17.3. y = 5e7 x−1 .

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

17.7. y =			x
				.
	2(3x + 2)
17.9. y =			.
		x

17.11.	y = 23x+5.
	y = 3		.
17.13.		e2x+1
17.15.	y = lg(3x +1).

17.17. y =			x
					.
		9	(4x + 9)
17.19. y =		4	.
		x
17.21. y = a2x+3.
17.23. y =				.
			e3x+1

17.25. y = lg(2x + 7).

	17.27. y =		x		.
		x +1
	17.29. y =	1	+ x	.
		1
			− x
	17.31. y = 32x+5.

17.4. y = 42xx ++ 73.

17.6. y = a3x .

17.8. y = lg(x + 4).

17.10. y = 2x + 5 .

13(3x +1)

17.12. y = sin(x +1) + cos2x.

17.14. y = 4 +15x .

5x +1

17.16. y = 75x.

17.18. y = lg(1+ x).

17.20. y =	5x +1
		.
	13(2x + 3)

17.22. y = sin (3x +1) + cos5x.

17.24. y = 11+12x .

6x + 5

17.26. y = 2kx.

17.28. y = log3 (x + 5).

17.30. y = 7x +1 .

17(4x + 3)

Задача 18. Найти производную указанного порядка.

18.1. y = (2x2 − 7)ln (x −1), yV = ?

18.2. y = (3 − x2 )ln2 x, yIII = ?

55

18.3. y = xcos x2 , yIII

= ?

18.5. y =

log2 x

,

yIII = ?

x3

18.7. y = x2 sin(5x - 3),

yIII

= ?

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

yIII = ?

18.11. y =

ln x

,

yIV

= ?

x3

18.13. y = e1−2x ×sin(2 + 3x),

yIV

= ?

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

yV = ?

18.17. y = (1- x - x2 )e(x−1) 2 ,

yIV

= ?

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 = (x2 + 3x +1)e3x+2 ,

yV

= ?

18.27. y =

ln(x - 2)

,

yV

= ?

x - 2

18.29. y = (5x -1)ln2 x,

yIII

= ?

18.31. y = (x3 + 3)e4x+3 ,

yIV

= ?

18.4. y =

ln (x -1)

,

yIII = ?

x -1

18.6. y = (4x3 + 5)e2x+1,

yV

= ?

18.8. y =

ln x

, yIV

= ?

x2

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

yIII = ?

18.12. y = (4x + 3)× 2− x ,

yV

= ?

18.14. y =

ln (3 + x)

,

yIII

= ?

3 + x

18.16. y = (x2 + 3)ln(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 = xln(1- 3x),

yIV

= ?

18.26. y = (5x - 8)× 2− x ,

yIV

= ?

18.28. y = e− x ×(cos2x - 3sin 2x),

yIV = ?

18.30. y =

log3 x

,

yIV

= ?

x2

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

56

ìx = cos2t,

19.1. í

îy = 2sec2 t.

ìïx = et cost,

19.3. í

ïîy = et sin t.

ìx = t + sint,

19.5. í

îy = 2 - cost.

ìïx = t ,

19.7. í

ïîy =1 1- t .

ìx = tgt,

19.9.í

îy =1sin 2t.

ïx = t ,

19.11.í

ïîy = 3t -1.ì

ì

ïx = t3 -1,

19.13. í

ïîy = lnt.

ìïx = t -1,

19.15. í

ïîy =1 t .

ïx = t - 3,

19.17.í

ïîy = ln(t - 2).ì

ìx = t + sint,

19.19. í

îy = 2 + cost.

ìx = cost,

19.21. í

îy = lnsint.

ìx = et ,

19.23.í

îy = arcsint.

ì

ïx = 1- t2 ,

19.2. í

ïîy =1t.

ìïx = sh2 t,

19.4. í

ïîy =1 ch2 t.

ìx =1 t ,

ï

19.6. íïîy =1 (1+ t2 ).

ìx = sint,

19.8. í

îy = sect.

ïx = t -1,

19.10.í

ïîy = t1- t .ì

ìx = cost (1+ 2cost ),

ï

19.12. í

ïîy = sint (1+ 2cost).

ìx = sht,

19.14. í

îy = th2 t.

ìïx = cos2 t,

19.16. í

ïîy = tg2 t.

ìx = sint,

19.18. í

îy = lncost.

ìx = t - sint,

19.20. í

îy = 2 - cost.

ìx = cost + t sint,

19.22. í

îy = sint - t cost.

ìïx = cost,

19.24. í

ïîy = sin4 (t2).

57

ìx = cht,

ï

19.25. í

ïîy = 3 sh2 t.

ìx = 2(t - sint),

ï

19.27. í

ïîy = 4(2 + cost ).

ìx =1 t2 ,

ï

19.29. íïîy =1 (t2 +1).

ìx = lnt,

19.31. í

îy = arctgt.

Задача 20. Показать,

ìx = arctgt,

19.26. í

îy = t2 2.

ìx = sint - t cost,

19.28.í

îy = cost + t sint.

ìx = cost + sin t,

19.30. í

îy = sin 2t.

что функция y удовлетворяет уравнению (1).

y = xe−x22 ,

20.1.	- x2 ) y. (1)
xy¢ = (1

y = 5e−2x + ex 3,

20.3.y¢ + 2y = ex . (1)

20.5.y = x1- x2 ,

yy¢ = x - 2x3. (1)

20.7.y = - 3x1+ c , y¢ = 3y2. (1)

y= x2 - cx,

20.9.(x2 + y2 )dx - 2xydy = 0. (1)

y = etg(x2) ,

20.11.y¢sin x = y ln y. (1)

20.2.y = sinx x ,

xy¢ + y = cos x. (1)

y= 2 + c1- 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

58

y =

b + x

,

20.13.

1+ bx

y - xy¢ = b(1+ x2 y¢).

(1)

y =

æ

1+ e

x

ö2

+1,

ln ç

÷

20.15.

è

2

ø

(1+ ex )yy¢ = ex .

(1)

y = -

2

-1,

20.17.

x2

1+ y2 + xyy¢ = 0.

(1)

20.19. y = a +

7x

,

ax +1

y - xy¢ = a(1+ x2 y¢).

(1)

y = 4

+

x

x +1,

20.21.

¢

-1

8xy

- y = y3

x +1.

(1)

20.23.y = x32+x 1 + x ,

x(x3 +1)y¢ + (2x3 -1)y =1

y = 32 + 3x - 3x2 ,

20.14. yy¢ = 1- 2x . (1)

y

y= tgln3x,

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

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

a2 + y2 + 2xax - x2 y¢ = 0. (1)

y = (x +1)ex2 ,

20.22.

y¢ - 2xy = 2xex2 . (1)

			y = ex+x2	+ 2ex ,
x3 - 2			20.24.
	.	(1)	y¢ - y = 2xex+x2		. (1)
x

y = −xcos x + 3x,

y =1

sin x + x

,

20.26. 2sin x × y

¢

+ y cos x =

20.25. xy¢ = y + x2 sin x. (1)

= y3 (xcos x - sin x). (1)

x

2

x

20.27. y =

+ x

,

20.28. y =

,

x -1

cos x

x(x -1) y¢ + y = x2 (2x -1). (1)

y¢ - y tg x = sec x. (1)

59

y = (x +1)n (ex -1),

20.29.	y¢ -	ny		= e	x	(1+ x)	n	. (1)
		x +1

20.31.y = -x4 - x2 , xyy¢ - y2 = x4. (1)

y= 2 sinx x + cos x,

20.30.xsin x × y¢ + (sin x - xcos x) y = = sin x ×cos x - x. (1)

60