Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.18

Задача 18. Найти производную указанного порядка.

18.1.

y'= 4xln(x-1)+(2x2-7)/(x-1)

y''= 4ln(x-1)+ 4x/(x-1) + 4x(x-1)-2x2+7 = 4ln(x-1) + 6x2-8x+7

(x-1)2 (x-1)2

y'''= 4/(x-1) + (12x-8)(x-1)2-2(x-1)( 6x2-8x+7) = 4x2-12x-2

(x-1)4 (x-1)3

y''''= (8x-12)(x-1)3-3(x-1)2(4x2-12x-2) = -4x2+16x+18

(x-1)6 (x-1)4

y'''''= (-8x+16)(x-1)4-4(x-1)3(-4x2+16x+18) = 8x2-40x-88

(x-1)8 (x-1)5

18.2.

y'= -2xln2x + 2lnx(3-x2)

x

y''= -2ln2x-4xlnx+-4x2lnx+2(3-x2)-2(3-x2)lnx =

x x2

= -2ln2x–4lnx - 2x2lnx+6lnx+2x2-6

x2

y'''= -4lnx – 4/x – (4xlnx+2x+6/x+4x)x2-2x(2x2lnx+6lnx+2x2-6) =

x x4

= 12lnx-4x2lnx-6x2-18

x3

18.3.

y'= cosx2-2x2sinx2

y''= -2xsinx2-4xsinx2-4x3cosx2 = -6xsinx2 – 4x3cosx2

y'''= -6sinx2-12x2cosx2-12x2cosx2+8x4sinx2 = 8x4sinx2-6sinx2-24x2cosx2

18.4.

y'= √(x-1)/(x-1) – ln(x-1)/2√(x-1) = 2-ln(x-1)

x-1 2(x-1)3/2

y''= -2√(x-1)-3√(x-1)(2-ln(x-1)) = 3√(x-1)ln(x-1)-8√(x-1)

4(x-1)3 4(x-1)3

y'''=((3ln(x-1))/(2√(x-1))+3√(x-1)/(x-1)-8/2√(x-1))(x-1)3-3(x-1)2(3√(x-1)ln(x-1)-8√(x-1)) =

4(x-1)6

= 46√(x-1)-15√(x-1)ln(x-1)

8(x-1)4

18.5.

y'= x2/lnx-3x2log2x = 1-3ln2log2x

x6 ln2 x4 ln2

y''= -3x3-4x3(1-3ln2log2x) = 12ln2log2x-7

x8 ln2 x5 ln2

y'''= 12x4+5x4(12ln2log2x-7) = 60ln2log2x-23

x10 ln2 x6 ln2

18.6.

y'= 12x2e2x+1+2(4x3+5)e2x+1= (8x3+12x2+10)e2x+1

y''= (24x2+24x) e2x+1+2(8x3+12x2+10)e2x+1=(16x3+48x2+24x+20) e2x+1

y'''= (48x2+96x+24)e2x+1+2(16x3+48x2+24x+20)e2x+1= 16(2x3+9x2+9x+4)e2x+1

y''''= 16((6x2+18x+9)e2x+1+2(2x3+9x2+9x+4)e2x+1)= 16(6x3+24x2+36x+17)e2x+1

y'''''= 16((18x2+48x+36)e2x+1+2(6x3+24x2+36x+17)e2x+1)= 16(12x3+72x2+120x+70)e2x+1

18.7.

y'= 2xsin(5x-3)+5x2cos(5x-3)

y''= 2sin(5x-3)+10xcos(5x-3)+10xcos(5x-3)-25x2sin(5x-3) = 2sin(5x-3)+20xcos(5x-3)-

-25x2sin(5x-3)

y'''= 10cos(5x-3)+20cos(5x-3)-100xsin(5x-3)-50xsin(5x-3)-125x2sin(5x-3)= 30cos(5x-3)-

-150xsin(5x-3)-125x2sin(5x-3)

