Задача 6. Знайти похідну.
6.1.
e x + 2 e 2 x + e x
y '= 1 - √ (e 2 x + e x +1) = 2 + e x + √ (e 2 x + e x +1) - e x √ (e 2 x + e x +1) -2 e 2 x - e x =
2 + e x +2 √ (e 2x + e x +1) 2 + e x +2 √ (e 2x + e x +1)
= (2-e x) √ (e 2x + e x +1) +2 + e x-2e x
2 + e x +2 √ (e 2x + e x +1)
6.2.
y '= 1 / 4 * e 2x (2-sin2x-cos2x) +1 / 8 * e 2x (-2cos2x +2 sin2x) = 1 / 8 * e 2x (4-2sin2x-2cos2x-2cos2x +2 sin2x) = 1 / 8 * e 2x (4-4cos2x) = e 2x * sin2x
6.3.
y '= 1 * 1 * 2e x = E x.
2 Січень + (e x -3) 2 4 e 2x-6e x +10
4
6.4.
y '= 1 * 1-2 x * -2 x ln2 (1 +2 x) - (1-2 x) 2 x ln2 = (2 x -1) 2 x ln4 = 2 x (2 x -1)
ln4 1 +2 x (1 +2 x) 2 ln4 (1 +2 x) 3 (1 +2 x) 3
6.5.
e x (√ (e x +1) +1) _ e x (√ (e x +1) -1)
y '= e x + √ (e x +1) +1 * 2 √ (e x +1) 2 √ (e x +1) =
√ (e x +1) √ (e x +1) -1 (√ (e x +1) +1) 2
= E x + E x √ (e x +1) + e x-e x √ (e x +1) + e x = √ (e x +1)
√ (e x +1) 2e x √ (e x +1)
6.6.
y "= 2 / 3 * 3 / 2 * √ (arctge x) * e x = E x √ (arctge x)
1 + e x 1 + e x
6.7.
y '= 2e x - 2e x = E x
2 (e 2x +1) 1 + e 2x 1 + e 2x
6.8.
6.9.
y "= 2 / ln 2 * ((2 x ln 2) / (2 √ (2 x -1)) - (2 x ln 2) / (1 +2 x -1)) = 2 x / √ (2 x -1) -2
6.10.
e x (√ (1 + e x) +1) _ e x (√ (1 + e x) -1)
y '= 2 √ (1 + e x) + 2 e x (x -2) _ √ (1 + e x) +1 * 2 √ (1 + e x) 2 √ (1 + e x) =
2 √ (1 + e x) √ (1 + e x) -1 (√ (1 + e x) +1)
= Xe x +2 _ 2e x √ (1 + e x) +2 e x = Xe x.
√ (1 + e x) e x √ (1 + e x) ( √ (1 + e x) +1) √ (1 + e x)
6.11.
y '= αe αx (αsinβx-βcosβx) + e αx (αβcosβx + β 2 sinβx) =
α 2 + β 2
= E αx (α 2 sinβx + β 2 sinβx) = e αx sinβx
α 2 + β 2
6.12.
y '= αe αx (βsinβx-αcosβx) + e αx (β 2 cosβx + αβsinβx) =
α 2 + β 2
= E αx (β 2 cosβx +2 αβsinβx-α 2 cosβx)
α 2 + β 2
6.13.
y '= ae ax * ┌ 1 + acos2bx +2 bsin2bx ┐ + e ax ┌-2absin2bx +4 b 2 cos2bx ┐ =
└ 2a 2 (a 2 +4 b 2) ┘ └ 2 (a 2 +4 b 2) ┘
= E ax / 2 * (1 + cos2bx) = e ax cos 2 bx
6.14.
y '= 1 - e x - E x = 1 - e x-e x-e 2x = 1 + e 2x.
(1 + e x) 2 1 + e x (1 + e x) 2 (1 + e x) 2
6.15.
3 / 6 * e x / 6 * √ (1 + e x / 3) + 1 / 3 * e x / 3 (1 + e x / 6)
y '= 1 - 2 √ (1 + e x / 3) _ 3 / 6 * e x / 6 =
(1 + e x / 6) √ (1 + e x / 3) 1 + e x / 3
= 1 - e x / 6 + e x / 2 + e x / 3 + e x / 2 _ E x / 6 = 1 - e x / 3-e x / 6.
