Расчетные задания (Кузнецов) / 02-Дифференцирование

13.25. y =	1 − x2	+arcsin x.	13.26. y = x	2	arccos x −	x2 +2	1	− x	2	.
	x					3

13.27. y =

x2 +2

−

1

ln

2 + x2 +2

.

x2

2

x

13.28. y =

x

(10 − x

2

)

4 − x

2

+6arcsin

x

.

4

2

13.29. y = arcsin

1

+

2 x2

+3x + 2,

2x +3 > 0.

2x +3

13.30. y = x arcsin

x

−

x +arctg

x.

x +1

13.31. y =

arcsin x

+

1

ln

1− x

.

2

1− x2

1+ x

Задача 14. Найти производную.

14.1.

y =

1

ln

(tg x +ctgα).

14.2. y = x cosα +sinαln sin (x −α).

sinα

14.3.

y =

1

sin ln x −(

2 −

1) cos ln x x 2+1.

2

2

14.4. y

= arctg

cos x

.

4 cos 2x

14.5. y =3

sin x

sin x

14.6. y =(a

2

+b

2

)

−1 2

a2

+b2 sin x

+2

.

arcsin

.

cos

2

x

cos

4

x

b

14.7. y =

7x (3sin 3x +cos3x ln 7)

.

14.8. y

= ln

sin x

.

cos x +

cos 2x

9 +ln2 7

14.9. y =

1

arctg(a cos x)+a lntg

x

.

+

2

a

1

a

)

2

(

14.10. y = −

1

−

1

+

1

ln

1 +sin x

.

3sin3 x

sin x

2

1−sin x

14.11. y = 1 + x2

)

earctg x .

14.12. y =

ctg x + x

.

(

1− x ctg x

1

2xsin

α

x4 +1 − x2

14.13. y =

arctg

2

.

14.14. y = arctg

, x >

0.

α

1 − x2

x

2sin 2

14.15. y =

6x (sin 4x ln 6 −4cos 4x)

.

14.16. y = arctg

2 tg x

.

16 +ln2 6

1

−tg x

14.17. y = arctg

2sin x

.

14.18. y =

5x (2sin 2x +cos 2x ln 5)

.

9cos2 x −4

4 +ln2 5

14.19. y = ln

2 + th x

.

14.20. y =

3x (4sin 4x +ln 3 cos 4x)

.

2 −th x

16 +ln2 3

14.21. y =

4x (ln 4 sin 4x −4cos 4x)

.

14.22. y =

cos x

−2cos x −3lntg

x

.

16 +ln2 4

sin2 x

2

14.23. y =

5

x

(sin 3x ln 5 −3cos3x)

14.24. y = x −ln 1+ex

)

−2e−

x

x

.

2

arctge

2

.

9 +ln2 5

(

	14.25. y =		2x (sin x +cos x ln 2)						.
		1				+ln2 2
	14.27. y =	2		cos x		+3	cos x	.
				sin4 x
							sin2 x

14.28.

14.29. y = 3x (ln 3 sin 2x −2cos 2x). ln2 3 +4

14.31. y =	tg x +	2 tg x +1	.
	tg x −
		2 tg x +1

14.26. y = ln (ctg x +ctgα). sinα

y =

cos x

4

2 tg (x 2)+1

+

arctg

.

3(2 +sin x)

3 3

3

14.30. y =

1

ln

1+cos x

−

1

−

1

.

1−cos x

cos x

3cos3 x

2

Задача 15. Найти производную y′x .

	3t	2	+3		1		,
x =
	3t
15.1.						3
y =sin				t				+t .
			3

	−t	2	,
x = 1
15.2.	1+t.
y = tg

x =			2t −t2 ,
				1
15.3.	y =						.
		3	(1−t )2
						2		+1),
x = ln (t + t
15.5.
			t	2	+1.
y =t

x = ctg (2et ),

y = ln (tget ).

x = arctget2 ,

15.9.

y = et +1.

x = ln 1 ,

1−t4

y = arcsin 1+−t2 .

1 t2

				2	),
x = arcsin ( 1−t
15.13.	(arccost )2 .
y =
x =	(1+cos2 t )2 ,
15.15.		cost
y =			.
		sin2 t

x = arccos1t ,

y = t2 −1 +arcsin 1t .

