Worksheet: Derivatives of Parametric Equations

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In this worksheet, we will practice finding the first derivative of a curve defined by parametric equations and finding the equations of tangents and normals to the curves.

Q1:

Given that 𝑥=3𝑡+1 and 𝑦=5𝑡−𝑡, find dd𝑦𝑥.

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

Q2:

Given that 𝑥=4𝑡+1 and 𝑦=4𝑡+5𝑡, find dd𝑦𝑥.

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

Q3:

Given that 𝑥=3𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A 5 𝑡 − 1 1 5 𝑒
  • B 1 5 ( 1 − 5 𝑡 )
  • C 1 − 5 𝑡 1 5 𝑒
  • D 1 5 𝑒 1 − 5 𝑡
  • E 1 5 ( 5 𝑡 − 1 )

Q4:

Given that 𝑥=5𝑡−4𝑡ln and 𝑦=4𝑡+53𝑡ln, find dd𝑦𝑥.

  • A ( 4 𝑡 + 5 ) ( 5 𝑡 − 4 ) 3 𝑡
  • B ( 4 𝑡 + 5 ) ( 5 𝑡 − 4 ) 𝑡
  • C 4 𝑡 + 5 5 𝑡 − 4
  • D 4 𝑡 + 5 3 ( 5 𝑡 − 4 )
  • E 5 𝑡 − 4 4 𝑡 + 5

Q5:

Given that 𝑥=𝑡cos and 𝑦=2𝑡sin, find dd𝑦𝑥.

  • A − 2 2 𝑡 𝑡 c o s s i n
  • B − 𝑡 2 2 𝑡 s i n c o s
  • C − 2 𝑡 𝑡 c o s s i n
  • D − 2 2 𝑡 𝑡 c o s s i n
  • E − 2 𝑡 𝑡 c o s s i n

Q6:

Given that 𝑥=2𝑡4+𝑡 and 𝑦=√4+𝑡, find dd𝑦𝑥.

  • A 4 ( 4 + 𝑡 )
  • B 1 6 ( 4 + 𝑡 )
  • C 1 8 ( 4 + 𝑡 )
  • D 1 1 6 ( 4 + 𝑡 )
  • E 1 6 ( 4 + 𝑡 )

Q7:

Given that 𝑦=√4𝑥−5 and 𝑧=5𝑥+9, determine 𝑦𝑦𝑥+𝑧𝑥dddd.

  • A14
  • B 1 4 𝑥 + 𝑦
  • C 1 4 𝑦 + 𝑧
  • D 6 𝑥
  • E 1 4 𝑥

Q8:

Given that 𝑥=√−𝑡+5 and 𝑦=√2𝑡+1, find dd𝑦𝑥 at 𝑡=0.

  • A − √ 5 1 0
  • B √ 5
  • C √ 5 2 0
  • D − 2 √ 5

Q9:

If 𝑦=𝑥√5+𝑥 and 𝑧=√5+𝑥5𝑥, find 5𝑧𝑦𝑥+𝑧𝑥dddd.

Q10:

Given that 𝑥=5𝑡𝑒 and 𝑦=3𝑡+4𝑡sin, find dd𝑦𝑥.

  • A 5 𝑒 ( 𝑡 + 1 ) ( 3 − 4 𝑡 ) c o s
  • B 3 + 4 𝑡 5 𝑒 ( 𝑡 + 1 ) c o s
  • C 3 + 4 𝑡 5 𝑒 ( 𝑡 − 1 ) c o s
  • D 3 − 4 𝑡 5 𝑒 ( 𝑡 + 1 ) c o s
  • E 5 𝑒 ( 𝑡 + 1 ) ( 3 + 4 𝑡 ) c o s

Q11:

Find the derivative of 7𝑥+4𝑥sin with respect to cos𝑥+1 at 𝑥=𝜋6.

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

Q12:

Find dd𝑦𝑥 at 𝜃=𝜋3, given 𝑥=5𝜃+72𝜃coscos and 𝑦=7𝜃+42𝜃sinsin.

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

Q13:

Find dd𝑦𝑥 at 𝜃=16, given that 𝑥=−92𝜋𝜃sin and 𝑦=−42𝜋𝜃cos.

  • A 2 √ 3 9
  • B − 4 √ 3 9
  • C − 2 9
  • D − 2 √ 3 9

Q14:

By using parametric differentiation, determine the derivative of 5𝑥+𝑥−2 with respect to 4𝑥+8.

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

Q15:

Given that 𝑦=−7𝑡+8, and 𝑧=−7𝑡+3, find the rate of the change of 𝑦 with respect to 𝑧.

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

Q16:

Find the derivative of 𝑥−6𝑥−9 with respect to √8𝑥+1 at 𝑥=3.

  • A − 5 4 8
  • B − 2 5 4 8
  • C − 5 9 6
  • D − 5 6

Q17:

Find the rate of change of (𝑥+2)(𝑥+7) with respect to 𝑥−2𝑥−7.

  • A 2 𝑥 + 9
  • B − 1 4 ( 𝑥 − 7 )
  • C − 1 0 𝑥 − 4 5 ( 𝑥 − 7 )
  • D − 2 𝑥 5 − 9 5 ( 𝑥 − 7 )

Q18:

Find the rate of change of ln−3𝑥−1 with respect to −6𝑥−5 at 𝑥=1.

  • A1
  • B − 4 8
  • C − 1 4
  • D − 4

Q19:

Find the equation of the tangent to the curve 𝑥=5𝜃sec and 𝑦=5𝜃tan at 𝜃=𝜋6.

  • A 2 𝑦 − 𝑥 = 0
  • B 𝑦 − 2 𝑥 + 5 √ 3 = 0
  • C 𝑦 + 2 𝑥 − 2 5 √ 3 3 = 0
  • D − 2 𝑦 − 𝑥 + 2 0 √ 3 3 = 0

Q20:

Suppose 𝑥=−35𝜃+13sec and 𝑦=−35𝜃−14tan. Find dd𝑦𝑥 when 𝜃=𝜋4.

  • A 1 2
  • B1
  • C − 5 2
  • D − 1 2

Q21:

If 𝑥=−8𝑡−8 and 𝑦=√𝑡, find dd𝑦𝑥 at 𝑡=1.

Q22:

A curve has parametric equations 𝑥=7𝑚+5𝑚+𝑚+4 and 𝑦=6𝑚−6𝑚−8. Find 𝑚 for which the tangent is horizontal.

  • A 1 2
  • B − 1 7 , − 1 3
  • C − 1 3
  • D − 1 7

Q23:

Determine dd𝑦𝑥 at 𝑡=0, given that 𝑥=(𝑡−2)(4𝑡+3), and 𝑦=3𝑡−4(𝑡−3).

  • A20
  • B − 9
  • C 4 5
  • D 5 4

Q24:

Find the rate of change of √2−𝑥 with respect to 𝑥5𝑥+3 at 𝑥=−1.

  • A 3 4
  • B 1 1 2
  • C 4 3
  • D12

Q25:

Find an equation of the tangent to the curve 𝑥=𝑒𝜋𝑡sin, 𝑦=𝑒 at the point corresponding to the value 𝑡=0.

  • A 𝑦 = 2 𝑥 + 1
  • B 𝑦 = 1 𝜋 𝑥 + 1
  • C 𝑦 = 2 𝜋 𝑥 + 1
  • D 𝑦 = 𝜋 2 𝑥 + 1
  • E 𝑦 = 𝑥 + 1

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