Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.8
Задача 8. Найти производную.
8.1.
y'= 6sin3xcos3xcos6x+3(1-cos6x)sin6x = sin12x+2sin6x-sin12x = tg6x_
3cos2 6x 3cos2 6x cos6x
8.2.
y'= 6cos3xsin3xsin6x+3(cos6x+1)cos6x = 3sin2 6x+3cos2 6x+3cos6x = 1+cos6x = 1+cos6x
6sin2 6x 6sin2 6x 2sin2 6x 1-cos12x
8.3.
y'= 8sin4xcos4xcos8x+4(1-cos8x)sin8x = 2sin16x+4sin8x-2sin16x = tg8x_
4cos2 8x 4cos2 8x cos8x
8.4.
y'= 8cos4xsin4xsin8x+4(cos8x+1)cos8x = 4sin2 8x+4cos2 8x+4cos8x = 1+cos8x = 1+cos8x
8sin2 8x 8sin2 8x 2sin2 8x 1-cos16x
8.5.
y'= cossin5*4sin2xcos2xcos2x+2(1-cos4x)sin4x = cossin5*sin8x+2sin4x-sin8x = cossin5*tg4x_
2cos2 4x 2cos2 4x cos4x
8.6.
y'= -sincos3*4cos2xsin2xsin4x+2(cos4x+1)cos4x=sincos3*2sin24x+2cos2 4x+2cos4x=sincos3*1+cos4x= sincos3*1+cos4x
4sin2 4x 4sin2 4x 2sin2 4x 1-cos8x
8.7.
y'=cosln7*14sin7xcos7xcos14x+7(1-cos14x)sin14x=cosln7*sin28x+2sin14x-sin28x=cosln7* tg14x_
7cos2 14x 2cos2 14x cos14x
8.8.
y'= 16cos8xsin8xsin16x+8(cos16x+1)cos16x = 8sin2 16x+8cos2 16x+8cos16x =
16sin2 16x 16sin2 16x
= 1+cos16x = 1+cos16x
2sin2 16x 1-cos32x
8.9.
y'= 12sin6xcos6xcos12x+6(1-cos12x)sin12x = sin24x+2sin12x-sin24x = tg12x_
6cos2 12x 2cos2 12x cos12x
8.10.
y'= 20cos10xsin10xsin20x+10(cos20x+1)cos20x =10sin2 20x+10cos2 20x+10cos20x=
20sin2 20x 20sin2 20x
= 1+cos20x = 1+cos20x
2sin2 20x 1-cos40x
8.11.
y'= 20sin10xcos10xcos20x+10(1-cos20x)sin20x =
10cos2 20x
= 5sin40x+10sin20x-5sin40x = tg20x_
10cos2 20x cos20x
8.12.
y'= 24cos12xsin12xsin24x+12(cos24x+1)cos24x= 12sin2 24x+12cos2 24x+12cos24x=
24sin2 24x 24sin2 24x
= 1+cos24x = 1+cos24x
2sin2 24x 1-cos48x
8.13.
y'= 10sin5xcos5xcos10x+5(1-cos10x)sin10x =
5cos2 10x
= sin20x+2sin10x-sin20x = tg10x_
2cos2 10x cos10x
8.14.
y'= -cosctg3*28cos14xsin14xsin28x+14(cos28x+1)cos28x = -cosctg3*14sin2 28x+14cos2 28x+14cos28x =
28sin2 28x 28sin2 28x
= -cosctg3*1+cos28x = -cosctg3*1+cosx
2sin2 28x 1-cos56x
8.15.
y'= costg1/3*30sin15xcos15xcos30x+15(1-cos30x)sin30x =
15cos2 30x
= costg1/3*sin60x+2sin30x-sin60x = costg1/3* tg30x_
2cos2 30x cos30x
8.16.