(sin t ),

y = arccos(cost ).x = arcsin

		2t −t								2		,
x =
15.6.
y = arcsin (t −1).
x = ln (ctg t ),
15.8.		1
y =									.
		2					t
	cos
								1−t
x = ln																,
							1+t
15.10.
		1−t								2		.
y =
x =		1−t2 ,
15.12.					t
y =														.
		1 −t2
					t
x =													,
		1−t2
15.14.			1				+						1 −t2
y = ln																	.
													t
			1−t
x = ln												,
			1				+t
15.16.
		1 −t								2		.
y =
		1
x =							,
		ln t
15.18.			1				+					1				−t2
y = ln																	.
													t

			t ,
x = arcsin
15.19.	1 +		t .
y =
x =t	t2 +1,
15.21.		1 +	1 +t	2
y = ln					.
			t

x = ln (1 −t2 ),

15.23.

y = arcsin 1−t2 .

					1−sin t
x = ln									,
				1+sin t
15.25.	1
						2
y =			tg				t	+ln cost.
	2
x = lntg t,
			1
15.27.
y =							.
	sin			2		t

	sec2 t	,
x = e
15.29.
y = tg t ln cost + tg t −t.
x = ln (t + 1+t2 ),
15.31.					1+		1+t2
			2
y =	1+t			−ln				.
							t

x =

(arcsin t )2 ,

15.20.

t

y =

.

1 −t2

x = arctg t,

2

15.22.

1+t

y = ln

.

t +1

t +1

x = arctg

,

t

−

1

15.24.

1−t

2

.

y = arcsin

t −t2

1−t

x =

−arctg

,

t

15.26.

t −

1−t arcsin t.

y =

t

2

ln t

+ln 1−t2 ,

x =

2

1−t

15.58.

t

y =

arcsin t +ln

1−t2 .

1−t

2

t

2

x =

arcsin t +ln

1−t

,

1−t2

15.30.

t

y =

.

1−t

2

Задача 16. Составить уравнения касательной и нормали к кривой в точке,

соответствующей значению параметра t =t0 .

	3
x = a sin		t,					x = 3 cost,
16.1.			t		=π	3.	16.2.	t		=π 3.
y = a cos3 t,				0			y =sin t,		0

x = a (t −sin t ),
16.3.	y = a (1−cost ),																		t				=	π 3.
																					0
			2t +t2
x =											,
							3
			1+t
16.5.									2
			2t −t
y =											,					t0 =1.
				1+t			3
x =t (t cost −2sin t ),
16.7.	y =t (t sin t +2cost ),																								t		=π 4.
																										0
x = 2ln (ctg t )+ctg t,
16.9.																	t0
y = tg t +ctg t,																					=π 4.
x = at cost,
16.11.																	t0 =π 2.
y = at sin t,
																	t
x = arcsin																					,
															1		+t2
16.13.																	1
y = arccos																						,		t		=1.
																									0
															1+t			2
x =				1 +t					,
					t2
16.15.				3							2
y =					2t	2	+						t		,		t0		= 2.
x = a (t sin t +cost ),
16.17.		y = a (sin t −t cost ),																							t		=π	4.
																										0
							2			,
x =1 −t
16.19.																	= 2.
y =t −t3 , t																0

									2		,
x = 2t −t
16.4.	3t −t3 , t																				=1.
y =																			0
																			t
x = arcsin																							,
																1+t2
16.6.																				1
y = arccos																								,	t		= −1.
																										0
																1+t						2
			3at
x =									,
	1+t							2
16.8.		3at2
y =										,						t0					= 2.
	1 +t							2
x =				1	t2 −									1			t4 ,
			2											4
16.10.
			1				2							1						3
y =						t			+									t			, t0 = 0.
			2											3
x =sin t,
16.12.																t0					=π 6.
y = cost,
x =			1+ln t													,
						t2
16.14.				3 +2ln t
y =																				,		t0			=1.
								t

			3
x = a sin				t,
16.16.							t		=π 6.
y = a cos3 t,								0
x =	t +1		,
	t
16.18.	t −1
y =			,		t0		= −1.
	t
					2	),
x = ln (1 +t
16.20.									t		=1.
y =t −arctg t,										0

x =t (1−sin t ),
16.21.
y =t cost, t0	= 0.

