Worksheet: Second Derivatives of Parametric Equations

In this worksheet, we will practice finding higher-order derivatives (d²y/dx²) of parametric equations by applying the chain rule.

Q1:

Given that 𝑥 = 𝑡 + 5 3 and 𝑦 = 𝑡 3 𝑡 2 , find d d 2 2 𝑦 𝑥 .

  • A 𝑡 2 ( 3 𝑡 )
  • B 2 ( 3 𝑡 ) 𝑡
  • C 9 𝑡 2 ( 3 𝑡 ) 5
  • D 2 ( 3 𝑡 ) 9 𝑡 5
  • E 2 ( 3 𝑡 ) 3 𝑡 ( 2 𝑡 3 ) 3

Q2:

Given that 𝑥 = 2 𝑒 2 𝑡 and 𝑦 = 𝑡 𝑒 2 𝑡 , find d d 2 2 𝑦 𝑥 .

  • A 8 𝑒 4 𝑡 3 6 𝑡
  • B 2 ( 4 𝑡 3 )
  • C 3 4 𝑡 8 𝑒 6 𝑡
  • D 4 𝑡 3 8 𝑒 6 𝑡
  • E 2 ( 3 4 𝑡 )

Q3:

Given that 𝑥 = 𝑡 + 1 2 and 𝑦 = 𝑒 1 𝑡 , find d d 2 2 𝑦 𝑥 .

  • A 4 𝑡 𝑒 ( 𝑡 1 ) 3 𝑡
  • B 𝑒 ( 𝑡 1 ) 𝑡 𝑡
  • C 𝑒 ( 𝑡 1 ) 2 𝑡 𝑡 3
  • D 𝑒 ( 𝑡 1 ) 4 𝑡 𝑡 3
  • E 𝑡 1 2 𝑡 2

Q4:

Given that d d 𝑧 𝑥 = 5 𝑥 6 and d d 𝑦 𝑥 = 2 𝑥 1 2 , determine d d 2 2 𝑧 𝑦 at 𝑥 = 1 .

  • A 1
  • B14
  • C 1 4
  • D9

Q5:

Find d d 2 2 𝑦 𝑥 if 𝑥 = 𝑒 4 𝑛 and 𝑦 = 2 𝑛 4 .

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

Q6:

If 𝑥 = 8 8 𝑧 s e c and 5 𝑦 = 7 8 𝑧 t a n , find d d 2 2 𝑦 𝑥 .

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

Q7:

If 𝑦 = ( 𝑥 + 4 ) 4 𝑥 1 2 and 𝑧 = ( 𝑥 5 ) ( 𝑥 + 4 ) , find ( 2 𝑥 1 ) 𝑦 𝑧 3 2 2 d d .

  • A 9 6 𝑥 + 1 6 𝑥 + 1 2 0 𝑥 8 4 𝑥 + 1 6 4 3 2
  • B 4 8 𝑥 + 7 2 𝑥 + 4 4 𝑥 3 4 3 2
  • C 7 2 𝑥 1 0 4 𝑥 + 3 0 2
  • D 2 4 𝑥 + 2 4 𝑥 + 3 4 2

Q8:

Determine d d 2 2 𝑦 𝑥 , given that 𝑥 = 6 𝑛 l n 5 and 𝑦 = 8 𝑛 3 .

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

Q9:

Given that 𝑥 = 3 𝑡 + 1 3 and 𝑦 = 3 𝑡 𝑡 2 , find d d 2 2 𝑦 𝑥 .

  • A 𝑡 2 ( 1 3 𝑡 )
  • B 2 ( 1 3 𝑡 ) 𝑡
  • C 8 1 𝑡 2 ( 1 3 𝑡 ) 5
  • D 2 ( 1 3 𝑡 ) 8 1 𝑡 5
  • E 2 ( 1 3 𝑡 ) 9 𝑡 ( 6 𝑡 1 ) 3

Q10:

Given that 𝑥 = 3 𝑡 + 1 3 and 𝑦 = 5 𝑡 𝑡 2 , find d d 2 2 𝑦 𝑥 .

  • A 𝑡 2 ( 1 5 𝑡 )
  • B 2 ( 1 5 𝑡 ) 𝑡
  • C 8 1 𝑡 2 ( 1 5 𝑡 ) 5
  • D 2 ( 1 5 𝑡 ) 8 1 𝑡 5
  • E 2 ( 1 5 𝑡 ) 9 𝑡 ( 1 0 𝑡 1 ) 3

Q11:

Given that 𝑥 = 𝑒 𝑡 and 𝑦 = 4 𝑡 𝑒 𝑡 , find d d 2 2 𝑦 𝑥 .

  • A 𝑒 4 ( 2 𝑡 3 ) 3 𝑡
  • B 4 ( 2 𝑡 3 )
  • C 4 ( 3 2 𝑡 ) 𝑒 3 𝑡
  • D 4 ( 2 𝑡 3 ) 𝑒 3 𝑡
  • E 4 ( 3 2 𝑡 )

Q12:

Given that 𝑥 = 2 𝑡 + 4 2 and 𝑦 = 5 𝑒 4 5 𝑡 , find d d 2 2 𝑦 𝑥 .

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

Q13:

Given that d d 𝑧 𝑥 = 7 𝑥 + 7 and d d 𝑦 𝑥 = 3 𝑥 1 2 , determine d d 2 2 𝑧 𝑦 at 𝑥 = 0 .

  • A 7
  • B42
  • C 4 2
  • D7

Q14:

Given that d d 𝑧 𝑥 = 𝑥 + 6 and d d 𝑦 𝑥 = 2 𝑥 + 5 2 , determine d d 2 2 𝑧 𝑦 at 𝑥 = 1 .

  • A1
  • B 2 6 3 4 3
  • C 2 6 3 4 3
  • D 3 7

Q15:

If 𝑥 = 4 6 𝑧 s e c and 𝑦 = 7 6 𝑧 t a n , find d d 2 2 𝑦 𝑥 .

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

Q16:

If 𝑥 = 2 5 𝑧 s e c and 3 𝑦 = 5 𝑧 t a n , find d d 2 2 𝑦 𝑥 .

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

Q17:

If 𝑦 = ( 𝑥 + 2 ) 3 𝑥 + 2 2 and 𝑧 = ( 𝑥 + 2 ) ( 𝑥 + 3 ) , find ( 2 𝑥 1 ) 𝑦 𝑧 3 2 2 d d .

  • A 7 2 𝑥 6 0 𝑥 + 1 8 𝑥 + 2 7 𝑥 + 6 4 3 2
  • B 3 6 𝑥 5 4 𝑥 + 1 4 𝑥 + 1 6 3 2
  • C 5 4 𝑥 3 0 𝑥 8 2
  • D 1 8 𝑥 + 1 8 𝑥 1 6 2

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.