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Математический анализ

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4. Непрерывность

Выяснить характер точек разрыва функций.

		−		1						1
4.1.		−		2	.		4.2.			1	.
	f (x) = e			x				f (x) = arctg
	f (x) = e			x				f (x) = arctg		x
									1	x
				x					1
4.3. f (x) =				x		.	4.4.	f (x) = e x+1 .
4.3. f (x) =			+ x)2			.	4.4.	f (x) = e x+1 .
	(1		+ x)2

4.5.	f (x) = xsin			1		.

				x
4.7.	f (x) =	1			.
			1
		1+ 2	x
4.9.	f (x) =	1					.
			1

4.6.	f (x) =	1 + x			.

	1 + x3
		cos	π
4.8.	f (x) =		x	.

		cos	p
			x

− 1

4.10. f (x) = e x+4 .

1 + e x−1

	−		1
4.11. f (x) = e			x3 .

4.13. f (x) = e	x+ 1
				x .
4.15. f (x) =				arctg	1 .
		x
					x
	1

4.12. f (x) = arctg x 1- 2 .

4.14. f (x) = xsin px . 4.16. f (x) = 2x1-1 .

4.17. f (x) =

2 x -1

4.18.

f (x) =

ln x

2 x +1

4.19. f (x) =

x + 2

4.20. f (x) = 1- cos x .

x + 2

4.21. f (x) =

ln(1+ x) - ln(1- x)

4.22.

f (x) =

- e−x

- 9

1+ x3 .

4.23. f (x) =

7 + x

4.24. f (x) =

x2 - 4

1+ x

x2 -1

−

4.25. f (x) =

4.26.

f (x) = e

−1 .

x3 - 3x +

4.28. f (x) = e2x .

4.27. f (x) = 1 - cos 2x .

4.29. f (x) =

(1+ x)5 -1

4.30. f (x) = xarctg

1 .

Найти при каком значении параметра a функции f(x) будут непрерывны. Дать геометрическую иллюстрацию для различных значений a.

, x < 0,

ìx2

- 5x + 6

, x ¹ 2,

4.31.

x - 2

f (x) = í

, x ³ 0.

4.32. f (x) = í

, x = 2.

îa + x

xÎ[0;1],

ì x(2 - x)

x ¹ 0,

4.33.

4.34.

f (x) = í

xÎ

(1;2]

f (x) = í

x = 0.

îa(2 - x),

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x+1

x < 0,

4.35.

ï e

f (x) = í

x ³ 0.

+ 2x

îa

4.37.

xÎ[0;1),

f (x) = í

+ 4x

- 2,

xÎ[1;3]

îax

4.39.

ìsin x + a,

xÎ[-p / 2;0],

f (x) = í

cos x

xÎ(0;p

/ 2].

xÎ[0;1],

4.40.

f (x) = í

xÎ(1;

×arctg x

îa

- x

x £1,

4.41.

ï1

f (x) = í

x >

x £1,

4.43.

f (x) = í

x >1.

î2x + a

x £1,

4.45.

f (x) = í

x >1.

îa(x +

xÎ[0;1],

4.47.

f (x) = í

x >

îax

xÎ

é0;

ù,

tg x

4.48.

f (x) = í

4 û .

ïsin(x - a)

æ p

xÎç

;

ú.

4.49.

xÎ[-1;1],

f (x) = í(x -

xÎ(1;3]. .

a)2 ,

4.50.

xÎ[0;1],

f (x) = í

+ ln x,

xÎ(1;e].

îa

Î[-1;1],

4.51.

ï x

f (x) = í

x+a ,

xÎ(1;2].

î3

4.53.

x Î[-1;1],

f (x) = í

î2x−a ,

xÎ(1;2].

4.55.

x -1

xÎ[-1;1],

f (x) = í

xÎ(1;3].

îln(2x - a),

x Î[0;1],

4.56.

f (x) = í

xÎ(1;

3].

×arctg x

îa

x−3

x Î[0;3],

4.57.

ï e

f (x) = í

xÎ

(3;5].

îax + 5

4.58.

x -1

x Î[-1;1],

f (x) = í

x Î(1;e].

+ ln x,

îa

4.59.

ìarcsin x,

xÎ[0;1],

f (x) = í

xÎ(1;2].