	1+t			3
x =						,
	2			1
	t		−
16.22.			t
y =						, t0 = 2.
	t	2	−1

4

x =3cost,

x =t −t

,

16.23.

t0

=π 4.

16.24.

−t3 ,

t

=1.

y = 4sin t,

y =t2

0

3

x =t

+1,

x = 2cost,

16.25.

+t +

1,

t

=1.

16.26.

t0 = −π 3.

y =t2

0

y =sin t,

x = 2 tg t,

x =t3

+1,

16.27.

y = 2sin

2 t

+sin 2t, t

=π 4.

16.28.

,

t

= −2.

0

y =t2

0

x =sin t,

x =sin t,

16.29.

t0

16.30.

t0

y = at ,

= 0.

y = cos 2t,

=π 6.

t

x = 2e

,

16.31.

t

= 0.

y = e−t ,

0

Задача 17. Найти производную n -го порядка.

17.1. y = x eax .

17.2. y =sin 2x +cos(x +1).

17.3. y = 5 e7 x−1 .

17.4. y =

4x +7

.

2x +3

17.5. y = lg (5x +2).

17.6. y = a3x .

17.7. y =

x

.

17.8. y = lg (x +4).

2

(3x +2)

17.9. y =

x.

17.10. y =

2x +5

.

13(3x +1)

17.11.y = 23x+5.

17.13.y = 3 e2 x+1 .

17.15. y = lg (3x +1).

17.12.y =sin (x +1)+cos 2x.

17.14.y = 45+x15+1x .

17.16.y = 75 x.

17.17. y =		x		17.18. y = lg (1 + x).
			.
	9(4x +9)
17.19. y =	4	.		17.20. y =	5x +1
							.
	x				13(2x +3)
17.21. y = a2 x+3.				17.22. y =sin (3x +1)+cos5x.
17.23. y =		e3x+1 .		17.24. y =	11+12x	.
					6x +5

17.25. y = lg (2x

17.27. y = x x+1.

17.29. y =11+− xx .

17.31. y = 32 x+5.

Задача 18.

+7).	17.26.	y = 2kx.
	17.28.	y = log3 (x +5).

17.30. y =	7x +1
		.
	17(4x +3)

Найти производную указанного порядка.

18.1. y =(2x2 −7)ln (x −1),

yV

=?

18.2. y =(3 − x2 )ln2 x,

yIII

=?

18.3. y = x cos x2 ,

yIII =?

18.4. y =

ln (x −1)

,

yIII =?

x −1

18.5. y =

log2 x

,

yIII =?

18.6. y =(4x3 +5)e2 x+1,

yV

=?

3

x

18.7. y = x2 sin (5x −3),

yIII

=?

18.8. y =

ln x

, yIV

=?

x2

18.9. y =(2x +3)ln2 x,

yIII =?

18.10. y =(1+ x2 )arctg x,

yIII =?

18.11. y =

ln x

,

yIV =?

18.12. y =(4x +3) 2−x ,

yV

=?

x3

18.13. y = e1−2 x sin (2 +3x),

yIV

=?

18.14. y =

ln

(3 + x)

,

yIII

=?

3 + x

18.15. y =(2x3 +1)cos x,

yV

=?

18.16. y =(x2 +3)ln

(x −3),

yIV =?

	(					)		e(x−1) 2 ,		yIV =?
18.17. y =		1 − x − x2
18.19. y =(x +7)ln (x +4),										yV	=?
18.21. y =		ln (2x +5)					, yIII =?
			2x +5
18.23. y =		ln x		, yIII			=?
	(		x5					)
18.25. y =										yV	=?
			x2 +3x +1 e3x+2 ,
18.27. y =		ln (x −2)			,			yV	=?
			x −2
y = e−x (cos 2x −3sin 2x),										yIV	=?
18.29. y =(			5x −1)ln2 x,						yIII	=?
18.31. y =(x3 +3)e4 x+3 ,									yIV	=?

