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Worksheet: Finding the Derivatives of Parametric Functions

Q1:

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

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

Q2:

Given that π‘₯ = 4 𝑑 + 1 2 and 𝑦 = 4 𝑑 + 5 𝑑 2 , find d d 𝑦 π‘₯ .

  • A 8 𝑑 8 𝑑 + 5
  • B 8 𝑑 ( 8 𝑑 + 5 )
  • C 4 𝑑 + 5 4 𝑑
  • D 8 𝑑 + 5 8 𝑑
  • E 4 𝑑 ( 4 𝑑 + 5 )

Q3:

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

  • A 1 5 𝑒 1 βˆ’ 5 𝑑 1 0 𝑑
  • B 1 5 ( 1 βˆ’ 5 𝑑 )
  • C 5 𝑑 βˆ’ 1 1 5 𝑒 1 0 𝑑
  • D 1 βˆ’ 5 𝑑 1 5 𝑒 1 0 𝑑
  • E 1 5 ( 5 𝑑 βˆ’ 1 )

Q4:

Given that π‘₯ = 5 𝑑 βˆ’ 4 𝑑 l n and 𝑦 = 4 𝑑 + 5 3 𝑑 l n , find d d 𝑦 π‘₯ .

  • A 5 𝑑 βˆ’ 4 4 𝑑 + 5
  • B ( 4 𝑑 + 5 ) ( 5 𝑑 βˆ’ 4 ) 𝑑 2
  • C 4 𝑑 + 5 3 ( 5 𝑑 βˆ’ 4 )
  • D 4 𝑑 + 5 5 𝑑 βˆ’ 4
  • E ( 4 𝑑 + 5 ) ( 5 𝑑 βˆ’ 4 ) 3 𝑑 2

Q5:

Given that π‘₯ = 2 𝑑 4 + 𝑑 and 𝑦 = √ 4 + 𝑑 , find d d 𝑦 π‘₯ .

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

Q6:

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

  • A βˆ’ 5 𝑑
  • B 5 3 6 𝑑 3
  • C 5 𝑑
  • D βˆ’ 5 3 6 𝑑 3
  • E βˆ’ 5 6 𝑑 ( 6 𝑑 + 5 ) 2

Q7:

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

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

Q8:

Given that π‘₯ = 2 𝑑 βˆ’ 3 3 𝑑 l n and 𝑦 = 3 𝑑 + 4 2 𝑑 l n , find d d 2 2 𝑦 π‘₯ .

  • A βˆ’ ( 2 𝑑 βˆ’ 3 ) 1 7 𝑑 3
  • B βˆ’ 1 7 𝑑 ( 2 𝑑 βˆ’ 3 )
  • C βˆ’ 4 𝑑 ( 2 𝑑 βˆ’ 3 )
  • D βˆ’ 1 7 𝑑 ( 2 𝑑 βˆ’ 3 ) 3
  • E βˆ’ 1 7 ( 2 𝑑 βˆ’ 3 ) 2

Q9:

Given that 𝑦 = √ 4 π‘₯ βˆ’ 5 2 and 𝑧 = 5 π‘₯ + 9 2 , determine 𝑦 ο€½ 𝑦 π‘₯  + 𝑧 π‘₯ d d d d .

  • A14
  • B 6 π‘₯
  • C 1 4 π‘₯ + 𝑦
  • D 1 4 π‘₯
  • E 1 4 𝑦 + 𝑧

Q10:

Given that π‘₯ = √ βˆ’ 𝑑 + 5 and 𝑦 = √ 2 𝑑 + 1 , find d d 𝑦 π‘₯ at 𝑑 = 0 .

  • A √ 5
  • B βˆ’ √ 5 1 0
  • C √ 5 2 0
  • D βˆ’ 2 √ 5

Q11:

If 𝑦 = π‘₯ √ 5 + π‘₯ 2 and 𝑧 = √ 5 + π‘₯ 5 π‘₯ 2 , find 5 𝑧 𝑦 π‘₯ + 𝑧 π‘₯ 2 d d d d .

Q12:

Given that π‘₯ = 5 𝑑 𝑒 𝑑 and 𝑦 = 3 𝑑 + 4 𝑑 s i n , find d d 𝑦 π‘₯ .

  • A 3 βˆ’ 4 𝑑 5 𝑒 ( 𝑑 + 1 ) c o s 𝑑
  • B 5 𝑒 ( 𝑑 + 1 ) ( 3 + 4 𝑑 ) 𝑑 c o s
  • C 5 𝑒 ( 𝑑 + 1 ) ( 3 βˆ’ 4 𝑑 ) 𝑑 c o s
  • D 3 + 4 𝑑 5 𝑒 ( 𝑑 + 1 ) c o s 𝑑
  • E 3 + 4 𝑑 5 𝑒 ( 𝑑 βˆ’ 1 ) c o s 𝑑

Q13:

Find the derivative of 7 π‘₯ + 4 π‘₯ s i n with respect to c o s π‘₯ + 1 at π‘₯ = πœ‹ 6 .