18.8.

y'= x-2xlnx = 1-2lnx

x4 x3

y''= -2x2-3x2(1-2lnx) = -5+6lnx

x6 x4

y'''= 6x3-4x3(6lnx-5) = 26-24lnx

x8 x5

y''''= -24x4-5x4(26-24lnx) = 120lnx-154

x10 x6

18.9.

y'= 2ln2x+2lnx(2x+3)

x

y''= 4lnx/x+ 2(2x+3)+4xlnx-4xlnx-6lnx = 4xlnx+4x+6-6lnx

x2 x2

y'''= (4lnx+8-6/x)x2-2x(4xlnx+4x+6-6lnx) = 12lnx-4xlnx-18

x4 x3

18.10.

y'= 2xarctgx+1

y''= 2arctgx+2x/(1+x2)

y'''= 2/(1+x2)+ 2(1+x2)-4x2 = 4/(1+x2)2

(1+x2)2

18.11.

y'= x2-3x2lnx = 1-3lnx

x6 x4

y''= -3x3-4x3(1-3lnx) = -7+12lnx

x8 x5

y'''= 12x4-5x4(12lnx-7) = -23-60lnx

x10 x6

y''''= -60x5+6x5(60lnx+23) = 360lnx+78

x12 x7

18.12.

y'= 4*2-x-(4x+3)2-xlnx

y''= -4ln2*2-x-4ln2*2-x+(4x+1)ln22*2-x= 2-xln2(4xln2+ln2-8)

y'''= -2-xln22(4xln2+ln2-8)+4*2-xln22= 2-xln22(12-4xln2-ln2)

y''''= -2-xln32(12-4xln2-ln2)-4*2-xln32= 2-xln32(4xln2+ln2-16)

y'''= -2-xln42(4xln2+ln2-16)+4*2-xln42= 2-xln42(ln2-4xln2-12)

18.13.

y'= -2e1-2xsin(2+3x)+3e1-2xcos(2+3x)= e1-2x(3cos(2+3x)-2sin(2+3x))

y''= -2e1-2x(3cos(2+3x)-2sin(2+3x))+e1-2x(-9sin(2+3x)-6cos(2+3x))= e1-2x(-12cos(2+3x)-5sin(2+3x))

y'''= -2e1-2x(-12cos(2+3x)-5sin(2+3x))-e1-2x(-36sin(2+3x)+15cos(2+3x))= e1-2x(46cos(2+3x)+9sin(2+3x))

y''''= -2e1-2x(46cos(2+3x)+9sin(2+3x))+e1-2x(-27sin(2+3x)+138cos(2+3x))= e1-2x(120cos(2+3x)-119sin(2+3x))

18.14.

y'= 1-ln(3+x)

(3+x)2

y''= -(3+x)-2(3+x)(1-ln(3+x)) = -3+2ln(3+x)

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

y'''= 2(3+x)2-3(3+x)2(2ln(3+x)-3) = 11-6ln(3+x)

(3+x)6 (3+x)4

18.15.

y'= 6x2cosx-2x3sinx-sinx

y''= 12xcosx-12x2sinx-2x3cosx-cosx

y'''= 12cosx-36xsinx-18x2cosx+2x3sinx+sinx

y''''= 2x3cosx+18x2sinx+6xsinx+cosx-48sinx-72xcosx

y'''''= 24x2cosx-2x3sinx+108xsinx+6xcosx+5sinx-120cosx

18.16.

y'= 2xln(x-3)+x2+3

x-3

y''= 2ln(x-3) + 2x/(x-3) + 2x2-6x-x2-3 = 2ln(x-3) + 3x2-12x-3

(x-3)2 (x-3)2

y'''= 2/(x-3) + (6x-12)(x-3)-2(x-3)( 3x2-12x-3) = -4x2+18x+12

(x-3)4 (x-3)3

y''''= (-8x+18)(x-3)3-2(x-3)2(-4x2+18x+12) = 6x-78

(x-3)6 (x-3)4

18.17.

