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Реферат: 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'= -2xln2 x + 2lnx(3-x2 )

x

y''= -2ln2 x-4xlnx +-4x2 lnx+2(3-x2 )-2(3-x2 )lnx =

x x2

= -2ln2 x–4lnx - 2x2 lnx+6lnx+2x2 -6

x2

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

x x4

= 12lnx-4x2 lnx-6x2 -18

x3

18.3.

y'= cosx2 -2x2 sinx2

y''= -2xsinx2 -4xsinx2 -4x3 cosx2 = -6xsinx2 – 4x3 cosx2

y'''= -6sinx2 -12x2 cosx2 -12x2 cosx2 +8x4 sinx2 = 8x4 sinx2 -6sinx2 -24x2 cosx2

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-3x2 log2 x = 1-3ln2log2 x

x6 ln2 x4 ln2

y''= -3x3 -4x3 (1-3ln2log2 x) = 12ln2log2 x-7

x8 ln2 x5 ln2

y'''= 12x4 +5x4 (12ln2log2 x-7) = 60ln2log2 x-23

x10 ln2 x6 ln2

18.6.

y'= 12x2 e2x+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)+5x2 cos(5x-3)

y''= 2sin(5x-3)+10xcos(5x-3)+10xcos(5x-3)-25x2 sin(5x-3) = 2sin(5x-3)+20xcos(5x-3)-

-25x2 sin(5x-3)

y'''= 10cos(5x-3)+20cos(5x-3)-100xsin(5x-3)-50xsin(5x-3)-125x2 sin(5x-3)= 30cos(5x-3)-

-150xsin(5x-3)-125x2 sin(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'= 2ln2 x+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 -3x2 lnx = 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-x lnx

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

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

y''''= -2-x ln3 2(12-4xln2-ln2)-4*2-x ln3 2= 2-x ln3 2(4xln2+ln2-16)

y'''= -2-x ln4 2(4xln2+ln2-16)+4*2-x ln4 2= 2-x ln4 2(ln2-4xln2-12)

18.13.

y'= -2e1-2x sin(2+3x)+3e1-2x cos(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-2x3 sinx-sinx

y''= 12xcosx-12x2 sinx-2x3 cosx-cosx

y'''= 12cosx-36xsinx-18x2 cosx+2x3 sinx+sinx

y''''= 2x3 cosx+18x2 sinx+6xsinx+cosx-48sinx-72xcosx

y'''''= 24x2 cosx-2x3 sinx+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-4x2 sin2x-2sin2x

x4 x3

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

x6

= 6x2 sin2x-8x2 cos2x-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-x ln3= 3-x (3-3xln3+7ln3)

y''= -3-x ln3(3-3xln3+7ln3)+3*3-x ln3= 3-x ln2 3(7-3x)

y'''= -3-x ln3 3(7-3x)-3*3-x ln2 3= 3-x ln2 3(3xln3-7ln3-3)

y'''= -3-x ln3 3(3xln3-7ln3-3)+3*3-x ln3 3= 3-x ln3 3(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/2 sin2x+2ex/2 cos2x= 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 -5x4 lnx = 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-x ln2(5x-8)+5*2-x = 2-x (5-5xln2+8ln2)

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

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

y''''= -2-x ln2 2(-13ln2+10+5xln2)+5*2-x ln2 2= 2-x ln2 2(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'= 5ln2 x+2lnx(5x-1)

x

y''= 10lnx/x+2(5x-1)+2xlnx(5x-1) = 10x2 lnx+8xlnx+10x-2

x2 x2

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

x4 x2

18.30.

y'= 1-2ln3log3 x

x3 ln3

y'' = -2x2 -3x2 (1-2ln3log3 x) = -5-6ln3log3 x

x6 ln3 x4 ln3

y''' = -6x3 +4x3 (5+6ln3log3 x) = 14+24ln3log3 x

x8 ln3 x5 ln3

y'''' = 24x4 -5x4 (14+24ln3log3 x) = -46-120ln3log3 x

x10 ln3 x6 ln3

18.31.

y'= 3x2 e4x+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)