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Реферат: Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.6
Задача 6 . Найти производную.
6.1.
ex + 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+ex +2√(e2x +ex +1) 2+ex +2√(e2x +ex +1)
= (2-e x )√(e 2x +e x +1)+2+e x -2e x
2+ex +2√(e2x +ex +1)
6.2.
y' = 1/4*e2x (2-sin2x-cos2x)+1/8*e2x (-2cos2x+2sin2x)=1/8*e2x (4-2sin2x-2cos2x-2cos2x+2sin2x)=1/8*e2x (4-4cos2x)=e2x *sin2x
6.3.
y' = 1 * 1 * 2e x = e x .
2 1 + (e x -3) 2 4 e2x -6ex +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+2x (1+2x )2 ln4(1+2x )3 (1+2x )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) =
√(ex +1) √(ex +1)-1 (√(ex +1)+1)2
= e x + e x √(e x +1)+e x -e x √(e x +1)+e x = √(ex +1)
√(ex +1) 2ex √(ex +1)
6.6.
y' = 2/3*3/2*√(arctgex ) * e x = e x √(arctge x )
1+ex 1+ex
6.7.
y' = 2e x - 2e x = e x
2(e2x +1) 1+e2x 1+e2x
6.8.
6.9.
y' = 2/ln2*((2x ln2)/(2√(2x -1))-(2x ln2)/(1+2x -1))=2x/√(2x -1)-2
6.10.
e x (√(1+ e x )+1) _ e x (√(1+ e x )-1)
y'= 2√(1+ex )+2 e x ( x -2) _ √(1+ e x )+1 * 2√(1+ e x ) 2√(1+ e x ) =
2√(1+ex ) √(1+ex )-1 (√(1+ex )+1)
= xe x +2 _ 2e x √(1+e x )+2e x = xe x .
√(1+ex ) ex √(1+ex )( √(1+ex )+1) √(1+ex )
6.11.
y'= αe αx (αsinβx-βcosβx)+e αx (αβcosβx+β 2 sinβx) =
α2 +β2
= e αx (α 2 sinβx+β 2 sinβx) = eαx sinβx
α2 +β2
6.12.
y'= αe αx (βsinβx-αcosβx)+e αx (β 2 cosβx+αβsinβx) =
α2 +β2
= e αx (β 2 cosβx+2αβsinβx-α 2 cosβx)
α2 +β2
6.13.
y'= aeax * ┌ 1 + acos2bx+2bsin2bx ┐+ eax ┌ -2absin2bx+4b 2 cos2bx ┐=
└ 2a 2(a2 +4b2 ) ┘ └ 2(a2 +4b2 ) ┘
= eax /2*(1+cos2bx)= eax cos2 bx
6.14.
y' = 1 – e x - e x = 1 - e x -e x -e 2x = 1 + e 2x .
(1+ex )2 1+ex (1+ex )2 (1+ex )2
6.15.
3/6*ex/6 *√(1+ex/3 ) + 1/3*e x/3 (1+e x/6 )
y'= 1 - 2√(1+e x/3 ) _ 3/6*e x/6 =
(1+ex/6 )√(1+ex/3 ) 1+ex/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+ex/6 )(1+ex/3 ) 2(1+ex/3 ) 2(1+ex/6 )(1+ex/3 )
6.16.
y' = 1 - 8e x/4 = 1 - 2e x/4 .
4(1+ex/4 )2 (1+ex/4 )2
6.17.
ex + e 2x
y'= √(e 2x -1) _ e -x = e x (e x +√(e 2x -1)) _ e -x *e x = e x -1 .
ex +√(e2x -1) √(1-e-2x ) (ex +√(e2x -1))√(e2x -1) √(e2x -1) √(e2x -1)
6.18.
e 2x
y'= 1+e-x arcsinex – e -x *e x + √(1-e 2x ) =
√(1-e2x ) 1+√(1-e2x )
= 1+e-x arcsinex - 1 + e 2x =
√(1-e2x ) (1+√(1-e2x )) √(1-e2x )
= e-x arcsinex
6.19.
y'= 1- e x +e-x/2 arctgex/2 – e -x/2 *e x/2 _ ex/2 arctgex/2 =
1+ex 1+ex 1+ex
= 1- ex + 1 + arctgex/2 * 1-ex = arctgex/2 * 1-ex .
1+ex 1+ex ex/2 (1+ex ) ex/2 (1+ex )
6.20.
y'= 3x2 ex3 (1+x3 )-3ex3 x2 = 3x5 ex3
(1+x3 )2 (1+x3 )2
6.21.
y'= b *memx √a = emx .
m√(ab)(b+ae2mx ) √b b+ae2mx
6.22.
y'= e3 ^√x /3√x(3 √x2 -23 √x+2)+3e3^√x (2/(33 √x)-2/(33 √x2 ))= e3^√x
3^√x= кубический корень из х
6.23.
( ex +2e2x _ ex )(√(1+ex +e2x )-ex +1) _ ( ex +2e2x _ ex )(√(1+ex +e2x )-ex -1)
y'= √(1+ex +e2x )-ex +1 * 2√(1+ex +e2x ) 2√(1+ex +e2x ) =
√(1+ex +e2x )-ex -1 (√(1+ex +e2x )-ex +1)2
= ex (1+2e2x -2√(1+ex +e2x )) = 1 .
(ex (1+2e2x -2√(1+ex +e2x )))√(1+ex +e2x ) √(1+ex +e2x )
6.24.
y'= cosxesinx (x-1/cosx)+esinx (1-sinx/cos2 x)= esinx (xcosx-sinx/cos2 x)
6.25.
y'= ex /2((x2 -1)cosx+(x-1)2 sinx)+ex /2(2xcosx-(x2 -1)sinx+2(x-1)sinx+(x-1)2 cosx)=
= ex /2(x-1)(5x+3)cosx
6.26.
y'= ex +e-x = e3x +ex .
1+(ex -e-x )2 e4x -e2x +1
6.27.
y'= e3^√x /3 √x2 (3 √x5 -53 √x4 +20x-603 √x2 +1203 √x-120)+e3^√x (53 √x2 -203 √x+20-120/3 √x+120/3 √x2 )= e3^√x (x-40)
6.28.
y'= -3e3x sh3 x+3e3x sh2 xchx = e3x (chx-shx)
3sh6 x sh4 x
6.29.
y'= -e-x + e2x = √(e4x -e2x )-√(e-2x -1) = √(e2x -1)-√(1-e2x )
√(1-e-2x ) √(1-e2x ) √(1-e-2x )*√(1-e2x ) √(1-e-2x ) √(1-e2x )
6.30.
y'= xe-x2 (x4 +2x2 +2)-1/2*e-x2 (4x3 +4x)= x5 e-x2
6.31.
y'= 2xex2 (1+x2 )-2ex2 x = 2x3 ex2
(1+x2 )2 (1+x2 )2