Задача 10
. Найти производную.
10.1.
y'= 1
*2-√5
thx
*√5/
ch
2
x
*(2-√5
thx
)+ √5/
ch
2
x
*(2+√5
thx
)
=
4√5 2+√5thx (2-√5thx)2
= 1 _
ch2
x(2-√5thx)
10.2.
y'= ch5
x-4ch3x
sh2
x
+ 3ch3
x-6chxsh2
x
+ 3chx
= 1
+ 3-3sh2
x
+ 3 _
4ch8
x 8ch4
x 8(1+sh2
x) 4ch5
x 8ch3
x 8chx
10.3.
1-√(thx)
+ 1+√(thx) _
y'= 1/2* 1-√(thx)
* 2√(thx)ch2
x 2√(thx)ch2
x
_ 1
=
1+√(thx) (1-√(thx))2
2√(thx)ch2
x
= √thx _
(1-th2
x)(ch2
x)
10.4.
√2-thx
+ √2+thx
2-th2
x
+ 2th2
x
y'= 3
*√2-thx
* ch2
x ch2
x
_ ch2
x ch2
x
=
8√2 √2+thx (√2-thx)2
4(2-th2
x)2
= 1 _
2ch2
x(2-th2
x)2
10.5.
y'= 1
+ 1-√2
thx
* √2(1-√2thx+1+√2thx)
=
2ch2
x 4√2(1+√2thx) ch2
x(1-√2thx)2
= 1-th2
x _
ch2
x(1-√2thx)2
10.6.
y'= _ 1
_ sh3
x-2shxch2
x
= 2ch3
x+2chx-sh2
x
4thxch2
x 2sh4
x 4sh3
xchx
10.7.
y'= a-√(1+a2
)thx
* √(1+a2
)thx(a-√(1+a2
)thx+a+√(1+a2
)thx)
=
2a√(1+a2
)(a+√(1+a2
)thx) (a-√(1+a2
)thx)2
= thx
= thx
= thx _
(a2
-(1+a2
)th2
x)ch2
x a2
ch2
x-(1+a2
)sh2
x a2
-sh2
x
10.8.
y'= 1-√2cthx
* √2(-1+√2cthx+1+√2cthx)
= √2cthx
=
18√2(1+√2cthx) sh2
x(1-√2cthx)2
9sh2
x(1-√2cthx)
= -√2cthx _
9(1+ch2
x)
10.9.
y'= 1
* ch2x/√(sh2x)-√(sh2x)(shx-chx)
=
1+sh2x/(shx-chx)2
chx-shx
= (chx-shx)(ch2x-sh2x(shx-chx))
√(sh2x)(ch2
x+sh2
x)
10.10.
y'= 2+sh2x
* -ch2x(2+sh2x)-ch2x(1-sh2x)
= ch2x _
6(1-sh2x) (2+sh2x)2
12-6sh2x-sh2
2x
10.11.
y'= 4
√(1-thx)3
* 1-thx+1+thx
= 1 _
44
√(1+thx)3
ch2
x(1-thx) 4ch2
x√(1+thx) 4
√(1-th2
x)
10.12.
y'= chx(1+chx)-sh2
x
= 1 _
(1+chx)2
1+chx
10.13.
y'= shx√(sh2x)-chxch2x/√(sh2x)
= shx-chxcth2x
sh2x
10.14.
y'= 3ch3x√(ch6x)-3sh6xsh3x/√(ch6x)
= 3sh3x-3th6xsh3x
ch6x
10.15.
y'= 16shxch3
x*ln(chx)+8ch3
xshx-16ch3
xshxln(chx)
= 4thx
2ch4
x
10.16.
y'= 2shxchx(12sh2
x+1)-24sh3
xchx
= 4chx
3sh4
x 3sh3
x
10.17.
y'= 2chxsh2
x-ch3
x
+ 3
= shx-1
+ 3 _
2ch4
x ch2
x√(1-th2
x) 2ch3
x ch2
x√(1-th2
x)
10.18.
y'= 1
* shx(1+3chx)-3shx(3+chx)
=
√8√(1-(3+chx)2
/(1+3chx)2
) (1+3chx)2
= -8shx
= -1 _
8(1+3chx)√(ch2
x-1) 1+3chx
10.19.
y'= 4-√8th(x/2)
* √8(4-√8th(x/2)+4+√8th(x/2))
=
√8(4+√8th(x/2)) 2ch2
(x/2)(4-√8th(x/2))2
= 1 _
ch2
(x/2)(4-√8th(x/2))
10.20.
y'= 1
_ shx(sh2
x-3chx-ch2
x)
= 1+chx _
8ch2
(x/2)th(x/2) 4sh2
x(3+chx) shx(3+chx)
10.21.
y'= -(3+5chx)(3shx(3+5chx)-5shx(5+3chx))
= 1
4(3+5chx)2
√(9+30chx+25ch2
x-25-30chx-9ch2
x)
10.22.
y'= -16ch5
xshx-4ch3
x(1-8ch2
x)
= -4ch2
xshx-1+8ch2
x
4ch8
x ch5
x
10.23.
y'= -2/sh2
x+1/sh4
x+ch3
x-2chxsh2
x
+ 5chx
= -2/sh2
x+1/sh4
x+1-sh2
x
+ 5_
2ch4
x 2+2sh2
x 2ch3
x 2chx
10.24.
y'= -16
+ sh4
x+3sh2
xch2
x
= 1-4sh2
x
3sh2
2x 3ch2
xsh6
x ch2
xsh4
x
10.25.
y'= chx
_ 1-sh2
x
= 1
_ 1-sh2
x
2+2sh2
x 2ch3
x 2ch2
x 2ch3
x
10.26.
y'= 3
+ shx – sh3
x-2shxch2
x
= 1+sh2
x
4ch2
(x/2)th(x/2) 2sh4
x shx
10.27.
y'= 2chxsh2
x-ch3
x
+ 2shxchx
_ 3chx
= sh2
x-1
+ 2chx
_ 3 _
2ch4
x sh4
x 2+2sh2
x 2ch3
x sh3
x 2chx
10.28.
y'= ch3
x-2chxsh2
x
+ chx
= 1 _
2ch4
x 2+2sh2
x ch3
x
10.29.
y'= ch3
x-2chxsh2
x
+ chx
= 1 _
2ch4
x 2+2sh2
x ch3
x
10.30.
y'= 2ch2
xshx-sh3
x
_ 1
= 1/sh3
x
2sh4
x 4ch2
(x/2)th(x/2)
10.31.
y'= -2
_ sh4
x-3ch2
xsh2
x
= 1/sh4
x
3sh2
x 3sh6
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