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 + 2e^4x + 2e^-2xarcsine^2x – 2e^2xe^-2x =

√(1-e^4x)(1+√(1-e^4x)) √(1-e^4x)

= 2e^-2xarcsine^2x

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√(2x-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 + 3e^6x + 3e^-3xarcsin(e^3x) – 3e^-3xe^3x =

√(1-e^6x)(1+√(1-e^6x)) √(1-e^6x)

= 3e^-3xarcsin(e^3x)

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)³arcsin(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)³arcsin(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 + 5e^10x + 5e^-5xarcsin(e^5x) – 5e^-5xe^5x =

√(1-e^10x)(1+√(1-e^10x)) √(1-e^10x)

= 5e^-5xarcsin(e^5x)

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)³arcsin(1/(3x-2)) –

√(9x²-12x+3)

- 9(3x-2)⁴ = 12(3x-2)³arcsin(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'= 5e^5x(1+√(e^10x-1)) _ 5e^-5x =

√(e^10x-1)(1+√(e^10x-1)) √(1-e^-10x)

= 5√(e^5x-1)

√(e^5x+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)³arcsin(1/(2x-1)) – 2(2x-1)⁵ =

3√(x²-x) (2x-1)²√(4x²-4x)

= 8(2x-1)³arcsin(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 + 4e^4x+4e^8x/√(e^8x-1) = 4√(e^4x-1)

√(1-e^-8x) e^4x+√(e^8x-1) √(e^4x+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)³arcsin(1/(3x+1)) – 3(3x+1)⁵ + (6x+2)√(9x²+6x) +

√(9x²+6x)(3x+1)²

+ (3x²+2x+1)(9x+3) = 12(3x+1)³arcsin(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'= 3e^3x(e^3x+√(e^6x-1)) _ 3e^-3x =

√(e^6x-1)(e^3x+√(e^6x-1)) √(1-e^-6x)

= 3√(e^3x-1)

√(e^3x+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 + 2e^2x+2e^4x/√(e^4x-1) = 2√(e^2x-1)

√(1-e^-4x) e^2x+√(e^4x-1) √(e^2x+1)