Задача 12
. Найти производную.
12.1. 
y'= 2x√(x2
-4)
+ x(x2
+8)
+ x/8*arcsin(2/x) – 2x2
=
24 24√(x2
-4) 16x2
√(1-4/x2
)
= x3
-x
+ x/8*arcsin(2/x)
8√(x2
-4)
12.2. 
y'= 4(16x2
+8x+3)-(4x+1)(32x+8)
+ 4
=
(16x2
+8x+3)2
2(1+(4x+1)2
/2)
= 16 _
(16x2
+8x+3)2
12.3. 
y'= 2 + 2e4x
+ 2e-2x
arcsine2x
– 2e2x
e-2x
=
√(1-e4x
)(1+√(1-e4x
)) √(1-e4x
)
= 2e-2x
arcsine2x
12.4. 
y'= (9x-6)arctg(3x-2)
+ 3√(9x2
-12x+5)
_ 3+(9x-6)/√(9x2
-12x+5)
=
√(9x2
-12x+5) 1+(3x-2)2
3x-2+√(9x2
-12x+5)
= (9x-6)arctg(3x-2)
√(9x2
-12x+5)
12.5. 
y'= -2√(2
x-x2
)
+ 2-2x
+ (x-1)((1-x)/√(2x-x2
)-1-√(2x-x2
))
=
(x-1)2
(x-1)√(2x-x2
) (x-1)2
(1+√(2x-x2
))
= -1
_ 2
_ 1_
(1+√(2x-x2
))√(2x-x2
) √(2x-x2
)(x-1)2
(x-1)
12.6. 
y'= 2xarcsin(3/x)
_ 3x2
+ 2x√(x2
-9)
_ x(x2
+18)
=
81 81x2
√(x2
-9) 81x2
√(x2
-9) 81x2
√(x2
-9)
= 2xarcsin(3/x)
+ x3
-39x _
81 81x2
√(x2-9)
12.7. 
y'= 6
+ 3(3x2
-2x+1)-(6x-2)(3x-1)
= 4 _
2(2+(3x-1)2
) 3(3x2
-2x+1)2
3(3x2
-2x+1)2
12.8. 
y'= 3 + 3e6x
+ 3e-3x
arcsin(e3x
) – 3e-3x
e3x
=
√(1-e6x
)(1+√(1-e6x
)) √(1-e6x
)
= 3e-3x
arcsin(e3x
)
12.9. 
y'= 16x-4+4√(16x2
-8x+2)
_ (16x-4)arctg(4x-1)
_ 4√(16x2
-8x+2)
=
(4x-1+√(16x2
-8x+2)√(16x2
-8x+2) √(16x2
-8x+2) 2+16x2
-8x
= (4-16x)arctg(4x-1)
√(16x2
-8x+2)
12.10. 
y'= (2x+1)((-1-2x)/√(-x-x2
)-2-4√(-x-x2
))
+ (-2-4x)(2x+1)/√(-x-x2
)-8√(-x-x2
)
=
(2x+1)2
(1+2√(-x-x2
)) (2x+1)2
= 4x+4x2
_ 3 _
(2x+1)√(-x-x2
)(1+2√(-x-x2
)) (2x+1)√(-x-x2
)
12.11. 
y'= 4(2x+3)3
arcsin(1/(2x+3)) – 2(2x+3)4
+ 2/3*(8x+12)√(x2
+3x+2) +
√(4x2
+12x+8)
+ 2(4x2
+12x+11)(2x+3)
= 4(2x+3)3
arcsin(1/(2x+3)) – 8/3*(2x+3)√(x2
+3x+2)
3√(x2
+3x+2)
12.12. 
y'= x2
+4x+6-(2x+4)(x+2)
+ 2
= 4 _
(x2
+4x+6)2
2(2+(x+2)2
) (x2
+4x+6)2
12.13. 
y'= 5 + 5e10x
+ 5e-5x
arcsin(e5x
) – 5e-5x
e5x
=
√(1-e10x
)(1+√(1-e10x
)) √(1-e10x
)
= 5e-5x
arcsin(e5x
)
12.14. 
y'= (x-4)arctg(x-4)
+ √(x2
-8x+17)
_ √(x2
-8x+17)+x-4
=
√(x2
-8x+17) x2
-8x+17 (√(x2
-8x+17)+x-4)√(x2
-8x+17)
= (x-4)arctg(x-4)
√(x2
-8x+17)
12.15. 
y'= (2-x)((2-x)2
/√(-3+4x-x2
)+1+√(-3+4x-x2
))
+ 2(4-2x)(2-x)/√(-3+4x-x2
)+2√(-3+4x-x2
)
=
(2-x)2
(1+√(-3+4x-x2
)) (2-x)2
= x2
-5x+7 _
(2-x)√(-3+4x-x2
)
12.16. 
