Расчетные задания (Кузнецов) / 02-Дифференцирование

1

a

3

x (3

x +2).

6.21. y =

arctg emx

.

6.22. y =3e

x

2 −2 3

m

ab

b

6.23. y = ln

1+ex

+e2 x −ex −1

sin x

1

.

6.24. y = e

x −

.

1+ex

+e2 x −ex +1

cos x

6.25. y =

ex

(x2 −1)cos x +(x −1)2 sin x .

2

6.26. y = arctg (ex

−e−x ).

6.27. y =3e3 x (3

x5

−53 x4

+20x −60 3 x2

+120 3 x −120).

6.28. y = −

e3x

.

3sh3

x

6.29. y = arcsin e−x

− 1−e2 x .

6.30. y = −

1

e−x2 (x4 +2x2 +2).

2

=ex2

6.31.y 1+ x2 .

Задача 7. Найти производную.

a2 + x2 ).

7.1. y = x ln (

x + x +a )− x +a.

7.2. y = ln (x +

7.3. y = 2

x −4ln (2 + x ).

7.4. y = ln

x2

.

−ax4

1

2

2

7.5. y = ln (

x +

x +1).

7.6. y = ln

a

+ x

.

2

2

a

− x

7.7. y = ln2 (x +cos x).

7.8. y = ln3 (1+cos x).

7.9. y = ln

x2

7.10. y = lntg

π

+

x

.

.

1 − x

2

2

4

7.11. y = ln

4

1+2x

.

7.12. y = x +

1

ln

x −

2

+aπ 2 .

2

2

1−2x

x +

7.13. y = ln sin	2x +4	.	7.14. y = log	log	tg x.
	x +1		16	5

7.15. y = log4 log2 tg x.

7.17. y = ln cos			2x +3				.
			x +1
7.19. y = loga				1				.
			1
				− x4
7.21. y = ln arcsin					1−e2 x .
7.23. y = ln (bx +				a2 +b2 x2 ).
7.25. y = ln						1
	arccos								.
						x
7.27. y = ln	5 + tg				(x 2)
										.
	5 −tg				(x 2)

7.29. y = ln lnsin (1+1 x).

7.31. y = ln ln2 ln3 x.

Задача 8. Найти производную.

	8.1. y =sin 3 +	1 sin2			3x	.
		3 cos 6x

8.3. y = tg lg 1 + 1 sin2 4x . 3 4 cos8x

8.5. y = cossin 5 sin2 2x . 2cos 4x

8.7. y = cos ln 7 sin2 7x . 7 cos14x

8.9. y = ctg (cos 2)+ 1 sin2 6x . 6 cos12x

	8.11. y =	1			1		+		1 sin2 10x
				cos tg								.
		3			2			10 cos 20x

7.16.y = x(cos ln x +sin ln x)2.

7.18.y = lg ln (ctg x).

7.20. y =	1		ln (		2 tg x + 1+2 tg2 x ).
2
7.22. y = ln arccos						1 −e4 x .
7.24. y = ln			x2	+1		+ x	2	.
			x2	+	1	− x	2
7.26. y = ln (ex +					1+e2 x ).

= ln x 7.28. y ln sin (1 x).

7.30. y = ln ln3 ln2 x.

8.2. y = cos ln 2 − 1 cos2 3x . 3 sin 6x

8.4. y = ctg 3 5 − 1 cos2 4x . 8 sin 8x

8.6. y = sin cos3 cos2 2x . 4sin 4x

8.8. y = cos(ctg 2)− 1 cos2 8x . 16 sin16x

8.10. y = 3 ctg 2 −				1 cos2 10x					.
			20					sin 20x
8.12. y = ln sin	1	−		1				cos2 12x			.
	2			24 sin 24x

8.13. y =8sin (ctg 3)

+

1

sin2 5x

.

8.14. y =

cos(ctg 3) cos2 14x

.

5 cos10x

28sin 28x

1

2

15x

1

2

16x

cos tg

sin

sin tg

cos

8.15.

y =

3

.

8.16. y =

7

.

15cos30x

32sin 32x

1

sin

2

17x

2

ctg sin

3

5 ctg 2 cos

18x

8.17.

y =

.

8.18. y =

.

17 cos34x

36sin 36x

8.19.

y =

tg (ln 2) sin2 19x

.

8.20. y = ctg (cos5)−

1

cos2 20x

.

19cos38x

40 sin 40x

8.21. y =

tg 4 +

sin2 21x

.