2 (1 + e x / 6) (1 + e x / 3) 2 (1 + e x / 3) 2 (1 + e x / 6) (1 + e x / 3)
6.16.
y '= 1 - 8e x / 4 = 1 - 2e x / 4 .
4 (1 + e x / 4) 2 (1 + e x / 4) 2
6.17.
e x + e 2x
y '= √ (e 2x -1) _ e-x = E x (e x + √ (e 2x -1)) _ e-x * e x = e x -1.
e x + √ (e 2x -1) √ (1-e-2x) (e x + √ (e 2x -1)) √ (e 2x -1) √ (e 2x -1) √ (e 2x -1 )
6.18.
e 2x
y '= 1 + e-x arcsine x - e-x * e x + √ (1-e 2x) =
√ (1-e 2x) 1 + √ (1-e 2x)
= 1 + e-x arcsine x - 1 + e 2x =
√ (1-e 2x) (1 + √ (1-e 2x)) √ (1-e 2x)
= E-x arcsine x
6.19.
y '= 1 - e x + E-x / 2 arctge x / 2 - e-x / 2 * e x / 2 _ e x / 2 arctge x / 2 =
1 + e x 1 + e x 1 + e x
= 1 - e x + 1 + arctge x / 2 * 1-e x = Arctge x / 2 * 1-e x.
1 + e x 1 + e x e x / 2 (1 + e x) e x / 2 (1 + e x)
6.20.
y '= 3x 2 e x3 (1 + x 3)-3e x3 x 2 = 3x 5 e x3
(1 + x 3) 2 (1 + x 3) 2
6.21.
y '= b * me mx √ a = e mx.
m √ (ab) (b + ae 2mx) √ b b + ae 2mx
6.22.
y '= e 3 ^ √ x / 3 √ x (3 √ x 2 -2 3 √ x +2) +3 e 3 ^ √ x (2 / (3 березня √ x) -2 / (3 березня √ x 2) ) = e 3 ^ √ x
3 ^ √ x = кубічний корінь з х
6.23.
(E x +2 e 2x _ E x) (√ (1 + e x + e 2x)-e x +1) _ (e x +2 e 2x _ E x) (√ (1 + e x + e 2x)-e x -1)
y '= √ (1 + e x + e 2x)-e x +1 * 2 √ (1 + e x + e 2x) 2 √ (1 + e x + e 2x) =
√ (1 + e x + e 2x)-e x -1 (√ (1 + e x + e 2x)-e x +1) 2
= E x (1 +2 e 2x -2 √ (1 + e x + e 2x)) = 1.
(E x (1 +2 e 2x -2 √ (1 + e x + e 2x))) √ (1 + e x + e 2x) √ (1 + e x + e 2x)
6.24.
y '= cosxe sinx (x-1/cosx) + e sinx (1-sinx/cos 2 x) = e sinx (xcosx-sinx/cos 2 x)
6.25.
y '= e x / 2 ((x 2 -1) cosx + (x-1) 2 sinx) + e x / 2 (2xcosx-(x 2 -1) sinx +2 (x-1) sinx + (x-1 ) 2 cosx) =
= E x / 2 (x-1) (5x +3) cosx
6.26.
y '= e x + e-x = e 3x + e x.
1 + (e x-e-x) 2 e 4x-e 2x +1
6.27.
y '= e 3 ^ √ x / 3 √ x 2 (3 √ x 5 -5 3 √ x 4 +20 x-60 3 √ x 2 +120 3 √ x-120) + e 3 ^ √ x (5 березня √ x 2 -20 3 √ x +20-120 / 3 √ x +120 / 3 √ x 2) = e 3 ^ √ x (x-40)
6.28.
y '=-3e 3x sh 3 x +3 e 3x sh 2 xchx = e 3x (chx-shx)
3sh 6 x sh 4 x
6.29.
y '=-e-x + e 2x = √ (e 4x-e 2x) - √ (e-2x -1) = √ (e 2x -1) - √ (1-e 2x)
√ (1-e-2x) √ (1-e 2x) √ (1-e-2x) * √ (1-e 2x) √ (1-e-2x) √ (1-e 2x)
6.30.
y '= xe-x2 (x 4 +2 x 2 +2) -1 / 2 * e-x2 (4x 3 +4 x) = x 5 e-x2
6.31.
y '= 2xe x2 (1 + x 2)-2e x2 x = 2x 3 e x2
(1 + x 2) 2 (1 + x 2) 2