Задача 19. Найти производную второго порядка от функции, заданной параметрически.

19.1.

x'= -2sin2t= -4sintcost

y'= 4sint/cos³t

y''_xx= 4sint = -1 _

16sin²tcos⁵t 4sintcos⁵t

19.2.

x'= -t/√(1-t²)

y'= -1/t²

y''_xx= (1-t²)² t⁴

19.3.

x'= e^tcost-e^tsint= e^t(cost-sint)

y'= e^tsint+e^tcost= e^t(sint+cost)

y''_xx= e^t(sint+cost) = sint+cost

e^2t(cost-sint)² e^t(cost-sint)²

19.4.

x'= 2shtcht

y'= -2sht/ch³t

y''_xx= -2sht = -1_

4shtch⁴t 2ch⁴t

19.5.

x'= 1+cost

y'= sint

y''_xx= sint/(1+cost)²

19.6.

x'= -1/t²

y'= -2t/(1+t²)²

y''_xx= -2t³ _

(1+t²)²

19.7.

x'= 1/2√t

y'= 1/√(1-t)³

y''_xx= 4t _

√(1-t)³

19.8.

x'= cost

y'= sint/cos²t

y''_xx= sint/cos⁴t

19.9.

x'= 1/cos²t

y'= -2cos2t/sin²2t

y''_xx= -2cos2tcos⁴t

sin²2t

19.10.

x'= 1/2√(t-1)

y'= (2-t)/(1-t)^3/2

y''_xx= 4(t-1)(2-t) = 2t-8

(1-t)^3/2 √(1-t)

19.11.

x'= 1/2√t

y'= 1/³√(t-1)²

y''_xx= 4t/³√(t-1)²

19.12.

x'= -sint/(1+2cost)²

y'= (cost+2)/(1+2cost)²

y''_xx= (cost+2)(1+2cost)⁴ = (cost+2)(1+2cost)²

sin²t(1+2cost)² sin²t

19.13.

x'= 3t² / 2√(t³-1)

y'= 1/t

y''_xx= 2(t³-1)

3t⁵

19.14.

x'= cht

y'= 2tht/ch²t

y''_xx= 2tht/ch⁴t

19.15.

x'= 1/2√(t-1)

y'= -1/2√t³

y''_xx= -2t+2

√t³

19.16.

x'= -2cost sint

y'= 2sint/cos³t

y''_xx= 2sint = 1/2cos⁴t

4cos⁴tsint

19.17.

x'= 1/2√(t-3)

y'= 1/(t-2)

y''_xx= 4(t-3)/(t-2)

19.18.

x'= cost

y'= -sint/cost

y''_xx= -sint/cos³t

19.19.

x'= 1+cost

y'= -sint

y''_xx= -sint/(1+cost)²

19.20.

x'= 1-cost

y'= sint

y''_xx= sint/(1-cost)²

19.21.

x'= -sint

y'= cost/sint

y''_xx= -cost/sin³t

19.22.

x'= -sint+sint+tcost= tcost

y'= cost-cost+tsint= tsint

y''_xx= sint/cos²t

19.23.

x'= e^t

y'= 1/√(1-t²)

y''_xx= e^t/√(1-t²)

19.24.

x'= -sint

y'= 2sin3(t/2)cos(t/2)

y''_xx= -2sin³(t/2)cos(t/2)/sint= -sin²(t/2)

19.25.

x'= sht

y'= 2cht/3³√sht

y''_xx= 2cht/3³√sh⁴t

19.26.

x'= 1/(1+t²)

y'= t

y''_xx= t(1+t²)²

19.27.

x'= 2-2cost

y'= -4sint

y''_xx= -2sint/(1-cost)

19.28.

x'= cost-cost+tsint= tsint

y'= -sint+sint+tcost= tcost

y''_xx= cost/sin²t

19.29.

x'= -2/t³

y'= -2t/(t²+1)²

y''_xx= -t⁷/2(t²+1)²

19.30.

x'= cost-sint

y'= 2cos2t

y''_xx= 2cos2t/( cost-sint)= 2cost+2sint

19.31.

x'= 1/t

y'= 1/(1+t²)

y''_xx= t²/(1+t²)