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