8.22. y = cos(ln13)−

1

cos2 22x

.

21cos 42x

44 sin 44x

1

sin2 23x

1

1 cos2 24x

8.23.

y = ln cos

+

.

8.24. y = ctg sin

−

.

3

23cos 46x

13

48 sin 48x

8.25. y =sin ln 2 +

sin2 25x

.

8.26. y =

3

cos

2 −

1 cos2 26x

.

25cos50x

52

sin 52x

8.27. y = 7

tg (cos 2)

+

sin2 27x

.

8.28. y =sin 3 tg 2 −

cos2 28x

.

27 cos54x

56sin 56x

8.29. y = cos2 sin 3 +

sin2 29x

.

8.30. y =sin3 cos 2 −

cos2 30x

.

29cos58x

60sin 60x

8.31. y = tg cos(1 3)+

sin2 31x

.

31cos 62x

Задача 9. Найти производную.

9.1. y = arctg

tg x −ctg x

.

9.2. y = arcsin

x −2

.

2

5x

9.3. y =

2x −1

2 + x − x

2 +

9

arcsin

2x −1

.

8

3

4

9.4. y = arctg

1+ x2

−1

.

x

	9.5. y = arccos	x2	−4	.
		x4	+16

9.7. y = 14 ln xx +−11 − 12 arctg x.

9.8. y =

1

2

9.9. y = (1+ x)arctg x

+

1

.

x2

3x x

9.11.

y =

1

+

1+ x

arctg

x.

2

x

2x

9.13. y =

4

+ x4

x2

+

4

arctg

.

x3

2

x

9.15.

y =

1

1

−1 −

arccos x

.

2

x2

2x2

9.17. y =

x −3

6x − x2

−8 +arcsin

x

−1.

2

2

9.6. y =	2	arctg	3x −1	.
	3
			6x

(x −4) 8x − x2 −7 −9arccos

x −1

.

6

9.10. y =

x3

arccos x −

2 + x2

1− x2 .

9

3

9.12. y =

3 + x

x(2 − x)+3arccos

x

.

2

2

9.14. y = arcsin

x

+arctg

x.

x +1

9.16. y = 6arcsin

x

−

6 + x

x(4 − x).

2

2

9.18.

y =

(1+ x)arctg

x −

x .

x

9.19. y =

2 1− x arcsin

x

+

2

.

x

x

9.20. y =

2x −5

5x −4 − x

2 +

9

arcsin

x −1

.

4

3

4

9.21. y = arctg x +

5

ln

x2

+1

.

9.22.

y = arcsin

x −2

.

6

x2

+4

(x −1)

2

9.23. y =

1− x2 − x arcsin 1 − x2 .

9.24. y =

x +

1

arctg

x +

8

arctg

x

.

3

2

3

9.25. y = arctg

1− x

y =(2x

2

+6x +5)arctg

x +1

− x.

.

9.26.

1−

x

x +2

9.27. y =

x

arcsin 2x +

1

ln (1

−4x

2

).

2 1−4x2

8

9.28. y =

2x

2

− x +

1

x2 −1

−

x3

−

3

x.

arctg

2

x 3

2 3

2

	9.29. y =(x +2 x +2)arctg	x	− x.
		x +2

9.30. y = 1	+2x − x2 arcsin	x 2	−	2 ln (1+ x).
	1+ x

9.31. y = arctg tg (x2)+1. 2

Задача 10. Найти производную.

10.1. y =

1

ln

2 +

5 th x

.

10.2. y =

sh x

+

3sh x

+

3

arctg (sh x).

4

5

2 −

4ch4 x

8ch2 x

8

5 th x

1

1+ th x

10.4. y =

3

2 + th x

th x

10.3. y =

ln

−arctg th x.

ln

−

.

2

1−

th x

8 2

2 −th x

4(2 −th2 x)

10.5. y =

1

th x

+

1

ln

1+

2 th x

.

2

4 2

1−

2 th x

10.7. y =

1

ln

a +

1+a2 th x

.

2a 1

a −

1+a2 th x

+a2

10.9. y = arctg

sh 2x

.

ch x −sh x

10.11. y = 4 11+−thth xx .

10.13. y =	ch x	.
	sh 2x

10.15. y =1+8ch2 x ln (ch x). 2ch2 x

10.6. y = −			1								x			−			ch x
						ln th															.
			2								2					2sh		2		x
10.8. y =			1					ln		1	+			2 cth x					.
						2					−
18									1					2 cth x
10.10. y =		1		ln			1		−sh 2x						.
		6					2
									+sh 2x

10.12. y =1+shchx x .