y'= - sintg1/7*32cos16xsin16xsin32x+16(cos32x+1)cos32x = - sintg1/7*16sin2 32x+16cos2 32x+16cos32x =
32sin2 32x 32sin2 32x
= - sintg1/7*1+cos32x = - sintg1/7*1+cos32x
2sin2 32x 1-cos64x
8.17.
y'= ctgsin1/3*34sin17xcos17xcos34x+17(1-cos34x)sin34x =
17cos2 34x
= ctgsin1/3*sin68x+2sin34x-sin68x = ctgsin1/3* tg34x_
2cos2 34x cos34x
8.18.
y'= - 5√ctg2*36cos18xsin18xsin36x+18(cos36x+1)cos36x = - 5√ctg2*18sin2 36x+18cos2 36x+cos36x =
36sin2 36x 36sin2 36x
= - 5√ctg2*1+cos36x = - 5√ctg2*1+cos36x
2sin2 36x 1-cos72x
8.19.
y'= tgln2*38sin19xcos19xcos38x+19(1-cos38x)sin38x =
19cos2 38x
= tgln2*sin76x+2sin38x-sin76x = tgln2* tg38x_
2cos2 38x cos38x
8.20.
y'= 40cos20xsin20xsin40x+20(cos40x+1)cos40x = 20sin2 40x+20cos2 40x+20cos40x =
40sin2 40x 40sin2 40x
= 1+cos40x = 1+cos40x
2sin2 40x 1-cos80x
8.21.
y'= 42sin21xcos21xcos42x+21(1-cos42x)sin42x =
21cos2 42x
= sin84x+2sin42x-sin84x = tg42x_
2cos2 42x cos42x
8.22.
y'= 44cos22xsin22xsin44x+22(cos44x+1)cos44x = 22sin2 44x+22cos2 44x+22cos44x =
44sin2 44x 44sin2 44x
= 1+cos44x = 1+cos44x
2sin2 44x 1-cos88x
8.23.
y'= 46sin23xcos23xcos46x+23(1-cos46x)sin46x =
23cos2 46x
= sin92x+2sin46x-sin92x = tg46x_
2cos2 46x cos46x
8.24.
y'= 48cos24xsin24xsin48x+24(cos48x+1)cos48x = 24sin2 48x+24cos2 48x+24cos48x =
48sin2 48x 48sin2 48x
= 1+cos48x = 1+cos48x
2sin2 48x 1-cos96x
8.25.
y'= 50sin25xcos25xcos50x+25(1-cos50x)sin50x =
25cos2 50x
= sin100x+2sin50x-sin100x = tg50x_
2cos2 50x cos50x
8.26.
y'= 52cos26xsin26xsin52x+26(cos52x+1)cos52x = 26sin2 52x+26cos2 52x+26cos52x =
52sin2 52x 52sin2 52x
= 1+cos52x = 1+cos52x
2sin2 52x 1-cos104x
8.27.
y'= 54sin27xcos27xcos54x+27(1-cos54x)sin54x =
27cos2 54x
= sin108x+2sin54x-sin108x = tg54x_
2cos2 54x cos54x
8.28.
y'= 56cos28xsin28xsin56x+28(cos56x+1)cos56x = 28sin2 56x+28cos2 56x+28cos56x =
56sin2 56x 56sin2 56x
= 1+cos56x = 1+cos56x
2sin2 56x 1-cos112x
8.29.
y'= 58sin29xcos29xcos58x+29(1-cos58x)sin58x =
29cos2 58x
= sin116x+2sin58x-sin116x = tg58x_
2cos2 58x cos58x
8.30.
y'= 60cos30xsin30xsin60x+30(cos60x+1)cos60x = 30sin2 60x+30cos2 60x+30cos60x =
60sin2 60x 60sin2 60x
= 1+cos60x = 1+cos60x
2sin2 60x 1-cos120x
8.31.
y'= 62sin31xcos31xcos62x+31(1-cos62x)sin62x =
31cos2 62x
= sin124x+2sin62x-sin124x = tg62x_
2cos2 62x cos62x