Задача 10. Найти производную.

10.1.

y'= 1 *2-√5thx*√5/ch²x*(2-√5thx)+ √5/ch²x*(2+√5thx) =

4√5 2+√5thx (2-√5thx)²

= 1 _

ch²x(2-√5thx)

10.2.

y'= ch⁵x-4ch^3xsh²x + 3ch³x-6chxsh²x + 3chx = 1 + 3-3sh²x + 3 _

4ch⁸x 8ch⁴x 8(1+sh²x) 4ch⁵x 8ch³x 8chx

10.3.

1-√(thx) + 1+√(thx) _

y'= 1/2* 1-√(thx) * 2√(thx)ch²x 2√(thx)ch²x _ 1 =

1+√(thx) (1-√(thx))² 2√(thx)ch²x

= √thx _

(1-th²x)(ch²x)

10.4.

√2-thx + √2+thx 2-th²x + 2th²x

y'= 3 *√2-thx * ch²x ch²x _ ch²x ch²x =

8√2 √2+thx (√2-thx)² 4(2-th²x)²

= 1 _

2ch²x(2-th²x)²

10.5.

y'= 1 + 1-√2thx * √2(1-√2thx+1+√2thx) =

2ch²x 4√2(1+√2thx) ch²x(1-√2thx)²

= 1-th²x _

ch²x(1-√2thx)²

10.6.

y'= _ 1 _ sh³x-2shxch²x = 2ch³x+2chx-sh²x

4thxch²x 2sh⁴x 4sh³xchx

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)th²x)ch²x a²ch²x-(1+a²)sh²x a²-sh²x

10.8.

y'= 1-√2cthx * √2(-1+√2cthx+1+√2cthx) = √2cthx =

18√2(1+√2cthx) sh²x(1-√2cthx)² 9sh²x(1-√2cthx)

= -√2cthx _

9(1+ch²x)

10.9.

y'= 1 * ch2x/√(sh2x)-√(sh2x)(shx-chx) =

1+sh2x/(shx-chx)² chx-shx

= (chx-shx)(ch2x-sh2x(shx-chx))

√(sh2x)(ch²x+sh²x)

10.10.

y'= 2+sh2x * -ch2x(2+sh2x)-ch2x(1-sh2x) = ch2x _

6(1-sh2x) (2+sh2x)² 12-6sh2x-sh²2x

10.11.

y'= ⁴√(1-thx)³ * 1-thx+1+thx = 1 _

4⁴√(1+thx)³ ch²x(1-thx) 4ch²x√(1+thx) ⁴√(1-th²x)

10.12.

y'= chx(1+chx)-sh²x = 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'= 16shxch³x*ln(chx)+8ch³xshx-16ch³xshxln(chx) = 4thx

2ch⁴x

10.16.

y'= 2shxchx(12sh²x+1)-24sh³xchx = 4chx

3sh⁴x 3sh³x

10.17.

y'= 2chxsh²x-ch³x + 3 = shx-1 + 3 _

2ch⁴x ch²x√(1-th²x) 2ch³x ch²x√(1-th²x)

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)√(ch²x-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(sh²x-3chx-ch²x) = 1+chx _

8ch²(x/2)th(x/2) 4sh²x(3+chx) shx(3+chx)

10.21.

y'= -(3+5chx)(3shx(3+5chx)-5shx(5+3chx)) = 1

4(3+5chx)²√(9+30chx+25ch²x-25-30chx-9ch²x)

10.22.

y'= -16ch⁵xshx-4ch³x(1-8ch²x) = -4ch²xshx-1+8ch²x

4ch⁸x ch⁵x

10.23.

y'= -2/sh²x+1/sh⁴x+ch³x-2chxsh²x+ 5chx = -2/sh²x+1/sh⁴x+1-sh²x+ 5_

2ch⁴x 2+2sh²x 2ch³x 2chx

10.24.

y'= -16 + sh⁴x+3sh²xch²x = 1-4sh²x

3sh²2x 3ch²xsh⁶x ch²xsh⁴x

10.25.

y'= chx _ 1-sh²x = 1 _ 1-sh²x

2+2sh²x 2ch³x 2ch²x 2ch³x

10.26.

y'= 3 + shx – sh³x-2shxch²x = 1+sh²x

4ch²(x/2)th(x/2) 2sh⁴x shx

10.27.

y'= 2chxsh²x-ch³x + 2shxchx _ 3chx = sh²x-1 + 2chx _ 3 _

2ch⁴x sh⁴x 2+2sh²x 2ch³x sh³x 2chx

10.28.

y'= ch³x-2chxsh²x + chx = 1 _

2ch⁴x 2+2sh²x ch³x

10.29.

y'= ch³x-2chxsh²x + chx = 1 _

2ch⁴x 2+2sh²x ch³x

10.30.

y'= 2ch²xshx-sh³x _ 1 = 1/sh³x

2sh⁴x 4ch²(x/2)th(x/2)

10.31.

y'= -2 _ sh⁴x-3ch²xsh²x = 1/sh⁴x

3sh²x 3sh⁶x