Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.15

СОДЕРЖАНИЕ: Задача 15 . Найти производную 15.1. 6t*t y= cos(t /3+t)(t cos(t /3+t)(t +1)t cos(t /3+t) 15.2. (1-t2) 2(1+t)cos (1+t) -(1-t

Задача 15 . Найти производную .

15.1.

x= 6t*t³ -3t² (3t² +1) = -t² -1

3t⁶ t⁴

y= cos(t³ /3+t)(t² +1)

y_x = cos(t³ /3+t)(t² +1)t⁴ = -t⁴ cos(t³ /3+t)

-t² -1

15.2.

x= -t _

(1-t2)

y= 1 _

2(1+t)cos² (1+t)

y_x = -(1-t² ) = -(1-t² ) _

2t(1+t)cos² (1+t) 2tcos² (1+t)

15.3.

x= 1-t _

(2t-t² )

y= 2 _

3 ³ (1-t)⁵

y_x = 2(2t-t² ) = 2(2t-t² ) _

3 ³ (1-t)⁵ (1-t) 3 ³ (1-t)² (1-t)²

15.4.

x= cost = 1

(1-sin² t)

y= sint = 1

(1-cos² t)

y_x = 1

15.5.

x= 1+t/(t² +1) = 1 _

t+(t² +1) (t² +1)

y= (t² +1)+ t² = 2t² +1_

(t² +1) (t² +1)

y_x = (2t² +1)(t² +1) = 2t² +1

(t² +1)

15.6.

x= 1-t _

(2t-t² )

y= 1 = 1 _

(1-(t-1)² ) (2t-t² )

y_x = (2t-t² ) = 1_

(t2-t² )(1-t) 1-t

15.7.

x= -2e^t _ = -2e^t _

sin² (2e^t ) 4sin² e^t cos² e^t

y= e^t = e^t _

tge^t cos² e^t sine^t cose^t

y_x = 4e^t sin² e^t cos² e^t = -2sine^t cose^t

-2e^t sine^t cose^t

15.8.

x= -1 = -1 _

ctgt sin² t sint cost

y= 2sint

cos³ t

y_x = -cos³ t = -1/2*ctg² t

2sin² tcost

15.9.

x= e^t/2 _

2(1+e^t )

y= e^t _

2(1+e^t )

y_x = 2e^t (1+e^t ) = (e^t +e^2t )

2e^t/2 (e^t +1)

15.10.

x= (1+t) *(1+t) *-1-t-1+t = -1_

(1-t) 2(1-t) (1+t)² 1-t²

y= -t _

(1-t² )

y_x = t(1-t² ) = t(1-t² )

(1-t² )

15.11.

x= 2t3_

1-t4

y= (1+t² )(-2t(1+t² )-2t(1-t² )) = -2

(1+2t² +t⁴ -1+2t² -t⁴ )(1+t² )

y_x = -2(1-t⁴ ) = t⁴ -1

2t³ t³

15.12.

x= -t _

(1-t² )

y= (1-t² )+t² /(1-t² ) = 1 _

1-t² (1-t² )^3/2

y_x = -(1-t² )_ = 1_

t(1-t² )^3/2 t³ -t

15.13.

x= -t = -1 _

(1-1+t² )(1-t² ) (1-t² )

y= -2arccost

(1-t² )

y_x = 2arccost(1-t² ) = 2arccost

(1-t² )

15.14.

x= (1-t² )+t² /(1-t² ) = 1 _

1-t² (1-t² )^3/2

y= t *-t² /(1-t² )-1-(1-t² ) = -1

1+(1-t² ) t² t

y_x = (1-t² )

t²

15.15.

x= -4(1+cos² t)costsint

y= -sin³ t-2cos² tsint = -1-cos² t

sin⁴ t sin³ t

y_x = 1+cos² t = 1 _

4sin³ t(1+cos² t)costsint 4sin⁴ tcost

15.16.

x= (1+t)(-1-t-1+t) = -2_

(1-t)(1+t)² 1-t²

y= -t _

(1-t² )

y_x = -t (1-t² )_ = t(1-t² )

-2(1-t² ) 2

15.17.

