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Реферат: Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.8
Задача 8 . Найти производную.
8.1. 
y'= 6 sin 3 xcos 3 xcos 6 x +3(1- cos 6 x ) sin6x = sin12x +2 sin6x - 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'= 8 sin4xcos4xcos8x +4(1- cos8x ) sin8x = 2 sin16x +4 sin8x -2 sin16x = 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*4 sin2xcos2xcos2x +2(1- cos4x ) sin4x = cossin5*sin8x +2 sin4x - sin8x = cossin5*tg4x _
2cos2 4x 2cos2 4x cos4x
8.6. 
y'= -sincos3*4cos2xsin2xsin4x+2(cos4x+1)cos4x =sincos3*2sin2 4x+2cos2 4x+2cos4x =sincos3*1+cos4x = sincos3*1+cos4x
4sin2 4x 4sin2 4x 2sin2 4x 1-cos8x
8.7. 
y'=cosln7*14 sin7xcos7xcos14x +7(1- cos14x ) sin14x =cosln7*sin28x +2 sin14x - 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'= 12 sin6xcos6xcos12x +6(1- cos12x ) sin12x = sin24x +2 sin12x - 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'= 20 sin10xcos10xcos20x +10(1- cos20x ) sin20x =
10cos2 20x
= 5 sin40x +10 sin20x -5 sin40x = 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'= 10 sin5xcos5xcos10x +5(1- cos10x ) sin10x =
5cos2 10x
= sin20x +2 sin10x - 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 +2 sin30x - 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*34 sin17xcos17xcos34x +17(1- cos34x ) sin34x =
17cos2 34x
= ctgsin1/3*sin68x +2 sin34x - 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*38 sin19xcos19xcos38x +19(1- cos38x ) sin38x =
19cos2 38x
= tgln2*sin76x +2 sin38x - 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'= 42 sin21xcos21xcos42x +21(1- cos42x ) sin42x =
21cos2 42x
= sin84x +2 sin42x - 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'= 46 sin23xcos23xcos46x +23(1- cos46x ) sin46x =
23cos2 46x
= sin92x +2 sin46x - 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'= 50 sin25xcos25xcos50x +25(1- cos50x ) sin50x =
25cos2 50x
= sin100x +2 sin50x - 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'= 54 sin27xcos27xcos54x +27(1- cos54x ) sin54x =
27cos2 54x
= sin108x +2 sin54x - 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'= 58 sin29xcos29xcos58x +29(1- cos58x ) sin58x =
29cos2 58x
= sin116x +2 sin58x - 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'= 62 sin31xcos31xcos62x +31(1- cos62x ) sin62x =
31cos2 62x
= sin124x +2 sin62x - sin124x = tg62x _
2cos2 62x cos62x