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.

  • A9𝑡(10𝑡−1)
  • B3𝑡(5𝑡−1)
  • C3𝑡5𝑡−1
  • D9𝑡10𝑡−1
  • E10𝑡−19𝑡

Q2:

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

  • A4𝑡+54𝑡
  • B4𝑡(4𝑡+5)
  • C8𝑡8𝑡+5
  • D8𝑡+58𝑡
  • E8𝑡(8𝑡+5)

Q3:

Given that 𝑥=3𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A5𝑡−115𝑒
  • B15(1−5𝑡)
  • C1−5𝑡15𝑒
  • D15𝑒1−5𝑡
  • E15(5𝑡−1)

Q4:

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

  • A(4𝑡+5)(5𝑡−4)3𝑡
  • B(4𝑡+5)(5𝑡−4)𝑡
  • C4𝑡+55𝑡−4
  • D4𝑡+53(5𝑡−4)
  • E5𝑡−44𝑡+5

Q5:

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

  • A−22𝑡𝑡cossin
  • B−𝑡22𝑡sincos
  • C−2𝑡𝑡cossin
  • D−22𝑡𝑡cossin
  • E−2𝑡𝑡cossin

Q6:

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

  • A4(4+𝑡)
  • B16(4+𝑡)
  • C18(4+𝑡)
  • D116(4+𝑡)
  • E16(4+𝑡)

Q7:

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

  • A14
  • B14𝑥+𝑦
  • C14𝑦+𝑧
  • D6𝑥
  • E14𝑥

Q8:

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

  • A−√510
  • B√5
  • C√520
  • 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.

  • A5𝑒(𝑡+1)(3−4𝑡)cos
  • B3+4𝑡5𝑒(𝑡+1)cos
  • C3+4𝑡5𝑒(𝑡−1)cos
  • D3−4𝑡5𝑒(𝑡+1)cos
  • E5𝑒(𝑡+1)(3+4𝑡)cos

Q11:

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

  • A−4√3+14
  • B−72−√3
  • C4√3+14
  • D−14−4√3

Q12:

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

  • A√357
  • B19√34
  • C−√312
  • D√312

Q13:

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

  • A2√39
  • B−4√39
  • C−29
  • D−2√39

Q14:

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

  • A120𝑥+16𝑥
  • B15𝑥+2𝑥8𝑥
  • C5𝑥+𝑥4𝑥
  • D20𝑥+4𝑥

Q15:

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

  • A23𝑡
  • B𝑡
  • C3𝑡2
  • D1𝑡

Q16:

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

  • A−548
  • B−2548
  • C−596
  • D−56

Q17:

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

  • A2𝑥+9
  • B−14(𝑥−7)
  • C−10𝑥−45(𝑥−7)
  • D−2𝑥5−95(𝑥−7)

Q18:

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

  • A1
  • B−48
  • C−14
  • D−4

Q19:

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

  • A2𝑦−𝑥=0
  • B𝑦−2𝑥+5√3=0
  • C𝑦+2𝑥−25√33=0
  • D−2𝑦−𝑥+20√33=0

Q20:

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

  • A12
  • B1
  • C−52
  • D−12

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.

  • A12
  • B−17, −13
  • C−13
  • D−17

Q23:

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

  • A20
  • B−9
  • C45
  • D54

Q24:

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

  • A34
  • B112
  • C43
  • D12

Q25:

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

  • A𝑦=2𝑥+1
  • B𝑦=1𝜋𝑥+1
  • C𝑦=2𝜋𝑥+1
  • D𝑦=𝜋2𝑥+1
  • E𝑦=𝑥+1

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