î ax + p ,

	ì		2	,	xÎ[0;1],
4.36.	ï ax			,	xÎ[0;1],	.
4.36.	f (x) = í			,	xÎ(1;2]	.
	ï	- x		,	xÎ(1;2]
	î2	- x
4.38.	ì	x +1 ,			x <1,
4.38.	f (x) = í3 - ax2 ,				x ³1..
	î

	ì	x	2	,		x £ 3,
4.42.	ï	x		,		x £ 3,	.
4.42.	f (x) = í			,		x > 3.	.
	ï			,		x > 3.
	îax +1
4.44.	ì4 - 3x ,					x £ 0,		.
4.44.	f (x) = í				,	x > 0.		.
	î a × ex				,	x > 0.
4.46.	ì	x +1			,	x £1,
4.46.	f (x) = í3 - ax2 ,					x >1. .
	î

	ìsin x,			xÎ		é0;	p	ù,
	ìsin x,			xÎ		é0;		ù,
	ï					ê	ú
4.52.	ï					ë	2 û
4.52.	f (x) = í					æ p	ù .
	ïax			xÎ		ç	;p .
	ï					è 2	ú
	î					è 2	û
	ì	x	3		,	x Î[-1;1],
4.54.	ï	x			,	x Î[-1;1],			.
4.54.	f (x) = í				,	x Î(1;e].			.
	ï				,	x Î(1;e].
	îa + ln x

	ì	x−2	,	x £ 0,
4.60.	ï 2				.
	f (x) = í		,	x > 0.
	ï
	îa + 5x

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x ×31-x . x4 + x2

Найти точки разрыва функции, установить их характер, доопределить функцию по непрерывности в точках устранимого разрыва.

4.61. f (x) =

x2 + 3x

4.62. f (x) =

x2 + x - 2

x2 × (x - 3)

x2 + 2x + 1

4.63. f (x) =

x -1

4.64. f (x) =

x2 - x

x -1

×(x -

×(x - 2)

4.65. f (x) =

(x -1) ×ex

4.66. f (x) = e

x -1

x × x2 - 2x +1

x3 -1

4.67. f (x) = 2		1			x +1	.
		x-1

				x3 +1
4.69.			x3		- x2
	f (x) =						.

			x2 ×(x2 -1)

4.71. f (x) = 2|x|×(x-1) .

4.73.	f (x) =	x2	- 2x - 3	.

		(x -1) × x2 - 6x + 9
		(x -1) × x2 - 6x + 9
		1

4.68. f (x) =

4.70. f (x) =

x3 - x

1- 2x + x2 ×(x3 +1)

4.72. f (x) = ln

x - 2

x2 - 4x + 4

x - 2

4.74. f (x) = 2

x-2

x2 - 4

4.75.

f (x) =

×(x -1)

4.76. f (x) =

ln x2 ×(x + 2)

x4 - x3

- x +1

4 + 2x3 + 8x +16

x2 + x

x2 +1 ×(x2 -1)

4.77. f (x) = 2

x-1

4.78. f (x) =

×(x3 +

2 - 2x +1×(x +1)

ln(x - 2)2 ×(x +1)

4.79.

f (x) =

x-1

× x(x -1)

4.80.

f (x) =

x4 - x3

+ x2

2 x2 + 2x +1

2x-1

x(x-1)

×(x +1)

2x2 ×3

4.81.

4.82.

x-1

f (x) =

x3 +

x4 + x2

x -1

×(x3 +

×(x2 + x) .

4.83. f (x) =

4.84. f (x) = e

x-2

x2 + 2x +1

x2 ×(x +1)

4.85.

f (x) =

x2 + x

(x -1) x4 + 2x3 + x2

×(x2 -1)

4.86. f (x) =

(

x2 - x) × 2

x+1

4.87. f (x) =

x +1

x4 - 2x3 + x2

x +1

x2 - 2x +1

(x2

4.88.

3 x+1

+ x)

4.89.

2 - x) ×e

x+1

f (x) =

x3 + 2x2 + x

4 - 2x3 + x2

4.90.f (x) = 3x + 2 ×(x2 -1) .

x2 -1

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5. Производная

Найти f ′(x0) , вычислив

lim

x→0

5.1.

f (x) = (x - 2)2 + 3, x

=1.

5.3.

f (x) =

+ ln x, x0 =1.

5.5.

f (x) = x3 + 2x, x

=1.

5.7.

f (x) =

x + b3 , x

= 4 .

5.9.

f (x) =

, x0 = -1.

+ x2

5.11. f (x) = tg(x + 3), x

= p .

5.13. f (x) = eln(x2 +1) , x

=1.

5.15. f (x) = 3x+2 + 3, x

= 2 .

5.17. f (x) = 3

(x - 3

), x

= e .

5.19. f (x) = 6

+ 3x, x

=1.

