Worksheet: Derivatives of Parametric Equations

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=5𝑡𝑡, find dd𝑦𝑥.

  • A9𝑡(10𝑡1)
  • B3𝑡(5𝑡1)
  • C3𝑡5𝑡1
  • D9𝑡10𝑡1
  • E10𝑡19𝑡

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.

  • A5𝑡115𝑒
  • B15(15𝑡)
  • C15𝑡15𝑒
  • D15𝑒15𝑡
  • E15(5𝑡1)

Q4:

Given that 𝑥=5𝑡4𝑡ln and 𝑦=4𝑡+53𝑡ln, find dd𝑦𝑥.

  • A(4𝑡+5)(5𝑡4)3𝑡
  • B(4𝑡+5)(5𝑡4)𝑡
  • C4𝑡+55𝑡4
  • D4𝑡+53(5𝑡4)
  • E5𝑡44𝑡+5

Q5:

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

  • A22𝑡𝑡cossin
  • B𝑡22𝑡sincos
  • C2𝑡𝑡cossin
  • D22𝑡𝑡cossin
  • E2𝑡𝑡cossin

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=4𝑥5 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.

  • A510
  • B5
  • C520
  • D25

Q9:

If 𝑦=𝑥5+𝑥 and 𝑧=5+𝑥5𝑥, find 5𝑧𝑦𝑥+𝑧𝑥dddd.

Q10:

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

  • A5𝑒(𝑡+1)(34𝑡)cos
  • B3+4𝑡5𝑒(𝑡+1)cos
  • C3+4𝑡5𝑒(𝑡1)cos
  • D34𝑡5𝑒(𝑡+1)cos
  • E5𝑒(𝑡+1)(3+4𝑡)cos

Q11:

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

  • A43+14
  • B723
  • C43+14
  • D1443

Q12:

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

  • A357
  • B1934
  • C312
  • D312

Q13:

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

  • A239
  • B439
  • C29
  • D239

Q14:

By using parametric differentiation, determine the derivative of 5𝑥+𝑥2 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 𝑥6𝑥9 with respect to 8𝑥+1 at 𝑥=3.

  • A548
  • B2548
  • C596
  • D56

Q17:

Find the rate of change of (𝑥+2)(𝑥+7) with respect to 𝑥2𝑥7.

  • A2𝑥+9
  • B14(𝑥7)
  • C10𝑥45(𝑥7)
  • D2𝑥595(𝑥7)

Q18:

Find the rate of change of ln3𝑥1 with respect to 6𝑥5 at 𝑥=1.

  • A1
  • B48
  • C14
  • D4

Q19:

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

  • A2𝑦𝑥=0
  • B𝑦2𝑥+53=0
  • C𝑦+2𝑥2533=0
  • D2𝑦𝑥+2033=0

Q20:

Suppose 𝑥=35𝜃+13sec and 𝑦=35𝜃14tan. Find dd𝑦𝑥 when 𝜃=𝜋4.

  • A12
  • B1
  • C52
  • D12

Q21:

If 𝑥=8𝑡8 and 𝑦=𝑡, find dd𝑦𝑥 at 𝑡=1.

Q22:

A curve has parametric equations 𝑥=7𝑚+5𝑚+𝑚+4 and 𝑦=6𝑚6𝑚8. Find 𝑚 for which the tangent is horizontal.

  • A12
  • B17, 13
  • C13
  • D17

Q23:

Determine dd𝑦𝑥 at 𝑡=0, given that 𝑥=(𝑡2)(4𝑡+3), and 𝑦=3𝑡4(𝑡3).

  • A20
  • B9
  • C45
  • D54

Q24:

Find the rate of change of 2𝑥 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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