| 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
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