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(
-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)