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 = 3 𝑡 + 1 and 𝑦 = 5 𝑡 − 𝑡 , find d d 𝑦 𝑥 .

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

Q2:

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

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

Q3:

Given that 𝑥 = 3 𝑒 and 𝑦 = 𝑡 𝑒 , find d d 𝑦 𝑥 .

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

Q4:

Given that 𝑥 = 5 𝑡 − 4 𝑡 l n and 𝑦 = 4 𝑡 + 5 3 𝑡 l n , find d d 𝑦 𝑥 .

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

Q5:

Given that 𝑥 = 𝑡 c o s and 𝑦 = 2 𝑡 s i n , find d d 𝑦 𝑥 .

  • A − 𝑡 2 2 𝑡 s i n c o s
  • B − 2 2 𝑡 𝑡 c o s s i n
  • 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 = 2 𝑡 4 + 𝑡 and 𝑦 = √ 4 + 𝑡 , find d 𝑦 d 𝑥 .

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

Q7:

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

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

Q8:

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

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

Q9:

If 𝑦 = 𝑥 √ 5 + 𝑥 2 and 𝑧 = √ 5 + 𝑥 5 𝑥 2 , find 5 𝑧 𝑦 𝑥 + 𝑧 𝑥 2 d d d d .

Q10:

Given that 𝑥 = 5 𝑡 𝑒 and 𝑦 = 3 𝑡 + 4 𝑡 s i n , find d d 𝑦 𝑥 .

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

Q11:

Find the derivative of 7 𝑥 + 4 𝑥 s i n with respect to c o s 𝑥 + 1 at 𝑥 = 𝜋 6 .

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

Q12:

Find d d 𝑦 𝑥 at 𝜃 = 𝜋 3 , given 𝑥 = 5 𝜃 + 7 2 𝜃 c o s c o s and 𝑦 = 7 𝜃 + 4 2 𝜃 s i n s i n .

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

Q13:

Find d d 𝑦 𝑥 at 𝜃 = 1 6 , given that 𝑥 = − 9 2 𝜋 𝜃 s i n and 𝑦 = − 4 2 𝜋 𝜃 c o s .

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

Q14:

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

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

Q15:

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

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

Q16:

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

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

Q17:

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

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

Q18:

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

  • A − 4 8
  • B1
  • C − 4
  • D − 1 4

Q19:

Find the equation of the tangent to the curve 𝑥 = 5 s e c 𝜃 and 𝑦 = 5 t a n 𝜃 at 𝜃 = 𝜋 6 .

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

Q20:

Suppose 𝑥 = − 3 5 𝜃 + 1 3 s e c and 𝑦 = − 3 5 𝜃 − 1 4 t a n . Find d d 𝑦 𝑥 when 𝜃 = 𝜋 4 .

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

Q21:

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

Q22:

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

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

Q23:

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

  • A20
  • B 5 4
  • C − 9
  • D 4 5

Q24:

Find the rate of change of √ 2 − 𝑥 with respect to 𝑥 5 𝑥 + 3 at 𝑥 = − 1 .

  • A 1 1 2
  • B 3 4
  • C12
  • D 4 3

Q25:

Find an equation of the tangent to the curve 𝑥 = 𝑒 𝜋 𝑡 𝑡 s i n , 𝑦 = 𝑒 2 𝑡 at the point corresponding to the value 𝑡 = 0 .

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

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