x= 1 = 1 _

t² (1-1/t² ) t(t² -1)

y= t + t = 2t _

(t² -1) (t² -1) (t² -1)

y_x = 2t² (t² -1) = 2t²

(t² -1)

15.18.

x= 1_

tln² t

y= t *-t² /(1-t² )-1-(1-t² ) = -1

1+(1-t² ) t² t

y_x = -tln² t = -ln² t

t

15.19.

x= 1 _

2t(1-t)

y= 1 _

4t(1+t)

y_x = 2t(1-t) = (1-t)

4t(1+t) 2

15.20.

x= 2arcsint

(1-t² )

y= (1-t² )+t² /(1-t² ) = 1 _

1-t² (1-t² )^3/2

y_x = (1-t² ) = 1 _

2(1-t² )^3/2 arcsint 2(1-t² )arcsint

15.21.

x= (t² +1)+t² /(t² +1) = 2t² +1

(t² +1)

y= t *-t² /(1-t² )-1-(1-t² ) = -1

1+(1-t² ) t² t

y_x = -(t² +1) = -1 _

(2t² +1)(t² +1) 2t² +1

15.22.

x= 1/(1+t² )

y= (t+1)(t(t+1)/(t² +1)-(1+t² )) = t-1 _

(1+t² )(1+t) (t² +1)(1+t)

y_x = (t-1)(1+t² ) = (t-1)(1+t² )

(1+t² )(t+1) 1+t

15.23.

x= -2t/(1-t² )

y= -t = -1/(1-t² )

(1-1+t² )(1-t² )

y_x = 1-t² = (1-t² )

2t(1-t² ) 2t

15.24.

x= (t-1)² (t-1-t-1) = -1_

((t-1)² +(t+1)² )(t-1)² t² +1

y= -t = -1/(1-t² )

(1-1+t² )(1-t² )

y_x = t² +1_

(1-t² )

15.25.

x= (1+sint)(1+sint)(-cost(1+sint)-cost(1-sint)) = -1_

2(1-sint)(1-sint)(1+sint)² cost

y= tgt/cos² t-tgt= tg³ t

y_x = -tg³ tcost

15.26.

x= 1-2t _ tt(-t-1+t) = (1-t)

2(t-t² ) 2(t+1-t)(1-t)t² 2t

y= 1 + arcsint _ (1-t) = arcsint

2t 2(1-t) 2t(1-t) 2(1-t)

y_x = 2t arcsint = t arcsint

2(1-t) (1-t)

15.27.

x= 1 = 1 _

tgtcos² t sintcost

y= -2cost

sin³ t

y_x = -2cost = -2_

sin⁴ tcost sin⁴ t

15.28.

x= (2tlnt+t)(1-t² )+2t³ lnt – 2t = 2tlnt

(1-t² )² 2(1-t² ) (1-t² )

y= (1-t² )+t² /(1-t² ) arcsint + t/(1-t² ) – t/(1-t² ) = arcsint

1-t² (1-t² )^3/2

y_x = arcsint(1-t² )² = arcsint(1-t² )

2tlnt(1-t² )^3/2 2tlnt

15.29.

x= 2e^{sec^2t} sec² t tgt= 2e^{sec^2t} sint

cos³ t

y= lncost _ sint + 1/cos² t-1= lncost-sintcost+sin² t

cos² t cost cos² t

y_x = 1/2*e^{-sec^2t} ctgt(lncost-sintcost+sin² t)

15.30.

x= (1-t² )+t² /(1-t² ) arcsint + t/(1-t² ) – t/(1-t² ) = arcsint

1-t² (1-t² )^3/2

y= (1-t² )+t² /(1-t² ) = 1 _

1-t² (1-t² )^3/2

y_x = (1-t² )^3/2 = 1 _

(1-t² )^3/2 arcsint arcsint

15.31.

x= 1+t/(t² +1) = 1 _

t+(t² +1) (t² +1)

y= t _ t *-t² /(1-t² )-1-(1-t² ) = t + 1 = t² +2(1-t² )

2(1+t² ) 1+(1-t² ) t² 2(1+t² ) t 2t(1-t² )

y_x = (t² +2(1-t² ))(1+t² )

2t(1-t² )

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