Завдання 13. Знайти похідну.
13.1.
y '= √ (1 - x 2) arcsinx + x / √ (1 - x 2) + xarcsinx * x / √ (1 - x 2) _ x =
1-x 2 √ (1-x 2)
= Arcsinx
1-x 2
13.2.
y '= 4 +4 √ (1-4x 2) * 1 + √ (1-4x 2) +4 x 2 / √ (1-4x 2) + 4x 2 / √ (1-4x 2) + √ (1 - 4x 2) =
x (1 + √ (1-4x 2)) 2 x 2
= 8x +1 + √ (1-4x 2) _
x 2 √ (1-4x 2) (1 + √ (1-4x 2))
13.3.
y '= (6x 2 +5) √ (x 2 +1) + 2x 4 +5 x 2 + 3 +3 x / √ (x 2 +1) =
√ (x 2 +1) x + √ (x 2 +1)
= 8 (x 2 +1) 2
√ (x 2 +1)
13.4.
y '= 3x 2 arcsinx + x 3 / √ (1-x 2) +2 / 3 * x √ (1-x 2) - x 3 +2 x =
3 √ (1-x 2)
= 3x 2 arcsinx
13.5.
y '= -3 * 12 + 8x +2 = 32x 2 +16 x-7 _
√ (1-9 / (4x +1) 2) (4x +1) 2 √ (4x 2 +2 x-2) (4x +1) √ (4x 2 +2 x-2)
13.6.
y '= x / √ (1 + x 2) +1 / √ (1 + x 2) - 1 + x / √ (1 + x 2) = x / √ (1 + x 2)
x + √ (1 + x 2)
13.7.
y '= -12 + 9x +12 = 27x 2 +72 x +36 _
(3x +4) 2 √ (1-4 / (3x +4) 2) √ (9x 2 +24 x +12) (3x +4) 2 √ (9x 2 +24 x +12)
13.8.
y '= (6x 2 +1) √ (x 2 +1) + 2x 4 + x 2 - 1 + x / √ (x 2 +1) =
√ (x 2 +1) x + √ (x 2 +1)
= 8x 2 √ (x 2 +1)
13.9.
y '= 1 + x / √ (x 2 +1) _ x 2 / √ (x 2 +1) - √ (x 2 +1) = √ (x 2 +1)
x + √ (x 2 +1) x 2 x 2
13.10.
y '=-3-4x + 12 = 6 _ 4x +3 =
2 √ (1-3x-2x 2) 2 √ (34) √ (1 - (4x +3) 2 / 17) √ 2 √ (8-16x 2-24x) 2 √ (1-3x-2x 2)
=-2x _
√ (1-3x-2x 2)
13.11.
y '= 2x +5 + 3 / (2 √ (4 + x) +3 / (2 √ (1 + x))) = √ (x +4)
2 √ ((4 + x) (1 + x)) √ (4 + x) + √ (1 + x) √ (x +1)
13.12.
2x 2-x - √ (x 2-x +1)
y '= x * 2 √ (x 2-x +1) + 2 = 2x-1_
√ (x 2-x +1) x 2 1 + (2x-1) 2 / 3 x 3-x 2 + x
13.13.
y '= (x 2 +1) 2 * (4x 3-2x) (x 2 +1) 2-4x (x 2 +1) (x 4-x 2 +1) + 4 √ 3x =
12 (x 4-x 2 +1) (x 2 +1) 4 2 √ 3 (1 +3 / (2x 2 -1) 2)
= 2x 5-2x 4 +3 x 3-2x 2
(X 2 +1) (x 4-x 2 +1)
13.14.
y '= -32 + 4x +6 = 2 √ (4x 2 +12 x-7)
(2x +3) 2 √ (1-16 / (2x +3) 2) √ (4x 2 +12 x-7) 2x +3
13.15.
y '= -12 + 9x +3 = 3 √ (9x 2 +6 x-3)
(3x +1) 2 √ (1-4 / (3x +1) 2) √ (9x 2 +6 x-3) 3x +1
13.16.
y '= 3 √ (x-1) + 3x +2 _ 3 = 18x 2-8x-3
2 √ (x-1) 4x √ (x-1) 4x √ (x-1)
13.17.