  • A 4 √ 3 + 1 4
  • B βˆ’ 4 √ 3 + 1 4
  • C βˆ’ 7 2 βˆ’ √ 3
  • D βˆ’ 1 4 βˆ’ 4 √ 3

Q14:

Find d d 𝑦 π‘₯ at πœƒ = πœ‹ 3 , given π‘₯ = 5 πœƒ + 7 2 πœƒ c o s c o s and 𝑦 = 7 πœƒ + 4 2 πœƒ s i n s i n .

  • A √ 3 1 2
  • B βˆ’ √ 3 1 2
  • C 1 9 √ 3 4
  • D √ 3 5 7

Q15:

Find d d 𝑦 π‘₯ at πœƒ = 1 6 , given that π‘₯ = βˆ’ 9 2 πœ‹ πœƒ s i n and 𝑦 = βˆ’ 4 2 πœ‹ πœƒ c o s .

  • A βˆ’ 2 9
  • B βˆ’ 2 √ 3 9
  • C 2 √ 3 9
  • D βˆ’ 4 √ 3 9

Q16:

By using parametric differentiation, determine the derivative of 5 π‘₯ + π‘₯ βˆ’ 2 3 2 with respect to 4 π‘₯ + 8 2 .

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

Q17:

Given that 𝑦 = βˆ’ 7 𝑑 + 8 3 , and 𝑧 = βˆ’ 7 𝑑 + 3 2 , find the rate of the change of 𝑦 with respect to 𝑧 .

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

Q18:

Find the derivative of π‘₯ βˆ’ 6 π‘₯ βˆ’ 9 with respect to √ 8 π‘₯ + 1 at π‘₯ = 3 .

  • A βˆ’ 5 9 6
  • B βˆ’ 2 5 4 8
  • C βˆ’ 5 6
  • D βˆ’ 5 4 8

Q19:

Find the rate of change of ( π‘₯ + 2 ) ( π‘₯ + 7 ) with respect to π‘₯ βˆ’ 2 π‘₯ βˆ’ 7 .

  • A βˆ’ 1 4 ( π‘₯ βˆ’ 7 ) 2
  • B 2 π‘₯ + 9
  • C βˆ’ 1 0 π‘₯ βˆ’ 4 5 ( π‘₯ βˆ’ 7 ) 2
  • D ο€Ό βˆ’ 2 π‘₯ 5 βˆ’ 9 5  ( π‘₯ βˆ’ 7 ) 2

Q20:

Find the rate of change of l n ο€Ή βˆ’ 3 π‘₯ βˆ’ 1  4 with respect to ο€Ή βˆ’ 6 π‘₯ βˆ’ 5  2 at π‘₯ = 1 .

  • A βˆ’ 4 8
  • B1
  • C βˆ’ 4
  • D βˆ’ 1 4

Q21:

Find the equation of the tangent to the curve π‘₯ = 5 πœƒ s e c and 𝑦 = 5 πœƒ t a n at πœƒ = πœ‹ 6 .

  • A 2 𝑦 βˆ’ π‘₯ = 0
  • B 𝑦 + 2 π‘₯ βˆ’ 2 5 √ 3 3 = 0
  • C βˆ’ 2 𝑦 βˆ’ π‘₯ + 2 0 √ 3 3 = 0
  • D 𝑦 βˆ’ 2 π‘₯ + 5 √ 3 = 0

Q22:

Suppose π‘₯ = βˆ’ 3 5 πœƒ + 1 3 s e c 2 and 𝑦 = βˆ’ 3 5 πœƒ βˆ’ 1 4 t a n . Find d d 𝑦 π‘₯ when πœƒ = πœ‹ 4 .

  • A βˆ’ 1 2
  • B1
  • C βˆ’ 5 2
  • D 1 2

Q23:

If π‘₯ = βˆ’ 8 𝑑 βˆ’ 8 5 and 𝑦 = √ 𝑑 5 6 , find d d 𝑦 π‘₯ at 𝑑 = 1 .

Q24:

A curve has parametric equations π‘₯ = 7 π‘š + 5 π‘š + π‘š + 4 3 2 and 𝑦 = 6 π‘š βˆ’ 6 π‘š βˆ’ 8 2 . Find π‘š for which the tangent is horizontal.

  • A βˆ’ 1 3
  • B βˆ’ 1 7
  • C βˆ’ 1 7 , βˆ’ 1 3
  • D 1 2

Q25:

If 𝑦 = βˆ’ 5 π‘₯ βˆ’ 7 3 and 𝑧 = 3 π‘₯ + 1 6 2 , find 𝑑 𝑧 𝑑 𝑦 2 2 at π‘₯ = 1 .

  • A βˆ’ 2 5
  • B 2 5
  • C βˆ’ 5 2
  • D βˆ’ 2 7 5
  • E 2 7 5