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Реферат: 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/cos3 t
y''xx = 4sint = -1 _
16sin2 tcos5 t 4sintcos5 t
19.2. 
x'= -t/√(1-t2 )
y'= -1/t2
y''xx = (1-t2 )2 t4
19.3. 
x'= et cost-et sint= et (cost-sint)
y'= et sint+et cost= et (sint+cost)
y''xx = et (sint+cost) = sint+cost
e2t (cost-sint)2 et (cost-sint)2
19.4. 
x'= 2shtcht
y'= -2sht/ch3 t
y''xx = -2 sht = -1_
4shtch4 t 2ch4 t
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 = 4 t _
√(1-t)3
19.8. 
x'= cost
y'= sint/cos2 t
y''xx = sint/cos4 t
19.9. 
x'= 1/cos2 t
y'= -2cos2t/sin2 2t
y''xx = -2cos2tcos4 t
sin2 2t
19.10. 
x'= 1/2√(t-1)
y'= (2-t)/(1-t)3/2
y''xx = 4( t -1)(2- t ) = 2 t -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
sin2 t(1+2cost)2 sin2 t
19.13. 
x'= 3t2 / 2√(t3 -1)
y'= 1/t
y''xx = 2 ( t 3 -1)
3t5
19.14. 
x'= cht
y'= 2tht/ch2 t
y''xx = 2tht/ch4 t
19.15. 
x'= 1/2√(t-1)
y'= -1/2√t3
y''xx = - 2t + 2
√t3
19.16. 
x'= -2cost sint
y'= 2sint/cos3 t
y''xx = 2sint = 1/2cos4 t
4cos4 tsint
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/cos3 t
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/sin3 t
19.22. 
x'= -sint+sint+tcost= tcost
y'= cost-cost+tsint= tsint
y''xx = sint/cos2 t
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 √sh4 t
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/sin2 t
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 )