| Задача 7
. Найти производную.
7.1. 
√x
+ √x
y'= ln(√x+√(x+a))
+ 2√x 2√(x+a)
_ 1
=
2√x √x+√(x+a) 2√(x+a)
= ln(√x+√(x+a))
+ √x
.
2√x 2(√x+√(x+a))√(x+a)
7.2. 
y'= 1+x/√(a2
+x2
)
= x+√(a2
+x2
)
= 1
.
x+√(a2
+x2
) (x+√(a2
+x2
))√(a2
+x2
) √(a2
+x2
)
7.3. 
y'= 1
_ 2/√x
= 2+√x-2
= 1
.
√x 2+√x √x(2+√x) 2+√x
7.4. 
y'= √(1-ax4
)
* 2x√(1-ax4
)+2ax5
/√(1-ax4
)
= 2√(1-ax4
)+2ax4
x2
1-ax4
x-ax5
7.5. 
1
+ 1 _
y'= 2√x 2√(x+1)
= √(x+1)+√x
= 1
.
√x+√(x+1) 2√(x2
+x)( √x+√(x+1)) 2√(x2
+x)
7.6. 
y'= a2
-x2
* 2x(a2
-x2
)+2x(a2
+x2
)
= 4xa2
a2
+x2
(a2
-x2
)2
a4
-x4
7.7. 
y'= 2ln(x+cosx)* 1-sinx
.
x+cosx
7.8. 
y'= -3ln2
(1+cosx)* -sinx
.
1+cosx
7.9. 
y'= 1-x2
* 2x(1-x2
)+2x3
= 2
.
x2
(1-x2
)2
x(1-x2
)
7.10. 
y'= ctg(π/4+x/2)
= 2
= 2
.
2cos2
(π/4+x/2) sin(π/2+x) cosx
7.11. 
y'= 1-2x
* 2(1-2x)+2(1+2x)
= 1
.
4+8x (1-2x)2
2-8x2
7.12. 
_
y'= 1+ (x+√2)(x+√2-x+√2)
= 1+ 1
.
(x-√2)(x+√2)2
x2
-2
7.13. 
y'= cos((2x+4)/(x+1))
* 2x+2-2x-4
= -2ctg((2x+4)/(x+1))
sin((2x+4)/(x+1)) (x+1)2
(x+1)2
7.14. 
y'= 1
* 1
* 1
= 1
= lntgx _
ln16*log5
tgx tgx*ln5 cos2
x ln4*ln5*sin2x*log5
tgx 2sin2x*ln3
2
7.15. 
y'= 1
= lntgx
.
4ln2
2*cos2
x*tgx*log2
tgx 2sin2x*ln3
2
7.16. 
y'= 1/2*(coslnx+sinlnx+x(-1/x*sinlnx+1/x*coslnx))= coslnx
7.17. 
y'= -sin((2x+3)/(x+1))
*2x+2-2x-3
= ctg((2x+3)/(x+1))
cos((2x+3)/(x+1)) (x+1)2
(x+1)2
7.18. 
y'= -lge
= -2lge .
lnctgx*ctgx*sin2
x lnctgx*sin2x
7.19. 
y'= 4x3
= 2x3 .
2(1-x4
)lna lna(1-x4
)
7.20. 
1
* 4tgx _
y'= cos2
x 2√2cos2
x√1+2tg2
x
= 2tgx _
√2tgx+√(1+2tg2
x) cos4
x√(1+2tg2
x)( √2tgx+√(1+2tg2
x))
7.21. 
y'= 1
* 1
* -2e2x
= -ex
_
arcsin√(1-e2x
) √(1-1+e2x
) 2√(1-e2x
) √(1-e2x
)arcsin√(1-e2x
)
7.22. 
y'= 1
* 1
* -4e4x
= -2e2x
_
arccos√(1-e4x
) √(1-1+e4x
) 2√(1-e4x
) √(1-e4x
)arccos√(1-e4x
)
7.23. 
y'= b+b2
x/√(a2
+b2
x2
)
= b _
bx+√(a2
+b2
x2
) √(a2
+b2
x2
)
7.24. 
y'= √(x2
+1)-x√2
* (x/√(x2
+1)+√2)( √(x2
+1)-x√2)-(x/√(x2
+1)-√2)( √(x2
+1)+x√2)
=
√(x2
+1)+x√2 (√(x2
+1)-x√2)2
= (x+√(x2
+1))(√(x2
+1)-x√2)-(x-√2√(x2
+1))(√(x2
+1)+x√2)
=
√(x2
+1)(√(x2
+1)-x√2)2
= 2√2 _
√(x2
+1)(√(x2
+1)-x√2)2
7.25. 
y'= -1/(2√x3
)
= -1 _
arcos(1/√x) 2√x3
arccos(1/√x)
7.26. 
y'= ex
+e2x
/√(1+e2x
)
= ex
_
ex
+√(1+e2x
) √(1+e2x
)
7.27. 
√5-tg(x/2)
+√5+tg(x/2)
y'= √5-tg(x/2)
* 2cos2
(x/2) 2cos2
(x/2)
= √5 _
√5+tg(x/2) (√5-tg(x/2))2
(5-tg2
(x/2))cos2
(x/2)
7.28. 
sin(1/x)
+lnxcos(1/x)
y'= sin(1/x)
* x x2
= 1
+ ctg(1/x)
lnx sin2
(1/x) xlnx x2
7.29. 
y'= cos(1+1/x)
* -1/x2
= -ctg(1+1/x) _
lnsin(1+1/x) sin(1+1/x) x2
lnsin(1+1/x)
7.30. 
y'= 3ln2
ln2
x
*3ln2
x
*1
= 6 _
ln3
ln3
x ln3
x x xlnln2
xlnx
7.31. 
y'= 2lnln3
x
*3ln2
x
* 1
= 6 _
ln2
ln3
x ln3
x x xlnln3
xlnx
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