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 − x2

−x − 2

x

+ 2

12.21. y =

2

(4x2

− 4x + 3)

+ (

2x −1)4 arcsin

1

, 2x −1 > 0.

x2 − x

3

2x −1

12.22. y =

2x −1

+

1

arctg

2x

−

1.

4x2 − 4x + 3

2

2

12.23.y = arcsin (e−4x )+ ln(e4x + e8x −1).

12.24.y = ln(5x + 25x2 +1)− 25x2 +1arctg5x.

12.25. y =

2

+ ln

1+ −3 +12x − 9x2

.

−3 +12x − 9x2

3x − 2

3x − 2

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

1

+ (3x2 + 2x +1)

, 3x +1 > 0.

9x2

+ 6x

3x +1

12.27. y =

1

arctg 2x

+

1

+

2x +1

.

4x

2 + 4x + 3

2

2

12.28.y = ln(e3x + e6x −1)+ arcsin(e−3x ).

12.29.y = 49x2 +1arctg7x − ln(7x + 49x2 +1).

	12.30. y =	1			+ ln	1+ 1+ 4x2	.
				1− 4x2
		x
						2x

12.31. y = arcsin (e−2x )+ ln(e2x + e4x −1).

Задача 13. Найти производную.

.

13.1. y =

x

arcsin

x

+ ln

.

13.2. y = 4ln

x

−

1− 4x2

1− x2

1− x2

1

+ 1− 4x2

x2

13.3. y = x(2x2 + 5)

+ 3ln(x +

).

x2 +1

x2 +1

47

x2 + 2

13.4. y = x3 arcsin x +

1- x2 .

3

3

13.5. y = 3arcsin

+ 2

4x2 + 2x - 2,

4x +1 > 0.

4x +1

13.6. y =

arctg x - ln(x +

).

1+ x2

1+ x2

2

13.7. y = 2arcsin

+

9x2 + 24x +12,

3x + 4 > 0.

3x + 4

13.8. y = x(2x2 +1)

- ln(x +

).

x2 +1

x2 +1

13.9. y = ln(x +

)-

+

1+ x2

3

arcsin

4x

3.

x2 +1

. 13.10. y = 1- 3x - 2x2 +

x

2

2

17

13.11. y = (4 + x)(1+ x) + 3ln (4 + x + 1+ x ).

13.12. y = ln

x2

- x +1

+

2x -1

3arctg

.

x

3

1

x4 - x2 +1

1

13.13. y =

ln

-

arctg

3

.

12

(x2 +1)2

2

2x2 -1

3

4

13.14. y = 4arcsin

+

4x2 +12x - 7, 2x + 3 > 0.

2x + 3

2

13.15. y = 2arcsin

+

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

3x +1

13.16. y = (2 + 3x)

-

3

arctg

x -1

x -1.

2

13.17. y =

1

(x - 2)

+ ln(

+1).

x +1

x +1

3

1 ln

x2 +1

- x

.

13.18. y =

x2 +1 -

2

x2 +1 +1

1

1

1

x -1

æ

ö

13.19. y = ln

3

-

ç

+

÷arctg x.

x +1

2

2

x

2

-1

è

ø

48

13.20. y = xln (1- x + 1+ x )+ 12 (arcsin x - x).

				ln x				3
13.21. y = arctg	x2 -1	-				.	13.22. y = 3arcsin		+ x2 + 4x - 5.
								x + 2
				x2 -1

13.23.y = (3 - x)(2 + x) + 5arcsin x +5 2.

13.24.y = x(arcsin x)2 + 21- x2 arcsin x - 2x.

13.25. y =

1- x2

+ arcsin x.

x2 + 2

13.26. y = x2 arccos x -

1

- x2 .

x

3

x2

+ 2

1

+ x2 + 2

13.27. y =

-

ln

2

.

x2

x

2

x

(10 - x2 )

+ 6arcsin

x

.

13.28. y =

4 - x2

4

2

1

13.29. y = arcsin

+ 2 x2

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

2x + 3

13.30. y = xarcsin

x

-

+ arctg

.

x

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 = xcosα + sinα lnsin(x -α ).

sinα

14.3. y =		1			ésinln x - (
						2
2		2 ë

14.5. y = 3	sin x	+ 2	sin x	.
	cos2 x
			cos4 x

-1)×cosln xùû x2+1.

æ		cos x ö
14.4. y = arctgç					÷.
è	4		cos2x ø

	æ							ö
14.6. y = (a2 + b2 )−1 2			a	2	+ b	2	sin x
	×arcsinç							÷.
					b
	ç							÷
	è							ø

49

14.7. y =	7x (3sin3x + cos3x ×ln 7)		14.8. y = ln	sin x
		.					.
	9 + ln2 7			cos x +
					cos2x

		1		é
14.9. y =					êarctg(acos x) +
	a(1	+ a	2
				) ë

14.11. y = (1+ x2 )earctg x .

1

2xsin

α

14.13. y =

arctg

2

.

α

1- x2

2sin 2

14.15. y =

6x (sin 4x ×ln6 - 4cos4x)

16 + ln2 6

14.17. y = arctg

2sin x

.

9cos2 x - 4

+ th x

.

14.19. y = ln

2

2 - th x

14.21. y =

4x (ln 4 ×sin 4x - 4cos4x)

16 + ln2 4

14.23. y =

5x (sin3x ×ln5 - 3cos3x)

.

9 + ln2 5

alntg

x

ù

ú.

2

û

14.10. y = -

1

-

1

+

1

ln

1+ sin x

.

3sin3 x

sin x

2

1- sin x

14.12. y =

ctg x + x

.

1- xctg x

x4

+1 - x2

14.14. y = arctg

, x > 0.

x

. 14.16. y = arctg 1-2tgtg xx .

14.18. y = 5x (2sin 2x + cos2x ×ln5). 4 + ln2 5

14.20. y = 3x (4sin 4x + ln3×cos4x). 16 + ln2 3

.	14.22. y =	cos x	- 2cos x - 3lntg				x	.
		sin2 x
					2
	14.24. y = x - ln(1+ ex )			- 2e−	x				x
					2	arctge			2	.

14.25. y = 2x (sin x + cos x ×ln 2). 1+ ln2 2

14.27. y = 2 sincos4 xx + 3sincos2 xx .

14.28.

14.29. y = 3x (ln3×sin 2x - 2cos2x). ln2 3 + 4

14.26. y = ln (ctg x + ctgα ). sinα

y =	cos x			4								2tg		(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

50