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Lesson Worksheet: Derivatives of Parametric Equations Mathematics • Higher Education

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.


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

  • A23𝑑
  • B𝑑
  • C3𝑑2
  • D1𝑑


Suppose π‘₯=9π‘›βˆ’5 and 𝑦=4βˆšπ‘›+6π‘›οŠ¨. Find dd𝑦π‘₯ when 𝑛=4.

  • Aβˆ’499
  • B479
  • C499
  • D8659


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

  • A120π‘₯+16π‘₯
  • B15π‘₯+2π‘₯8π‘₯
  • C5π‘₯+π‘₯4π‘₯
  • D20π‘₯+4π‘₯


Find the equation of the normal to the curve π‘₯=4π‘›βˆ’6π‘›οŠ¨ and 𝑦=5π‘›οŠ¨ at 𝑛=1.

  • A𝑦+5π‘₯+5=0
  • B5π‘¦βˆ’π‘₯βˆ’27=0
  • Cβˆ’5π‘¦βˆ’π‘₯+23=0
  • Dπ‘¦βˆ’5π‘₯βˆ’15=0


Find the equation of the tangent to the curve π‘₯=5πœƒsec and 𝑦=5πœƒtan at πœƒ=πœ‹6.

  • A2π‘¦βˆ’π‘₯=0
  • Bπ‘¦βˆ’2π‘₯+5√3=0
  • C𝑦+2π‘₯βˆ’25√33=0
  • Dβˆ’2π‘¦βˆ’π‘₯+20√33=0


Given that π‘₯=5𝑑𝑒 and 𝑦=3𝑑+4𝑑sin, find dd𝑦π‘₯.

  • A5𝑒(𝑑+1)(3βˆ’4𝑑)cos
  • B3+4𝑑5𝑒(𝑑+1)cos
  • C3+4𝑑5𝑒(π‘‘βˆ’1)cos
  • D3βˆ’4𝑑5𝑒(𝑑+1)cos
  • E5𝑒(𝑑+1)(3+4𝑑)cos


If π‘₯=βˆ’8π‘‘βˆ’8 and 𝑦=βˆšπ‘‘οŽ€οŠ¬, find dd𝑦π‘₯ at 𝑑=1.


Find the derivative of 7π‘₯+4π‘₯sin with respect to cosπ‘₯+1 at π‘₯=πœ‹6.

  • Aβˆ’4√3+14
  • Bβˆ’72βˆ’βˆš3
  • C4√3+14
  • Dβˆ’14βˆ’4√3


Find an equation of the tangent to the curve π‘₯=1+βˆšπ‘‘, 𝑦=π‘’οοŽ‘ at the point (2,𝑒).

  • A𝑦=2𝑒π‘₯βˆ’3𝑒
  • B𝑦=𝑒π‘₯βˆ’π‘’
  • C𝑦=4𝑒π‘₯βˆ’7𝑒
  • D𝑦=2𝑒π‘₯+4𝑒
  • E𝑦=4𝑒π‘₯+9𝑒


Find the value of π‘š at which the curve π‘₯=8π‘š+5π‘š+π‘šβˆ’1, 𝑦=5π‘šβˆ’π‘š+2 has a vertical tangent.

  • A14, 16
  • Bβˆ’16, βˆ’14
  • C14
  • D110
  • E16

This lesson includes 55 additional questions and 351 additional question variations for subscribers.

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