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Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.19

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'= etcost-etsint= et(cost-sint)

y'= etsint+etcost= et(sint+cost)

y''xx= et(sint+cost) = sint+cost

e2t(cost-sint)2 et(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'= et

y'= 1/√(1-t2)

y''xx= et/√(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)