y '= 1 / 3 * √ (x +1) + x-2 + 1 = x + √ (x +1) _
6 √ (x +1) 2 √ (x +1) (√ (x +1) +1) 2 (√ (x +1) +1)
13.18.
y '= x _ √ (x 2 +1) +1 * (x / √ (x 2 +1) -1) (√ (x 2 +1) +1)-x (√ (x 2 +1) - x) / √ (x 2 +1) =
√ (x 2 +1) 2 (√ (x 2 +1)-x) (√ (x 2 +1) +1) 2
= 2x √ (x 2 +1) +3 x + √ (x 2 +1)
2 √ (x 2 +1) (√ (x 2 +1) +1)
13.19.
y '= 3 √ (x +1) * 3 √ (x +1) 2 * x +1- x +1 + xarctgx _ 1 / 2 +1 / (x 2 -1) =
3 √ (x-1) 3 3 √ (x-1) 2 (x +1) 2 (x 2 -1) 2 2 (x 2 +1)
= 5x 2 +8 + xarctgx
12 (x 4 -1) (x 2 -1) 2
13.20.
y '= ln (√ (1-x) + (1 + x)) + x (-1 / (2 √ (1-x) -1 / (2 √ (1 + x)))) + 1 - 1 / 2 =
√ (1-x) + (1 + x) 2 √ (1-x 2)
= Ln (√ (1-x) + (1 + x)) + √ (1-x) - 1 / 2
2 √ (1 + x)
13.21.
y '= x _ 1 / (x √ (x 2 -1))-xlnx / √ (x 2 -1) = 1 _ 1-x 2 lnx
(1 + x 2 -1) √ (x 2 -1) x 2 -1 x √ (x 2 -1) x 2 -1
13.22.
y '= -3 * 3 + x +2 = √ (x 2 +4 x-5)
√ (1-9 / (x +2) 2) (x +2) 2 √ (x 2 +4 x-5)
13.23.
y '= 3-x-2-x + 5 √ 5 = √ (3-x)
2 √ ((3-x) (2 + x)) 10 √ (x +2) √ (1 - (x +2) / 5) √ (x +2)
13.24.
y '= (arcsinx) 2 + 2xarcsinx _ 2xarcsinx + 2 √ (1-x 2) - 2 = (arcsinx) 2
√ (1-x 2) √ (1-x 2) √ (1-x 2)
13.25.
y '=-x 2 / √ (1-x 2) - √ (1-x 2) + 1 = √ (1-x 2)
x 2 √ (1-x 2)
13.26.
y '= 2xarccosx - x 2 _ 2x √ (1-x 2) + x (x 2 +2) = 2xarccosx - x 2 √ (1-x)
√ (1-x 2) 3 3 √ (1-x 2) √ (1 + x)
13.27.
y '= x 3 / √ (x 2 +2)-2x √ (x 2 +2) _ x * x 2 / √ (x 2 +2) - √ (x 2 +2) - √ 2 =
x 4 √ 2 (√ 2 + √ (x 2 +2)) x 2
= X 2-2x 2 -4 _ x 2-x 2 -2 - √ 2 √ (x 2 +2) = -4 / x 3
x 3 √ 2 (√ 2 + √ (x 2 +2)) x
13.28.
y '= (10-3x 2) √ (4-x 2) _ x (10x-x 3) + 3 =
4 квітня √ (4-x 2) √ (1-x 2 / 4)
= 64-32x 2-2x 4
4 √ (4-x 2)
13.29.
y '= -2 + 2x +3 = √ (4x 2 +12 x +8)
(2x +3) 2 √ (1-1 / (2x +3) 2) √ (x 2 +3 x +2) 2x +3
13.30.
y '= arcsin √ (x / (x +1)) + x (x +1- x) _ 1 + 1 =
√ (1-x / (x +1)) (x +1) 2 2 √ x x +1
= Arcsin √ (x / (x +1)) + x √ (x +1) _ 1 + 1 _
(X +1) 2 2 √ x x +1
13.31.
y '= √ (1-x 2) / √ (1-x 2) + xarcsinx / √ (1-x 2) + (1 + x) (-1-x-1 + x) =
1-x 2 (1-x) (1 + x) 2
= Xarcsinx
√ (1-x 2) 3