18.18. y =	1	sin 2x, yIII =?
	x
18.20. y =(3x −7) 3−x ,			yIV	=?
18.22. y = ex 2 sin 2x,			yIV =?
18.24. y = x ln (1−3x),			yIV	=?
18.26. y =(5x −8) 2−x ,			yIV	=?
18.28.

	18.30. y =	log3 x	, yIV	=?
		x2

Задача 19. Найти производную второго порядка y′xx′ от функции, заданной параметрически.

x = cos 2t,

19.1.

y = 2sec2 t.

x = et cost,

19.3.

y = et sin t.

x =t +sin t,

19.5.

y = 2 −cost.

	t ,
x =
19.7.	1 −t .
y =1

x = tg t,

y =1sin 2t .

	−t	2	,
x = 1

19.2.

y =1t .

x =sh2 t,

y =1ch2 t .

	x =1 t ,
19.6.		(1+t2 ).
	y =1
	x =sin t,
19.8.
	y =sect.
		t −1,
	x =
19.10.		1−t .
	y =t

x =

y =

x =

y =

x =

y =

x =

y =

x =

y =

x =

y =

x =

y =

x =

19.25.

y =

x =

19.27.

y =

x =

19.29.

y =

x =

19.31.

y =

t ,

3 t −1.

t3 −1, ln t.

t −1, 1 t .

t −3, ln (t −2).

t +sin t, 2 +cost.

cost, ln sin t.

et , arcsin t.

ch t,

3 sh2 t.

2(t −sin t ),

4(2 +cost ).

1t2 ,

1 (t2 +1).

ln t, arctg t.

x = cost (1+2cost ),

y =sin t(1 +2cost ).

x =sh t,

y = th2 t.

x = cos2 t,

19.16.

y = tg2 t.

x =sin t,

19.18.

y = ln cost.

x =t −sin t,

19.20.

y = 2 −cost.

x = cost +t sin t,

19.22.

y =sin t −t cost.

x = cost,

19.24. y =sin4 (t2).

x = arctg t,

19.26.

y =t2 2.

x =sin t −t cost,

y = cost +t sin t.

x = cost +sin t,

y =sin 2t.

	Задача 20. Показать, что функция y удовлетворяет уравнению (1).
	y = x e−x2 2 ,						20.2. y =	sin x		,
20.1.	′		2	)				x
	xy =	1 − x			y.	(1)	xy′+ y		= cos x. (1)
		(

y =5e−2 x +ex 3,

20.3.y′+2 y = ex . (1)

20.5.y = x 1− x2 ,

yy′ = x −2x3. (1)

20.7.y = −3x1+c , y′ =3y2 . (1)

y = x2 −cx,

20.9. (x2 + y2 )dx −2xydy = 0. (1)

	20.11.	y = etg(x 2)		,
		y′sin x = y ln y. (1)
		y =	b + x	,
	20.13.
		1+bx

y − xy′ =b(1+ x2 y′). (1)

20.15.y = ln 1+2ex 2 +1,

(1+ex )yy′ = ex . (1)

y = −

2

−1,

20.17.

x2

1+ y2 + xyy′= 0.

(1)

20.19. y = a +

7x

,

ax +1

)

′

(

2

′

y − xy

= a 1+ x

y

.

(1)

y = 4

x +

x +1,

20.21.

′

−1

8xy

− y = y3

x +1 .

(1)

y = 2 +c 1− x2 ,

20.4.(1− x2 )y′+ xy = 2x. (1)

20.6.y = cosc x ,

y′−tg x y = 0. (1) y = ln (c +ex ),

20.8.y′ = ex−y . (1)

y= x(c −ln x),

20.10.(x − y)dx + xdy = 0. (1)

20.12.y =11+− xx ,

y′ =1+ y2 . (1) 1+ x2

y= 3 2 +3x −3x2 ,

20.14.yy′ =1−y2x . (1)

y= tg ln 3x,

20.16. (1+ y2 )dx = xdy. (1)

	20.18.	y = 3 x −ln x −1,
		ln x + y3	−3xy2 y′ = 0. (1)
		y = a tg		a	−1,
	20.20.
				x

a2 + y2 +2x ax − x2 y′ = 0. (1)

20.22.y =(x +1)ex2 ,

y′−2xy = 2x ex2 . (1)