5.21. f (x) = (x + 5)3, x

= -3.

5.23. f (x) = 2e ex+1, x

= -1.

5.25. f (x) = 7 + lg x, x0 =10 .

5.27. f (x) = 3x +

, x

=1.

5.29. f (x) =

- cos3x, x

= p .

Найти f−′(0) и f+′(0) .

< 0,

5.31.

ï e

f (x) = í

³ 0.

î1 + x

5.33.

f (x) =

ì- x × sin x ,

x < 0,

sin x

x ³ 0.

5.35.

f (x) =

x £ 0,

> 0.

î3 x2 ,

5.37.

f (x) =

ìsin x +1,

x < 0,

cos x

x ³ 0.

5.39.

f (x) =

ì- x ,

x £ 0,

x > 0.

Найти f−′(1)

îtg x,

и f+′(1) .

f (x0 + Dx) - f (x0 ) .

5.2.	f (x) = 2sin 3x, x								= p .
								0				6
								0
5.4.	f (x) = x + ctg x, x									=			p	.
5.4.	f (x) = x + ctg x, x									=				.
									0			4
									0
5.6.	f (x) = 3						- 2, x					= 4 .
5.6.	f (x) = 3		x +1				- 2, x				0	= 4 .
											0
5.8.	f (x) = x + 3						, x		=1.
5.8.	f (x) = x + 3					x	, x	0	=1.
								0
5.10. f (x) = 2x+1 - 4, x										0		= -1.
										0
5.12. f (x) =		3			+ 8, x
		3		2							= 2 .
											= 2 .
			x3						0

5.14. f (x) = cos3(x - 2), x = p .

5.16. f (x) =

= 2 .

x + 2, x0

5.18. f (x) =

, x0

=1.

(x + 2)2

5.20. f (x) = 5×3

= 7 .

x +1, x

5.22. f (x) =

, x0

= -3 .

x + 7

5.24. f (x) = 2x +

, x

=1.

= p .

5.26. f (x) = x - ctg x, x

5.28. f (x) =

x -1, x = 2

5.30. f (x) = p2 - sin 3x, x0 = p6 .

5.32.	ì- sin x ,				x < 0,
	f (x) = í			,	x ³ 0.
	î sin x
	ì x +1 ,				x £ 0,
5.34. f (x) = í
			x +1,		x > 0.
	î3
	ì	x+1		,	x < 0,
5.36.	ï e
	f (x) = í				x ³ 0.
	ïe + 2x ,
	î
5.38.	ì4 - 3x ,				x < 0,
	f (x) = í	× ex		,	x ³ 0.
	î 4
5.40.	ì x - 2 ,				x £ 0,
	f (x) = í(x -1)2 ,				x > 0.
	î

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5.41.	ì			2x	,	x <1,
5.41.	f (x) = í					x ³1.
	î2(2 - x),					x ³1.
5.43.	ì x +1				,	x <1,
5.43.	f (x) = í3 - x2				,	x ³1.
	î
	ì		2	+ 4	,	x £1,
5.45.	ïx			+ 4	,	x £1,
5.45.	f (x) = í				,	x >1.
	ï	2x +1			,	x >1.
	î	2x +1

ïx2 , x <1,

5.47.f (x) = í x + 5

ï, x ³1.

î 6ì

5.49.	ì				\| x \|					,	x <1,
5.49.	f (x) = í(x - 2)2 ,										x ³1.
	î
	ì	x			3		,			x <1,
5.51.	ï	x					,			x <1,
5.51.	f (x) = í		x−1				,			x ³1.
	ï		x−1				,			x ³1.
	î3
5.53.	ì \| x \|						,			x <1,
5.53.	f (x) = í		x−1				,			x ³1.
	î2						,			x ³1.
5.55.	ì				x -1					,	x <1,
5.55.	f (x) = í										x ³1.
	îln(2x -1) ,										x ³1.
	ì			e		x−3				,	x < 3,
5.57.	ï			e						,	x < 3,
5.57.	f (x) = í		4			x + 5,
	ï-										x ³ 3.
	ï-		3								x ³ 3.
	î		3
	ì								p	x,	x £1,
5.59.	ï- cos								2	x,	x £1,
5.59.	f (x) = í								2	,	x >1.
	ï	p -						px		,	x >1.
	î	p -							2

Найти производные функций.

5.61.f (x) = 3x2 - 523 × x + 3ab2 .

5.62.f (x) = x4 - 13 x3 + 2.5x2 - 0.3x + 0.1.

