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Lesson: Second Derivatives of Parametric Equations

In this lesson, we will learn how to find higher-order derivatives (d²y/dx²) of parametric equations by applying the chain rule.

Sample Question Videos

07:26

Worksheet • 17 Questions • 1 Video

Q1:

Given that π‘₯ = 𝑑 + 5 3 and 𝑦 = 𝑑 βˆ’ 3 𝑑 2 , find d d 2 2 𝑦 π‘₯ .

  • A 2 ( 3 βˆ’ 𝑑 ) 9 𝑑 5
  • B 2 ( 3 βˆ’ 𝑑 ) 3 𝑑 ( 2 𝑑 βˆ’ 3 ) 3
  • C 2 ( 3 βˆ’ 𝑑 ) 𝑑
  • D 9 𝑑 2 ( 3 βˆ’ 𝑑 ) 5
  • E 𝑑 2 ( 3 βˆ’ 𝑑 )

Q2:

Given that π‘₯ = 3 𝑑 + 1 3 and 𝑦 = 3 𝑑 βˆ’ 𝑑 2 , find d d 2 2 𝑦 π‘₯ .

  • A 2 ( 1 βˆ’ 3 𝑑 ) 8 1 𝑑 5
  • B 2 ( 1 βˆ’ 3 𝑑 ) 9 𝑑 ( 6 𝑑 βˆ’ 1 ) 3
  • C 2 ( 1 βˆ’ 3 𝑑 ) 𝑑
  • D 8 1 𝑑 2 ( 1 βˆ’ 3 𝑑 ) 5
  • E 𝑑 2 ( 1 βˆ’ 3 𝑑 )

Q3:

Given that π‘₯ = 3 𝑑 + 1 3 and 𝑦 = 5 𝑑 βˆ’ 𝑑 2 , find d d 2 2 𝑦 π‘₯ .

  • A 2 ( 1 βˆ’ 5 𝑑 ) 8 1 𝑑 5
  • B 2 ( 1 βˆ’ 5 𝑑 ) 9 𝑑 ( 1 0 𝑑 βˆ’ 1 ) 3
  • C 2 ( 1 βˆ’ 5 𝑑 ) 𝑑
  • D 8 1 𝑑 2 ( 1 βˆ’ 5 𝑑 ) 5
  • E 𝑑 2 ( 1 βˆ’ 5 𝑑 )

Q4:

Given that π‘₯ = 2 𝑒 2 𝑑 and 𝑦 = 𝑑 𝑒 βˆ’ 2 𝑑 , find d d 2 2 𝑦 π‘₯ .

  • A 4 𝑑 βˆ’ 3 8 𝑒 6 𝑑
  • B 2 ( 3 βˆ’ 4 𝑑 )
  • C 2 ( 4 𝑑 βˆ’ 3 )
  • D 3 βˆ’ 4 𝑑 8 𝑒 6 𝑑
  • E 8 𝑒 4 𝑑 βˆ’ 3 6 𝑑

Q5:

Given that π‘₯ = 𝑒 𝑑 and 𝑦 = 4 𝑑 𝑒 βˆ’ 𝑑 , find d d 2 2 𝑦 π‘₯ .

  • A 4 ( 2 𝑑 βˆ’ 3 ) 𝑒 3 𝑑
  • B 4 ( 3 βˆ’ 2 𝑑 )
  • C 4 ( 2 𝑑 βˆ’ 3 )
  • D 4 ( 3 βˆ’ 2 𝑑 ) 𝑒 3 𝑑
  • E 𝑒 4 ( 2 𝑑 βˆ’ 3 ) 3 𝑑

Q6:

Given that π‘₯ = 𝑑 + 1 2 and 𝑦 = 𝑒 βˆ’ 1 𝑑 , find d d 2 2 𝑦 π‘₯ .

  • A 𝑒 ( 𝑑 βˆ’ 1 ) 4 𝑑 𝑑 3
  • B 𝑑 βˆ’ 1 2 𝑑 2
  • C 𝑒 ( 𝑑 βˆ’ 1 ) 𝑑 𝑑
  • D 𝑒 ( 𝑑 βˆ’ 1 ) 2 𝑑 𝑑 3
  • E 4 𝑑 𝑒 ( 𝑑 βˆ’ 1 ) 3 𝑑

Q7:

Given that π‘₯ = 2 𝑑 + 4 2 and 𝑦 = 5 𝑒 βˆ’ 4 5 𝑑 , find d d 2 2 𝑦 π‘₯ .

  • A 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 1 6 𝑑 5 𝑑 3
  • B 5 𝑑 βˆ’ 1 4 𝑑 2
  • C 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 𝑑 5 𝑑
  • D 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 8 𝑑 5 𝑑 3
  • E 1 6 𝑑 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 3 5 𝑑

Q8:

Determine d d 2 2 𝑦 π‘₯ , given that π‘₯ = 6 𝑛 l n 5 and 𝑦 = βˆ’ 8 𝑛 3 .

