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 e^2x+1 +2(4x³ +5)e^2x+1 = (8x³ +12x² +10)e^2x+1

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

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

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

y'''''= 16((18x² +48x+36)e^2x+1 +2(6x³ +24x² +36x+17)e^2x+1 )= 16(12x³ +72x² +120x+70)e^2x+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'= -2e^1-2x sin(2+3x)+3e^1-2x cos(2+3x)= e^1-2x (3cos(2+3x)-2sin(2+3x))

y''= -2e^1-2x (3cos(2+3x)-2sin(2+3x))+e^1-2x (-9sin(2+3x)-6cos(2+3x))= e^1-2x (-12cos(2+3x)-5sin(2+3x))

y'''= -2e^1-2x (-12cos(2+3x)-5sin(2+3x))-e^1-2x (-36sin(2+3x)+15cos(2+3x))= e^1-2x (46cos(2+3x)+9sin(2+3x))

y''''= -2e^1-2x (46cos(2+3x)+9sin(2+3x))+e^1-2x (-27sin(2+3x)+138cos(2+3x))= e^1-2x (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*e^x/2 sin2x+2e^x/2 cos2x= e^x/2 /2*(sin2x+4cos2x)

y''= e^x/2 /4*(sin2x+4cos2x)+ e^x/2 /2(2cos2x-8sin2x)= e^x/2 /4*(-15sin2x+8cos2x)

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

y''''= e^x/2 /16*(-45sin2x-24cos2x)+ e^x/2 /8(-90cos2x+48sin2x)= e^x/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'= 3e^3x+2 (x² +3x+1)+e^3x+2 (2x+3)= e^3x+2 (3x² +11x+6)

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

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

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

y'''''= 3e^3x+2 (81x² +459x+513)+e^3x+2 (162x+459)= e^3x+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 e^4x+3 +4e^4x+3 (x³ +3)= e^4x+3 (4x³ +3x² +12)

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

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

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