5.63.f (x) = ax2 + (b + d)−2 x + c .

5.65. f (x) = x - 1x + 43 .

5.67. f (x) =

+ m2 .

5.69. f (x) = 0.1x−

5.2

2.5

x1.4

5 x

5.71. f (x) =

×(x3 -

+1) .

5.73. f (x) = 0.1x−

5.2

2.5

1.4

5.75. f (x) = ax3 + bx2 + c . (a + b)x

5.77. f (x) = xx +-11 .

x <1,

5.42.

f (x) = í

x ³1.

î2 - x

, x £1,

5.44.

ïx

f (x) = í

, x >1.

îx

x <1,

5.46.

f (x) = í

x ³1.

î2x + e -

x <1,

5.48.

f (x) = í

x ³1.

5.50.

x <1,

f (x) = í

x ³1.

î1 + ln x ,

5.52.

ì2x +1,

x £1,

f (x) = í

x ,

x >1.

x <1,

5.54.

f (x) = í

x ³1.

ïln x +1,

x <1,

5.56.

ï2x

f (x) = í

x ³1.

î x

5.58.

ì| x -1|,

x <1,

f (x) = í

ln x

x ³1.

x−2

x <1,

5.60.

f (x) = í

+ 5x,

x ³1.

5.64. f (x) = 3x + 32 + x−2 .

5.66. f (x) = 0.8 ×

0.3

5x2

mx2

5.68.

f (x) =

5.70. f (x) = (x - 0.5)2 .

5.72. f (x) = 0.5 - (a - x)2 .

5.74. f (x) = (x - 0.5)2 .

+ n

ö3

5.76.

f (x) =

ç mx

÷ .

5.78.

f (x) =

x2 +1

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5.79.f (x) = 3x2 +1 .

x-1

5.81. f (x) = ax + b .
	cx + d
5.83. f (x) =	1- x3 .
	1+ x3
5.85. f (x) =	x2 - x +1		.

	a2 - 3
	1
5.87. f (x) =				.
	x2 + x +1
5.89. f (x) =	2x4	.
	b2 - x2

5.91. f (x) =	3				.
	(1- x2 )(1- 2x3)

5.93.f (x) = (x + 4)2 .

x+ 3

5.95. f (x) =	1 +		x	.
5.95. f (x) =				.
1 +		2x


5.97. f (x) =	1 - x2 .
æ		x öm
5.99. f (x) = ç			÷ .
5.99. f (x) = ç			÷ .
è	1 - x ø
5.101. f (x) =	1			.

		a2 - x2
		a2 - x2
5.103. f (x) =	1					.

		a - x4 - x8
		a - x4 - x8
5.105. f (x) =		x2			.

		x2 + a2
		x2 + a2

5.107.	f (x) = x5 - 4x3 + 2x - 3 .
5.109.	f (x) = ax2 + bx + c .
5.111.	f (x) = axm + bxm+n .

5.113. f (x) = px + ln 2 + x2 .

5.115. f (x) = x2 ×3x2 - 32 x3 .

5.117. f (x) =	a + bx			2		x	b	.
		-					2
	c + dx
				b
5.119. f (x) =	2	-	1		.
	2x -1
			x

5.80. f (x) =		x3 - 2x					.
5.80. f (x) =							.
		x2 + x +1
5.82. f (x) =		x5				.
5.82. f (x) =						.
		x3 - 2
5.84. f (x) =		2			.
5.84. f (x) =					.
		x3 -1
5.86. f (x) =	1-		x	3 .
			p

5.88.	1		.
	f (x) =
		x2 - 3x + 6

5.90.f (x) = x2 + x -1 .

x3 +1

5.92. f (x) =			ax + bx2										.
5.92. f (x) =													.
			am + bm2
5.94. f (x) =					x3
									.
			(1- x)2						.
		1 - 3
5.96.	f (x) =	1 - 3					2x				.

		1		+ 3 2x
		1		+ 3 2x
		æ						1		ö2
5.98. f (x) =		ç1- 2x						2			÷ .
		ç									÷
		è									ø
5.100. f (x) =						2									.
5.100. f (x) =				(x2 - x + 1)2											.
5.102.	f (x) =			3		1								.
5.102.	f (x) =			3		1 + x2								.
						1 + x2
5.104.	f (x) =				1 + x							.
5.104.	f (x) =											.
						1 - x

5.106. f (x)

5.108. f (x)

5.110. f (x)

5.112. f (x)

=		1	.

	x -	a2 + x2
	x -	a2 + x2

=13 x + x2 - 0.5x4 .