y'= 1/2*e(x-1)/2(1-x-x2)+ e(x-1)/2(-1-2x)= 1/2* e(x-1)/2(-1-5x-x2)

y''= 1/4*e(x-1)/2(-1-5x-x2)+ 1/2*e(x-1)/2(-5-2x)= 1/4* e(x-1)/2(-11-9x-x2)

y'''= 1/8*e(x-1)/2(-11-9x-x2)+ 1/4*e(x-1)/2(-9-2x)= 1/8* e(x-1)/2(-29-13x-x2)

y''''= 1/16*e(x-1)/2(-29-13x-x2)+ 1/8*e(x-1)/2(-13-2x)= 1/16* e(x-1)/2(-55-17x-x2)

18.18.

y'= 2xcos2x+sin2x

x2

y''= (2cos2x-4xsin2x+cos2x)x2-2x(2xcos2x+sin2x) = -xcos2x-4x2sin2x-2sin2x

x4 x3

y'''= (-cos2x+2xsin2x-8xsin2x-8x2cos2x-4cos2x)x3-3x2(-xcos2x-4x2sin2x-2sin2x) =

x6

= 6x2sin2x-8x2cos2x-2xcos2x+6sin2x

x4

18.19.

y'= ln(x+4) +(x+7)/(x+4)

y''= x+4+x+4-x-7 = x+1

(x+4)2 (x+4)2

y'''= (x+4)2-2(x+1)(x+4) = 2-x

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

y''''= -(x+4)3-3(x+4)2(2-x) = 2x-10

(x+4)6 (x+4)4

y'''''= 2(x+4)4-4(x+4)3(2x-10) = 48-6x

(x+4)8 (x+4)5

18.20.

y'= 3*3-x-(3x-7)3-xln3= 3-x(3-3xln3+7ln3)

y''= -3-xln3(3-3xln3+7ln3)+3*3-xln3= 3-x ln23(7-3x)

y'''= -3-xln33(7-3x)-3*3-xln23= 3-x ln23(3xln3-7ln3-3)

y'''= -3-xln33(3xln3-7ln3-3)+3*3-xln33= 3-x ln33(3xln3-7ln3+6)

18.21.

y'= 1-2ln(2x+5)

(2x+5)2

y''= -2(2x+5)-2(2x+5)( 1-2ln(2x+5)) = -4+4ln(2x+5)

(2x+5)4 (2x+5)3

y'''= 4(2x+5)2-3(2x+5)2(4ln(2x+5)-4) = -8-12ln(2x+5)

(2x+5)6 (2x+5)4

18.22.

y'= 1/2*ex/2sin2x+2ex/2cos2x= ex/2/2*(sin2x+4cos2x)

y''= ex/2/4*(sin2x+4cos2x)+ ex/2/2(2cos2x-8sin2x)= ex/2/4*(-15sin2x+8cos2x)

y'''= ex/2/8*(-15sin2x+8cos2x)+ ex/2/4(-16cos2x-30sin2x)= ex/2/8*(-45sin2x-24cos2x)

y''''= ex/2/16*(-45sin2x-24cos2x)+ ex/2/8(-90cos2x+48sin2x)= ex/2/16*(51sin2x-204cos2x)

18.23.

y'= x4-5x4lnx = 1-5lnx

x5 x

y''= -5-1+5lnx = 5lnx-6

x2 x2

y'''= 5x-2x(5lnx-6) = 17-10lnx

x4 x3

18.24.

y'= ln(1-3x)-3x/(1-3x)

y''= 1/(1-3x) – 3(1-3x)+9x = -3x-2

(1-3x)2 (1-3x)2

y'''= -3(1-3x)2+2(1-3x)(3x+2) = 15x+1

(1-3x)4 (1-3x)3

y''''= 15(1-3x)3-3(1-3x)2(15x+1) = 12-90x

(1-3x)6 (1-3x)4

18.25.

