x

+ 1

; 2) (a

x 8

= a

ln a;

3) (e

)

= e

x < 0 , , x+

D &. ' ' / -

' / / 0 0 & .

+ ' // 0 / 0

y = u(x)v(x) .

* . > / 0 y = xsin x .

= / 0 y = ex y = ln x ",

" ' + ' A +

A = eln A .

y = xsin x = e	ln(xsin x )	= esin x,ln x .
y = xsin x = e		= esin x,ln x .
= '
y8 = (esin x,ln x )8 = esin x,ln x , (sin x , ln x)8 =
= xsin x ((sin x)8 ln x + sin x , (ln x)8 )= xsin x			!	sin x "
= xsin x ((sin x)8 ln x + sin x , (ln x)8 )= xsin x			$ cos x , ln x +	%.
			&	x '

C " // 0 + / 0

// 0 .

165

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