=- 53x3 + x−3 .

	ax6		+ b		1
=				+ x 5 .

		a2 + b2

5.114.					2				5	+ x−3 .
	f (x) = 3x 3					- 2x			2
5.116.	f (x) =			a		-			b				.
								x × 3
			3 x2									x
5.118.					2x + 3
	f (x) =										.
			x2 - 5x + 5
5.120.	f (x) =		1 + x				+ 1 +								.
														x

		1 -				x

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Найти производные функций.

5.121. f (x) = (sin x + cos x )3 .

5.123. f (x) =		tg x	- cos x .

		x
5.125. f (x) =		sin x		+	x		.
		x			sin x
				x
5.127.	f (x) =					.
		sin x + cos x

5.129. f (x) = cos2 x + cos 2x .

5.131. f (x) = cos x - 13 cos3 x .

5.133. f (x) = 13 tg3 x - tg x + x .

5.135. f (x) = sec2 x + cos ec2 x .

5.137. f (x) = a cos

- bsin

5.139. f (x) = tg

x +1

sin x

5.141. f (x) = x2 × sin 1x .

5.122.	f (x) =		x		.
		1 - cos x
5.124.	f (x) = x sin x + cos x .
5.126.	f (x) =		sin x			.
		1 + cos x

5.128.	f (x) =	xsin x		.

		1+ tg x
5.130.	f (x) =	1 tg4 x + sin 2x .
		4
5.132.	f (x) = 3sin 2 x - sin3 x .
5.134.	f (x) = xsec2 x - tg x .
5.136.	f (x) = cos 2x + 2sin 2 x .
5.138.	f (x) = 2sin 3x × tg3x .
5.140.	f (x) =						×cos3 x .
			1+ 2 tg x
5.142.	f (x) = sin(sin x) + sin 2 x .


5.143. f (x) = cos3 4x + sin3 4x .									5.144.	f (x) = tg	x	- tg			x	.

										2		2
										f (x) = 3x - ctg 3								.
5.145. f (x) = 3		sin x - sin 3			x2		.		5.146.					1+ x2
	a
5.147. f (x) = (1+ sin2 x)4 + 4 tg x .									5.148.	f (x) = ctg x -							.
													1+ tg x
5.149. f (x) = sin x + cos2			1	-				.
						x			5.150.	f (x) = cos x - sin2 (cos x) .
				+
1						x

Найти производные функций, содержащих обратные тригонометрические функции.

f (x) = arcsin x

5.151. f (x) =

1 - x2 arcsin x2 .

5.152.

+ arcctg x .

arccos x

5.153. f (x) = x(arcsin

1 - x2 )2 .

5.154.

f (x) = arcsin x +

1 - x2 .

5.155. f (x) =

5.156.

+ arccos x .

f (x) = arcsin

x ×arctg x .

arcsin x

5.157. f (x) =

arccos x

+ ln x .

5.158.

f (x) =

+ x ×arctg

5.159. f (x) = (arctg x + arcsin x)n .

5.160.

f (x) = cos x × arccos 4

arcsin x

5.161. f (x) =

- arctg x .

5.162.

f (x) =

1 - x2 .

+ x2

1 - x2

5.163. f (x) =

- arccos x .

5.164.

f (x) = tg x ×

arcsin(x -1)

arctg x

5.165. f (x) = x2 × arccos 2x

1 .

5.166.

f (x) = arctg(x2 )× tg(x2 ) .

5.167. f (x) = arccos

arcsin

5.168.

f (x) = arcsin(sin(sin x)) .

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5.169.

5.171.

5.173.

5.175.

5.177.

f (x) =

+ arctg2 1 .

f (x) = (x + 1) arcsin

1 - x

1 + x

f (x) = arcsin

4x × sin x

1 - 2α × sin x

f (x) = arctg(x - 1+ x2 ) .

f (x) = cos

×arctg

5.170. f (x) = 2x1 - (arccos x)2 .

5.172. f (x) = 12 4arcsin x2 + 2x .

5.174. f (x) = arccos				b + a cos x				.

				a + b cos x
5.176. f (x) =	x		× arccos 1 - x .

		2			2
	1								.
5.178. f (x) =			tg x ×arcctg				2

2						x

5.179. f (x) =

+ 2x - arctg

5.180.

f (x) =

1 - x2 × arccos x2 .

Найти производные функций, содержащих логарифмические функции.

5.181. f (x) = ln x2 + x2 log3 x .

5.182.

f (x) = ln2 x + log2 2x .

