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)+(2×2-7)/(x-1) y”= 4ln(x-1)+ 4x/(x-1) + 4x(x-1)-2×2+7 = 4ln(x-1) + 6×2-8x+7 (x-1)2 (x-1)2 y”’= 4/(x-1) + (12x-8)(x-1)2-2(x-1)( 6×2-8x+7) = 4×2-12x-2 (x-1)4 (x-1)3 y””= (8x-12)(x-1)3-3(x-1)2(4×2-12x-2) = -4×2+16x+18 (x-1)6 (x-1)4 y””’= (-8x+16)(x-1)4-4(x-1)3(-4×2+16x+18) = 8×2-40x-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+2×2-6 x2 y”’= -4lnx – 4/x – (4xlnx+2x+6/x+4x)x2-2x(2x2lnx+6lnx+2×2-6) = x x4 = 12lnx-4x2lnx-6×2-18 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”= -3×3-4×3(1-3ln2log2x) = 12ln2log2x-7 x8 ln2 x5 ln2 y”’= 12×4+5×4(12ln2log2x-7) = 60ln2log2x-23 x10 ln2 x6 ln2 18.6. y’= 12x2e2x+1+2(4×3+5)e2x+1= (8×3+12×2+10)e2x+1 y”= (24×2+24x) e2x+1+2(8×3+12×2+10)e2x+1=(16×3+48×2+24x+20) e2x+1 y”’= (48×2+96x+24)e2x+1+2(16×3+48×2+24x+20)e2x+1= 16(2×3+9×2+9x+4)e2x+1 y””= 16((6×2+18x+9)e2x+1+2(2×3+9×2+9x+4)e2x+1)= 16(6×3+24×2+36x+17)e2x+1 y””’= 16((18×2+48x+36)e2x+1+2(6×3+24×2+36x+17)e2x+1)= 16(12×3+72×2+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”= -2×2-3×2(1-2lnx) = -5+6lnx x6 x4 y”’= 6×3-4×3(6lnx-5) = 26-24lnx x8 x5 y””= -24×4-5×4(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)-4×2 = 4/(1+x2)2 (1+x2)2 18.11. y’= x2-3x2lnx = 1-3lnx x6 x4 y”= -3×3-4×3(1-3lnx) = -7+12lnx x8 x5 y”’= 12×4-5×4(12lnx-7) = -23-60lnx x10 x6 y””= -60×5+6×5(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) + 2×2-6x-x2-3 = 2ln(x-3) + 3×2-12x-3 (x-3)2 (x-3)2 y”’= 2/(x-3) + (6x-12)(x-3)-2(x-3)( 3×2-12x-3) = -4×2+18x+12 (x-3)4 (x-3)3 y””= (-8x+18)(x-3)3-2(x-3)2(-4×2+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)x3-3×2(-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(3×2+11x+6) y”= 3e3x+2(3×2+11x+6)+e3x+2(6x+11)= e3x+2(9×2+39x+29) y”’= 3e3x+2(9×2+39x+29)+e3x+2(18x+39)= e3x+2(27×2+135x+126) y””= 3e3x+2(27×2+135x+126)+e3x+2(54x+135)= e3x+2(81×2+459x+513) y””’= 3e3x+2(81×2+459x+513)+e3x+2(162x+459)= e3x+2(243×2+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+10×3+8x2lnx+8×2+10×2 = 20xlnx+10x+8lnx+18 x4 x2 18.30. y’= 1-2ln3log3x x3ln3 y” = -2×2-3×2(1-2ln3log3x) = -5-6ln3log3x x6ln3 x4ln3 y”’ = -6×3+4×3(5+6ln3log3x) = 14+24ln3log3x x8ln3 x5ln3 y”” = 24×4-5×4(14+24ln3log3x) = -46-120ln3log3x x10ln3 x6ln3 18.31. y’= 3x2e4x+3+4e4x+3(x3+3)= e4x+3(4×3+3×2+12) y”= 4e4x+3(4×3+3×2+12)+e4x+3(12×2+6x)= e4x+3(16×3+24×2+6x+12) y”’= 4e4x+3(16×3+24×2+6x+12)+e4x+3(48×2+48x+6)= e4x+3(64×3+144×2+72x+54) y””= 4e4x+3(64×3+144×2+72x+54)+e4x+3(192×2+288x+72)= e4x+3(256×3+768×2+576x+288)