  • A βˆ’ 2 𝑛 2 5 3
  • B βˆ’ 1 2 𝑛 5 2
  • C βˆ’ 4 𝑛 5 3
  • D βˆ’ 8 𝑛 5 3

Q9:

Given that d d 𝑧 π‘₯ = 5 π‘₯ βˆ’ 6 and d d 𝑦 π‘₯ = 2 π‘₯ βˆ’ 1 2 , determine d d 2 2 𝑧 𝑦 at π‘₯ = 1 .

  • A9
  • B14
  • C βˆ’ 1 4
  • D βˆ’ 1

Q10:

Given that d d 𝑧 π‘₯ = βˆ’ 7 π‘₯ + 7 and d d 𝑦 π‘₯ = 3 π‘₯ βˆ’ 1 2 , determine d d 2 2 𝑧 𝑦 at π‘₯ = 0 .

  • A7
  • B42
  • C βˆ’ 4 2
  • D βˆ’ 7

Q11:

Given that d d 𝑧 π‘₯ = βˆ’ π‘₯ + 6 and d d 𝑦 π‘₯ = 2 π‘₯ + 5 2 , determine d d 2 2 𝑧 𝑦 at π‘₯ = βˆ’ 1 .

  • A 3 7
  • B βˆ’ 2 6 3 4 3
  • C 2 6 3 4 3
  • D1

Q12:

If π‘₯ = 8 8 𝑧 s e c and √ 5 𝑦 = 7 8 𝑧 t a n , find d d 2 2 𝑦 π‘₯ .

  • A 4 9 1 6 0
  • B 4 9 2 0
  • C 4 9 3 2 0
  • D 7 1 6 0
  • E 4 9 3 2

Q13:

If π‘₯ = 4 6 𝑧 s e c and √ 𝑦 = 7 6 𝑧 t a n , find d d 2 2 𝑦 π‘₯ .

  • A 4 9 8
  • B 4 9 1 6
  • C 4 9 2
  • D 7 8

Q14:

If π‘₯ = 2 5 𝑧 s e c and √ 3 𝑦 = 5 𝑧 t a n , find d d 2 2 𝑦 π‘₯ .

  • A 1 6
  • B 1 1 2
  • C 1 3
  • D 1 2

Q15:

Find d d 2 2 𝑦 π‘₯ if π‘₯ = βˆ’ 𝑒 4 𝑛 and 𝑦 = βˆ’ 2 𝑛 4 .

  • A 𝑛 2 𝑒 ( 4 𝑛 βˆ’ 3 ) 2 βˆ’ 8 𝑛
  • B 𝑒 𝑛 ( βˆ’ 8 𝑛 + 6 ) βˆ’ 4 𝑛 2
  • C 2 𝑒 𝑛 βˆ’ 4 𝑛 3
  • D βˆ’ 𝑒 𝑛 βˆ’ 4 𝑛 2

Q16:

If 𝑦 = ( π‘₯ + 4 ) ο€Ή βˆ’ 4 π‘₯ βˆ’ 1  2 and 𝑧 = ( π‘₯ βˆ’ 5 ) ( π‘₯ + 4 ) , find ( 2 π‘₯ βˆ’ 1 ) 𝑦 𝑧 3 2 2 d d .

  • A βˆ’ 2 4 π‘₯ + 2 4 π‘₯ + 3 4 2
  • B βˆ’ 4 8 π‘₯ + 7 2 π‘₯ + 4 4 π‘₯ βˆ’ 3 4 3 2
  • C βˆ’ 7 2 π‘₯ βˆ’ 1 0 4 π‘₯ + 3 0 2
  • D βˆ’ 9 6 π‘₯ + 1 6 π‘₯ + 1 2 0 π‘₯ βˆ’ 8 4 π‘₯ + 1 6 4 3 2

Q17:

If 𝑦 = ( βˆ’ π‘₯ + 2 ) ο€Ή 3 π‘₯ + 2  2 and 𝑧 = ( βˆ’ π‘₯ + 2 ) ( π‘₯ + 3 ) , find ( βˆ’ 2 π‘₯ βˆ’ 1 ) 𝑦 𝑧 3 2 2 d d .

  • A 1 8 π‘₯ + 1 8 π‘₯ βˆ’ 1 6 2
  • B βˆ’ 3 6 π‘₯ βˆ’ 5 4 π‘₯ + 1 4 π‘₯ + 1 6 3 2
  • C 5 4 π‘₯ βˆ’ 3 0 π‘₯ βˆ’ 8 2
  • D βˆ’ 7 2 π‘₯ βˆ’ 6 0 π‘₯ + 1 8 π‘₯ + 2 7 π‘₯ + 6 4 3 2
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