5.183. f (x) = x × ln x -

5.184.

f (x) =

+ 2x lg x3 .

lg(x + 2)

ln x

5.185. f (x) =

x -1

- ln

5.186.

f (x) = log4 x + sin x × ln x .

log2 x

5.187. f (x) =

2lg x - lg x2 + 2x

5.188.

f (x) = ln x - log

ln x

5.189. f (x) =

1 - ln x

+ lg 3

5.190.

f (x) =

ln x

x -1

1 + ln x

1 + x2

lg2 x

5.191. f (x) = xn ln x + log3

3 x2 .

5.192.

f (x) =

1 + ln

2 x +

log3x

5.193. f (x) = ln(1 - 2x)-

5.194.

f (x) = 2x ln x2 + log x (2x) .

log

2x 2

5.195. f (x) = ln sin x + ln(2cos x) .

5.196.

f (x) = log3 (x2 -1) - lg

5.197. f (x) = ln tg x + ln ctg x − x .

5.198.

f (x) = ln arccos 2x + lg x2 .

5.199. f (x) = ln sin x + lg cos2 x .

5.200.

f (x) = arctg[ln(ax)]+ lg x3 .

5.201. f (x) = lg(x) + (ln sin x)n .

5.202.

f (x) = log2 (log3 (log5 x)) .

f (x) = arcsin 2 [ln(a3 + x3 )].

5.203. f (x) = ln arctg

1+ x2 .

5.204.

5.205. f (x) =

3 ln sin

x + 3

- lg x .

5.206.

f (x) = lg2 x2 - log x (2x) .

5.207. f (x) = ln(ln(ln x)) - 4

5.208.

f (x) = ln(ln3 x) - sin ln x .

lg x

+ coslg 4

1 - sin x

5.209. f (x) = ln tg

5.210.

f (x) = lg ax - ln

1 + sin x

Найти производные функций, содержащих показательные функции.

5.211. f (x) = 2x ×32x - 4x2 .

5.212.

f (x) =10x + e2x−3 .

5.213. f (x) =

+ sin 3x−1 .

5.214.

f (x) =

- cos 2sin x .

5.215. f (x) = x ×10x

- log

100 .

5.216.

f (x) = x × ex

+ 5log5 x2 .

x3 + 2x

5.217. f (x) =

- 2x ×32x .

5.218.

f (x) =

- log3 32x−4 .

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5.219. f (x) = ex cos x + tg53x−2 .

5.220.

f (x) =

+ 2x+sin x .

sin x

5.221. f (x) = cos x - e3x−cos x .

5.222.

f (x) =

- sin(ln x) .

2ln x

5.223. f (x) = x3 - 3x + cos(ln 2x) .

5.224.

f (x) =

1+ ex

+ tg(ln x2 ) .

5.225. f (x) = (x2 - 2x + 3) × ex .

5.226.

f (x) =

1+ ex

- 3log3 (x

+1) .

1- ex

1-10x

5.227. f (x) =

1+10x - sin(32x ) .

5.228.

f (x) = cosex

1+ x2

5.229. f (x) = xex cos x - sin x) .

f (x) = e−x + 3

5.230.

42x−tg x

5.231. f (x) =102x−3 - a2x+tg x .

5.232.

f (x) = e

x+1

- b

sin x−1

5.233. f (x) = sin(2x ) + tg(3cos2x ) .

5.234.

f (x) = 3sin x + arcsin 2−x .

5.235. f (x) = asin3 x - (23x−4 - 3x) .

5.236.

f (x) = earcsin2x

- cos(3x) .

5.237. f (x) = 23x - arccos(37x ) .

5.238.

f (x) = e

+ 52cos x−tg x .

ln x

5.239. f (x) = sin(e

x2 +3x−2

) + 4

tg x

5.240.

f (x) =

1−sin4 3x

- cos(3

) .

Найти производные функций, содержащих гиперболические функции.

5.241. f (x) = sh3x + 3

5.242.

f (x) = ln chx - sin(chx) .

chx

5.243. f (x) = arctg(thx) + shx .

5.244.

f (x) = th(1 - x)- ch(ln x2 ) .

5.245. f (x) = sh2 x + ch2 (1 - x) .

5.246.

f (x) = ch(shx) + 2shx .

5.247. f (x) =

- tg(shx2 ) .

5.248. f (x) =10ch2x - sh3 x .

chx

5.249. f (x) = th(ln x) + ln(chx) .

5.250.

f (x) = x × shx - chx .

th3

5.251. f (x) = 4 (1 + th2 x)3 .