y'= (6x-4)√(9x2
-12x+3) + (3x2
-4x+2)(9x+6)
+ 12(3x-2)3
arcsin(1/(3x-2)) –
√(9x2
-12x+3)
- 9(3x-2)4
= 12(3x-2)3
arcsin(1/(3x-2)) - 6(3x-2)3
_
√(1-1/(3x-2)2
)(3x-2)2
√(9x2
-12x+3)
12.17. 
y'= 2
+ x2
-2x+3-(x-1)(2x-2)
= 4 _
2(3+x2
-2x) (x2
-2x+3)2
(x2
-2x+3)2
12.18. 
y'= 5e5x
(1+√(e10x
-1))
_ 5e-5x
=
√(e10x
-1)(1+√(e10x
-1)) √(1-e-10x
)
= 5√(e5x
-1)
√(e5x
+1)
12.19. 
y'= 2+(4x-6)/√(4x2
-12x+10)
_ (4x-6)arctg(2x-3)
_ 2√(4x2
-12x+10)
=
2x-3+√(4x2
-12x+10) √(4x2
-12x+10) √(4x2
-12x+10)
= (6-4x)arctg(2x-3)
√(4x2
-12x+10)
12.20. 
y'= (-2-x)((-2-x)2
/√(-3-4x-x2
)+1+√(-3-4x-x2
))
+ 2√(-3-4x-x2
)
+ 4+2x
=
(-2-x)2
(1+√(-3-4x-x2
)) (2+x)2
(2+x)√(-3-4x-x2
)
= -x _
(2+x)2
√(-3-4x-x2
)
12.21. 
y'= 2/3*(8x-4)√(x2
-x) + (4x2
-4x+3)(2x-1)
+ 8(2x-1)3
arcsin(1/(2x-1)) – 2(2x-1)5
=
3√(x2
-x) (2x-1)2
√(4x2
-4x)
= 8(2x-1)3
arcsin(1/(2x-1))
12.22. 
y'= 2(4x2
-4x+3)-4(2x-1)2
+ 4
= 8 _
(4x2
-4x+3)2
2(4x2
-4x+3) (4x2
-4x+3)2
12.23. 
y'= -4e-4x
+ 4e4x
+4e8x
/√(e8x
-1)
= 4√(e4x
-1)
√(1-e-8x
) e4x
+√(e8x
-1) √(e4x
+1)
12.24. 
y'= 5+25x/√(25x2
+1)
_ 25xarctg5x
_ 5√(25x2
+1)
= _ 25xarctg5x
5x+√(25x2
+1) √(25x2
+1) 25x2
+1 √(25x2
+1)
12.25. 
y'= -6√(-3+12x-9x2
)
+ 12-18x
+ (3x-2)((6-9x)(3x-2)/√(-3+12x-9x2
)-3-3√(-3+12x-9x2
))
=
(3x-2)2
(3x-2)√(-3+12x-9x2
) (1+√(-3+12x-9x2
))(3x-2)2
= -9x-2 _
(3x-2)2
√(-3+12x-9x2
)
12.26. 
y'= 12(3x+1)3
arcsin(1/(3x+1)) – 3(3x+1)5
+ (6x+2)√(9x2
+6x) +
√(9x2
+6x)(3x+1)2
+ (3x2
+2x+1)(9x+3)
= 12(3x+1)3
arcsin(1/(3x+1)) + 18x2
(3x+1)/√(x2
+3x+2)
√(9x2
+6x)
12.27. 
y'= 2
+ 8x2
+8x+6-16x2
-16x-4
= 5-4x2
-4x _
2(3+4x2
+4x) (4x2
+4x+3)2
(4x2
+4x+3)2
12.28. 
y'= 3e3x
(e3x
+√(e6x
-1))
_ 3e-3x
=
√(e6x
-1)(e3x
+√(e6x
-1)) √(1-e-6x
)
= 3√(e3x
-1)
√(e3x
+1)
12.29. 
y'= 49xarctg7x
+ 7√(49x2
+1)
_ 7+49x/√(49x2
+1)
= 49xarctg7x
√(49x2
+1) 49x2
+1 7x+√(49x2
+1) √(49x2
+1)
12.30. 
y'= -√(1-4x2
)
_ 4x
+ 2x(4x2
/√(1+4x2
)-1-√(1+4x2
))
= -1
_ 1 _
x2
x√(1-4x2
) 2x2
(1+√(1+4x2
)) x2
√(1-4x2
) x√(1+4x2
)
12.31. 
y'= -2e-2x
+ 2e2x
+2e4x
/√(e4x
-1)
= 2√(e2x
-1)
√(1-e-4x
) e2x
+√(e4x
-1) √(e2x
+1)
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