	10.14. y =	sh 3x		.
			ch 6x
	10.16. y = −		12sh2		x +1			.
			3sh		2	x

4ch4 x

10.17. y = −2shch2xx + 32 arcsin (th x).

	1		4	+	8 th		x
10.19. y =		ln					2		.
	8
			4	−	8 th		x
							2

	10.18. y =	1		arcsin						3 +ch x				.
		8
										1 +3ch x
	10.20. y =	1				x			−		1	ln	3 +ch x		.
			ln		th
		4
					2					4			sh x

10.21. y = −14 arcsin 53 ++5ch3ch xx .

10.22. y =1−8ch2 x .

10.23. y =

2

−

1

+

sh x

+

5

arctg (sh x).

sh x

3sh3 x

2ch2 x

2

10.24. y =

8

cth 2x −

1

.

3ch x sh3

x

3

10.25.y = 12 arctg (sh x)− 2shch2x x .

10.27.y = −2shch2x x − sh1x − 32 arctg (sh x).

	1	sh x
10.29. y =							+arctg (sh x) .
	2			2	x
		ch
10.31. y =	2	cth x −					ch x	.
	3
							3sh3 x

Задача 11. Найти производную.

11.1.y =(arctg x)(12)ln(arctg x) .

11.3.y =(sin x)5ex .

11.5. y =(ln x)3x .

11.7. y =(ctg 3x)2ex .

11.9. y =(tg x)4ex .

10.26. y =	3		x		+ch x −	ch x
		ln th							.
	2					2sh	2	x
			2

10.28. y =

sh x

+

1

arctg (sh x).

2ch2 x

2

10.30. y = −

ch x

−

1

x

ln th

.

2sh

2

x

2

2

11.2. y =(sin x )ln(sin x ).

11.4.y =(arcsin x)ex .

11.6.y = xarcsin x .

11.8.y = xetg x .

11.10. y =(cos5x)ex .

11.11. y =(xsin x)8ln(xsin x) . 11.13. y =(x3 +4)tg x .

11.15. y =(x2 −1)sh x .

11.17. y =(sin x)5 x2 .

11.19. y =19x19 x19 .

11.21. y =(sin x )e1 x .

11.23. y = xecos x .

11.25. y = xesin x .

11.27.y = xearctg x .

11.29.y = x29x 29x.

11.31.y = xex x9 .

Задача 12. Найти производную.

12.1. y =	1		(x	2	+8)	x	2	−4	+		x2		2
												arcsin				,
	24										16				x
12.2. y =			4x +1			+		1	arctg				4x +1	.
	16x2 +8x +3							2					2

11.12.y =(x −5)ch x .

11.14.y = xsin x3 .

11.16. y =(x4 +5)ctg x . 11.18. y =(x2 +1)cos x .

11.20. y = x3x 2x.

11.22.y = xectg x .

11.24.y = x2x 5x.

11.26.y =(tg x)ln(tg x)4 .

11.28.y =(x8 +1)th x .

11.30.y =(cos 2x)ln(cos 2 x)4 .

x> 0.

12.3. y = 2x −ln (1+

1−e4 x )−e−2 x arcsin (e2 x ).

12.4. y =

9x2 −12x +5 arctg (3x −2)−ln (3x −2 + 9x2 −12x +5 ).

12.5. y =

2

2x − x

2

+ln

1+

2x − x2

.

x −1

x −1

12.6. y =

x2

3

+

1

(x

2

+

18) x

2

−9,

x > 0.

arcsin

81

x

81

12.7. y =

1

arctg

3x −1

+

1

3x −1

.

2

3

3x2 −2x +1

2

12.8. y =3x −ln (1+

1−e6 x )−e−3x arcsin (e3x ).

12.9. y = ln (4x −1+

16x2 −8x +2 )− 16x2 −8x +2 arctg (4x −1).

12.10. y = ln

1 +

2

−x − x2

+

4

−x − x

2

.

2x +

1

2x +1

12.11. y =(2x +

3)

4

arcsin

1

+

2

(4x

2

+

12x +11)

x

2

+3x +2, 2x +3 > 0.

2x +3

3

12.12. y =

x + 2

+

1

arctg

x +2

.

x

2 +4x +6

2

2

12.13. y =5x −ln (1+

1−e10 x )−e−5 x arcsin (e5 x ).