5.252.

f (x) =

5.253. f (x) =

1 + thx

- ch3 x .

5.254.

f (x) =

thx + ln

1 +

thx

1 - thx

1 -

2thx

5.255. f (x) =

ch2x +

5.256.

f (x) = x2e3x csc hx .

xsh2x .

5.257. f (x) = lg3 (chx) + shx .

5.258.

f (x) =

+ sh

chx

x2 +

5.259.

5.261.

5.263.

5.265.

5.267.

f (x) = thx - ch2 x2 - x .

f (x) = arctg(thx) - shx3 .

f (x) = arcsin(shx) + echx . x

f (x) = ln(chx) + 1 .

2ch2 x f (x) = (th2 x)3 - sh(e x ) .

5.260.	f (x) =		3cthx		- lg(shx) .
5.260.	f (x) =		ln x		- lg(shx) .
			ln x
5.262.	f (x) = arcsin(shx) + chx4 .
5.264. f (x) =				thx		- 3shx .
5.264. f (x) =		1		- x2		- 3shx .
		1		- x2
5.266.			chx				æ	x ö
	f (x) =				- lnçcth				÷ .
	f (x) =		sh2 x		- lnçcth				÷ .
			sh2 x				è	2 ø
5.268.	æ shx + 2x2 ö
5.268.	f (x) = ç						÷ + sh2 x .
	ç			chx			÷
	è			chx			ø

æ3

chx

5.269.

f (x) = thx - ch

5.270.

f (x) = sh

x - ch

x + 2

÷ .

Найти производные заданных функций, используя понятие логарифмической производной.

5.271. f (x) = xx2 × 2x + 3 . 5.272. f (x) = x2x ×sin x .

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5.273.

5.275.

5.277.

5.279.

5.281.

5.283.

5.285.

5.287.

5.289.

5.291.

5.293.

f (x) = (sin x)cos x ×ex2 .

f (x) = (x +1)

×e2x+3 .

(x - 2)

2 × 3

f (x) =

x + 1

- 5)3

(x + 1)3 × 4

f (x) =

x - 2

5 (x - 3)2

×esin x .

f (x) = (x3 +1)

- 2x

öx

×e3x−2 .

f (x) =

1+ x

f (x) = 2x

× (2x - 3) .

f (x) =

x(x2 +1)

(x2 -

1)2

f (x) =

(x + 2)2

(x +1)3(x + 3)4

f (x) = tg x ×3

x2 +

f (x) =			x -1		.
	3
		(x + 2)2		(x + 3)3

5.274.	f (x) = (ln x)x ×cos x2 .
5.276.	f (x) = x3 ×ex2 ×sin 2x .
	f (x) = xln x ×3							.
5.278.					2x2 - 4x

			x sin x ×						.
5.280.	f (x) =					1 - ex
		1
5.282.	f (x) = x x × 4			4x3 + 3x			.
	æ		x öx
5.284.	f (x) = ç			÷ × sh2x .

	è	1+ x ø

5.286. f (x) =

5.288. f (x) =

5.290. f (x) =

(x2 +1)sin x × sh(ln x) .

(x +1)(2x +1)(3x4 ) .

chx3 ×	x(x -1)	.

	x - 2

5.292.				(x - 2)9
	f (x) =							.

			(x -1)		5	(x	11
							- 3)
5.294.	f (x) = x			×sin x × 4cos2 x+1 .
		x

x +1

f (x) = (cos x)sin x × tg(x -1) .

5.295. f (x) = x

5.296.

3x - 4

5.297.

öx

5.298.

f (x) = (arctg x)x ×3sin x .

f (x) = ç1

÷ × sh(3x + 2) .

5.299. f (x) = (tg 2x)ctg

cos3x

5.300. f (x) = sin xcos x ×3

x - 5

×3

5x -1

5 x2 + 4

Найти производные функций.

1 + x

5.301. f (x) =

- k

1 - k

1 - x

1 - x k

5.302. f (x) =

2 ln(1 + x)-

4 ln(1 + x2 )-

2(1 + x)

1 + x

5.303. f (x) =

- k

1 - k

1 - x

1 - x k

5.304.

5.305.

5.306.

5.307.

f (x) = xln

1+ x

+ x

ç x +

÷ - 2 1

lnç x

f (x) =

+ a

lnç x +

+ a

÷ .

+ x

f (x) =

- 3

sin x + ch3x

a - x

2 + 3x2

+ 3ln

x +

1- x2

f (x) =

1- x2

+2 ö .

1x ÷

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5.308.

5.309.