y'= 3e3x+2(x2+3x+1)+e3x+2(2x+3)= e3x+2(3x2+11x+6)

y''= 3e3x+2(3x2+11x+6)+e3x+2(6x+11)= e3x+2(9x2+39x+29)

y'''= 3e3x+2(9x2+39x+29)+e3x+2(18x+39)= e3x+2(27x2+135x+126)

y''''= 3e3x+2(27x2+135x+126)+e3x+2(54x+135)= e3x+2(81x2+459x+513)

y'''''= 3e3x+2(81x2+459x+513)+e3x+2(162x+459)= e3x+2(243x2+1539x+1998)

18.26.

y'= -2-xln2(5x-8)+5*2-x= 2-x(5-5xln2+8ln2)

y''= -2-xln2(5-5xln2+8ln2)-5*2-xln2= 2-xln2(8ln2-10-5xln2)

y'''= -2-xln2(5-5xln2+8ln2)-5*2-xln22= 2-xln2(-13ln2+10+5xln2)

y''''= -2-xln22(-13ln2+10+5xln2)+5*2-x ln22= 2-x ln22(13ln2-5-5xln2)

18.27.

y'= 1-ln(x-2)

(x-2)2

y''= -(x-2)2-2(x-2)(1-ln(x-2)) = -x+2ln(x-2)

(x-2)4 (x-2)3

y'''= 2(x-2)2-(x-2)3-3(x-2)2(-x+2ln(x-2)) = 2x+4-6ln(x-2)

(x-2)6 (x-2)4

y''''= 2(x-2)4-6(x-2)3-4(x-2)3(2x+4-6ln(x-2)) = 24ln(x-2)-6x+14

(x-2)8 (x-2)5

y'''''= 24(x-2)4-6(x-2)5-5(x-2)4(24ln(x-2)-6x+14) = 24x-34-120ln(x-2)

(x-2)10 (x-2)6

18.28.

y'= -e-x(cos2x-3sin2x)+e-x(-2sin2x-6cos2x)= e-x(sin2x-7cos2x)

y''= -e-x(sin2x-7cos2x)+e-x(14sin2x+2cos2x)= e-x(13sin2x+9cos2x)

y'''= -e-x(13sin2x+9cos2x)+e-x(-18sin2x+26cos2x)= e-x(-31sin2x+15cos2x)

y''''= -e-x(-31sin2x+15cos2x)+e-x(-30sin2x-62cos2x)= e-x(sin2x-77cos2x)

18.29.

y'= 5ln2x+2lnx(5x-1)

x

y''= 10lnx/x+2(5x-1)+2xlnx(5x-1) = 10x2lnx+8xlnx+10x-2

x2 x2

y'''= 20x3lnx+10x3+8x2lnx+8x2+10x2 = 20xlnx+10x+8lnx+18

x4 x2

18.30.

y'= 1-2ln3log3x

x3ln3

y'' = -2x2-3x2(1-2ln3log3x) = -5-6ln3log3x

x6ln3 x4ln3

y''' = -6x3+4x3(5+6ln3log3x) = 14+24ln3log3x

x8ln3 x5ln3

y'''' = 24x4-5x4(14+24ln3log3x) = -46-120ln3log3x

x10ln3 x6ln3

18.31.

y'= 3x2e4x+3+4e4x+3(x3+3)= e4x+3(4x3+3x2+12)

y''= 4e4x+3(4x3+3x2+12)+e4x+3(12x2+6x)= e4x+3(16x3+24x2+6x+12)

y'''= 4e4x+3(16x3+24x2+6x+12)+e4x+3(48x2+48x+6)= e4x+3(64x3+144x2+72x+54)

y''''= 4e4x+3(64x3+144x2+72x+54)+e4x+3(192x2+288x+72)= e4x+3(256x3+768x2+576x+288)