12.14. y =

x2 −8x +17 arctg (x −4)−ln (x −4 +

x2 −8x +17 ).

12.15. y = ln

1 + −3 +4x − x2

+

2

−3

+

4x − x

2

.

2 − x

2

− x

12.16. y =(3x2 −4x +2)

9x2 −12x

+3 +(3x −2)4 arcsin

1

, 3x −2 > 0.

3x −2

12.17. y =

1

arctg

x −1

+

x −1

.

x

2

−2x +

3

2

2

12.18. y = ln (e5 x +

e10 x −1)+arcsin (e−5 x ).

12.19. y = ln (2x −3 +

4x2 −12x +10 )−

4x2 −12x +10 arctg (2x −3).

12.20. y = ln

1 + −3 −4x − x2

−

2

−3

−

4x − x

2

.

−x

−2

x

+

2

12.21. y =

2

(4x

2

−4x +3)

x

2

− x

+(

2x −1)

4

arcsin

1

, 2x −1 > 0.

3

2x −1

12.22. y =

2x −1

+

1

arctg

2x −1

.

4x2 −4x +3

2

2

12.23. y = arcsin (e−4 x )+ln (e4 x +

e8 x −1).

12.24. y = ln (5x +

25x2 +1)−

25x2 +1arctg 5x.

12.25. y =

2

−3 +12x −

9x

2

+ln

1 + −3 +

12x −9x2

.

3x −

2

3x

−2

12.26. y =(3x +1)4 arcsin

1

+(3x2 + 2x +1)

9x2 +6x, 3x +1 > 0.

3x +1

12.27. y =

1

arctg

2x +1

+

2x +1

.

2

4x2 +4x +3

2

12.28. y = ln (e3x +

e6 x −1)+arcsin (e−3x ).

12.29. y =

49x2 +1arctg 7x −ln (7x + 49x2 +1).

12.30. y =

1

1

−4x

2

+ln

1 +

1+4x2

.

x

2x

e4 x −1).

12.31. y = arcsin (e−2 x )+ln (e2 x +

Задача 13. Найти производную.

13.1. y =

x arcsin x

+ln

1− x

2

y = 4ln

x

−

1−4x2

.

13.2.

.

1 − x2

x2

1 +

1−4x2

13.3. y = x(2x2 +5) x2 +1 +3ln (x + x2 +1).

13.4.

y = x

3

arcsin x +

x2

+

2

1− x

2

.

3

13.5. y =3arcsin

3

+2

4x2 +2x −2,

4x +1 > 0.

4x +1

1+ x2 ).

1+ x2 arctg x −ln (x +

13.6. y =

13.7. y = 2arcsin

2

+

9x2 +24x +12,

3x +

4 > 0.

3x +4

13.8. y = x(2x2 +1)

x2 +1 −ln (x +

x2 +1).

13.9. y = ln (x +

x2 +1)−

1 + x2

. 13.10. y =

1−3x −

2x2 +

3

arcsin

4x +3

.

x

2

17

2

13.11. y =

(

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

4 + x +

1+ x ).

13.12. y = ln

x2

− x +1

+

3 arctg

2x −1

.

x

3

13.13. y =

1

ln

x4

− x2 +1

−

1

arctg

3

.

(

)

2

12

x

2

+

1

2

2 3

2x

−1

13.14. y = 4arcsin

4

+

4x2 +12x −7, 2x +3 > 0.

2x +3

13.15. y = 2arcsin

2

+

9x2 +6x −3,

3x +1 > 0.

3x +1

13.16. y =(2 +3x)

x −1 −

3

arctg

x −1.

2

(

)

1

(x −2)

13.17. y =

x +1 +ln

x +1 +1 .

3

	13.19. y = ln 3	x −1		−	1		1	+
		x +1			2		2

13.18. y =	x	2	+1	−	1	ln	x2 +1 − x	.
					2		x2 +1 +1

1arctg x.

x2 −1

			13.20. y = x ln ( 1− x +			1+ x )+				1	(arcsin x − x).
									2
	13.21. y = arctg x2 −1 −	ln x	.	13.22. y =3arcsin		3		+	x2 +4x −5.
						x +2
		x2 −1
	13.23. y = (3 − x)(2 + x)+5arcsin			x +2	.
				5
			13.24. y = x(arcsin x)2 +2				1− x2 arcsin x −2x.