5.310.

5.311.

5.312.

5.313.

5.314.

5.315.

5.316.

5.317.

5.318.

5.319.

5.320.

5.321.

5.322.

5.323.

5.324.

5.325.

5.326.

5.327.

5.328.

f (x) = ln b + a cos x + b2 - a2 sin x + 3thx . a + bcos x

ö2

f (x) =

ç1-

+ x

+ 3lnç1

1+ x

÷ .

f (x) = x arcsin

+ arctg

1+ x

a - b

f (x) =

arctg

- b2

a + b 2

f (x) = ln(1+ sin2 x) - 2sin x ×arctg(sin x).

arctg

- 43x .

f (x) =

a2 - x2

(x +1)2

f (x) =

1 ln

arctg

x2 - x +1

x2 + x

f (x) =

arctg

x2 - x 2 +

2 2

x2 -

f (x) = x(arcsin x)2 + 2

1- x2 ×arcsin x - 2x .

x4 - x2 +1

f (x) =

arctg

(x2 +1)2

2x2

2 3

-1

1- 3

1+ 23

f (x) = ln

arctg

+ 3

1+ 3

3 - x

1+ x

f (x) =

1- 2x - x

2 + 2arcsin

- sin x3 .

+ x

f (x) =

1+ x4

arctg

1+ x4

4 1+ x4 - x

1 ln 1 -

1 - x2

1 - x

f (x) = 1 - x2 × ln

- 1 - x2 .

1 + x

1 +

1 - x

f (x) = xarctg x - 12 ln(1+ x2 )- 12 (arctg x)2 .

+ 2 - x

x2 + 2

f (x) =

arctg

x2 + 2 + x

1+ x4 - x

f (x) =

arctg

1+ x4

4 2

1+ x4 + x

1- x2

f (x) =

arctg

+ cos2 x .

1+ x2

1- x2

f (x) = em arcsin x [cos(m arcsin x)+ sin(m arcsin x)].

2 a

- b

a - b

f (x) =

x +

arctg

a + b

f (x) = (arccos x)2 êéln2 (arccos x)- ln(arccos x)+

úù .

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æ4

1+ x4

5.329.

f (x) =

1+ x

arctgç

4 1+ x4

-1

−x2

arcsinçe

−2x2 ö

5.330. f (x) =

lnç1 - e

÷ .

1 - e−2x

Найти производные 1-го и 2-го порядка параметрически заданных функций.

5.331. íìx = sect,

5.332.

−t

íïx = e

y = tg x.

î y = t

−t

cost,

5.333.

ïx = e

5.334.

ïx = e

−t

sin t.

ï y = et sin t.

îy = e

5.335.

5.336. íìx = ln t,

íïx = ln(1+ t

y = t

y = t3.

5.337.

ìx = arctg x,

5.338. íìx = sin t - t cost,

íï

y = cost + t sin t.

îy = ln(1+ t

5.339.

ìx = cos 2t,

5.340.

ìx = arctgt,

íï

y =

îy = sin

ìx = ln t,

íìx = sec2t,

5.341. íï

5.342.

ïy =

î y = tg 2t.

1- t

sin 2t,

5.343.

ïx = t

5.344.

ïx = e

cos 2t.

ïy = et

îy = e

5.345.

5.346. íìx = ln(1+ t),

íïx = ln(4 - t

y = t3 + 4.

îy = t

5.347.

ìx = arctg 2t,

5.348.

íìx = cost + t sin t,

íï

y = sin t - t cost.

îy = ln(1+ 4t

5.349.

ìx = sin 2t,

5.350.

íï

íïx = arctgt

= t

îy = cos

ìx = ln(1+ t),

5.352. íìx = ln t,

5.351. íï

ïy =

î y = t5.

1- t

t2 +1

ïx =

5.354. íï x = t + t ,

4(t -1)

5.353. íï

ïy = t

2 +

ïy =

(t +1)

ìx = t3 - 3t,

ì x = t +

5.355. íï

5.356. íï

ïy = t3 -

t2.

ïy = t

2 +

ïx = t

5.357.

íï

5.358. íï x = t

- t,

2t +

+ 2t.

ïy =

2 .

îy = t

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ì	t	2
ïx =	t			,
ïx =	t +		1	,
5.359. íï	t +		1
ïy =		t3			.
ïy =	(t +1)				.
î	(t +1)
ï				2

	ì			t
	ïx =						,
5.360.	ïx =	t	2				,
5.360.	í	t		-1
	ï		t2
	ï y =		t +		1	.
	î		t +		1

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