| 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
|
