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Worksheet: First-Order Derivatives Using the Chain Rule

Q1:

Evaluate d d 𝑦 π‘₯ at π‘₯ = 2 if 𝑦 = βˆ’ 𝑧 3 and 𝑧 = 6 π‘₯ βˆ’ 8 .

Q2:

Find d d 𝑦 π‘₯ , given that 𝑦 = 𝑧 2 and 𝑧 = 9 π‘₯ + 2 2 .

  • A 2 4 3 π‘₯ + 3 6 π‘₯ 3
  • B 3 6 π‘₯ + 7 2 π‘₯ 3
  • C 8 1 π‘₯ + 3 6 π‘₯ 3
  • D 3 2 4 π‘₯ + 7 2 π‘₯ 3

Q3:

Determine d d 𝑧 π‘₯ when π‘₯ = 1 2 if 𝑧 = 𝑦 3 + 2 𝑦 + 1 3 and 𝑦 = βˆ’ 2 π‘₯ + π‘₯ βˆ’ 3 2 .

  • A 𝑦 + 2 2
  • B βˆ’ 4 π‘₯ + 1
  • C βˆ’ 9 9
  • D βˆ’ 1 1

Q4:

If 𝑦 = ( 𝑧 + 1 1 ) 7 and 𝑧 = 5 π‘₯ + 8 2 , find d d 𝑦 π‘₯ .

  • A 3 5 π‘₯ ο€Ή 5 π‘₯ + 1 9  2 6
  • B 7 ο€Ή 5 π‘₯ + 1 9  2 6
  • C 1 0 π‘₯ ο€Ή 5 π‘₯ + 1 9  2 6
  • D 7 0 π‘₯ ο€Ή 5 π‘₯ + 1 9  2 6

Q5:

Evaluate d d 𝑦 π‘₯ at π‘₯ = √ 3 if 𝑦 = ( βˆ’ 2 + π‘₯ ) ( βˆ’ 2 βˆ’ π‘₯ ) 4 4 .

  • A βˆ’ 2 √ 3
  • B 4 √ 3
  • C βˆ’ √ 3
  • D βˆ’ 8 √ 3

Q6:

Evaluate d d 𝑦 π‘₯ at π‘₯ = 4 if 𝑦 = 𝑧 βˆ’ 5 𝑧 + 1 2 and 𝑧 =  ( π‘₯ βˆ’ 3 ) 3 2 .

  • A βˆ’ 9 2
  • B βˆ’ 8 3
  • C βˆ’ 6
  • D βˆ’ 2

Q7:

Find d d 𝑦 π‘₯ , given that 𝑦 = ( 𝑧 + 9 ) 3 and 𝑧 = π‘₯ βˆ’ 9 4 .

  • A π‘₯ 1 1
  • B 1 2 π‘₯ 1 3
  • C π‘₯ 1 3
  • D 1 2 π‘₯ 1 1

Q8:

Evaluate d d 𝑦 π‘₯ at π‘₯ = 0 if 𝑦 = 5 √ 𝑧 and 𝑧 = π‘₯ + 1 6 π‘₯ + 1 .

  • A 1 1 2 5 6
  • B 1 1 0 √ 𝑧
  • C 8 5 8
  • D βˆ’ 7 5 8

Q9:

Evaluate d d 𝑦 𝑧 at 𝑧 = 3 if 𝑦 = π‘₯ βˆ’ 3 π‘₯ + 3 and π‘₯ = 3 𝑧 + 4 .

  • A βˆ’ 8 1 1 2 8
  • B 8 1 1 2 8
  • C βˆ’ 9 1 2 8
  • D 9 1 2 8

Q10:

Evaluate d d 𝑦 π‘₯ at π‘₯ = βˆ’ 2 if 𝑦 = 𝑧 + 1 𝑧 βˆ’ 1 and 𝑧 = π‘₯ βˆ’ 1 π‘₯ + 1 .

  • A 2 ( π‘₯ + 1 ) 2
  • B1
  • C βˆ’ 1 4
  • D βˆ’ 1

Q11:

Evaluate d d 𝑦 π‘₯ at π‘₯ = βˆ’ 3 if 𝑦 = 𝑧 βˆ’ 2 𝑧 + 2 and 𝑧 = π‘₯ + 2 π‘₯ βˆ’ 2 .

  • A βˆ’ 4 ( π‘₯ βˆ’ 2 ) 2
  • B 1 6 1 2 1
  • C βˆ’ 1 6 8 1
  • D βˆ’ 1 6 1 2 1

Q12:

Evaluate d d 𝑦 π‘₯ at π‘₯ = 4 if 𝑦 = 𝑧 + 3 𝑧 + 1 3 and 𝑧 = π‘₯ βˆ’ 1 0 π‘₯ βˆ’ 3 .

  • A 1 0 4 9
  • B βˆ’ 1 0 7
  • C βˆ’ 1 0 4 9
  • D 1 0 7

Q13:

Given that 𝑦 = 𝑧 βˆ’ 8 𝑧 + 1 6 4 2 and 𝑧 = 2 5 π‘₯ s i n , determine d d 𝑦 π‘₯ .

  • A 3 2 0 5 π‘₯ s i n 3
  • B 3 2 0 5 π‘₯ 5 π‘₯ c o s s i n 3
  • C βˆ’ 3 2 0 5 π‘₯ c o s 3
  • D βˆ’ 3 2 0 5 π‘₯ 5 π‘₯ c o s s i n 3

Q14:

Given that 𝑦 = βˆ’ 4 𝑧 + 3 2 𝑧 s i n and 𝑧 = βˆ’ 2 π‘₯ + πœ‹ , find d d 𝑦 π‘₯ at π‘₯ = 0 .

Q15:

If 𝑦 = ( βˆ’ 8 𝑧 + 1 ) 3 and 𝑧 = 1 6 2 π‘₯ c o s , find d d 𝑦 π‘₯ when π‘₯ = πœ‹ 4 .

  • A βˆ’ 1 6 3
  • B βˆ’ 2 3
  • C βˆ’ 2 4
  • D8

Q16:

If 𝑦 = ( 7 𝑧 + 3 ) 4 and 𝑧 = 1 7 2 π‘₯ c o t , find d d 𝑦 π‘₯ at π‘₯ = 3 πœ‹ 8 .

Q17:

Given that 𝑦 = πœ‹ 𝑧 3 6 c o t and 𝑧 = 6 √ π‘₯ , determine d d 𝑦 π‘₯ at π‘₯ = 4 .

  • A βˆ’ πœ‹ 9
  • B βˆ’ πœ‹ 1 6
  • C 3 4
  • D βˆ’ πœ‹ 1 8

Q18:

Find d d 𝑦 π‘₯ at πœƒ = πœ‹ 6 , given π‘₯ = 7 5 πœƒ + 3 3 πœƒ c o s c o s 6 and 𝑦 = 3 2 πœƒ + 7 3 πœƒ s i n s i n 6 .

  • A βˆ’ 1 0 5 2
  • B βˆ’ 3 5 6
  • C βˆ’ 2 9 2
  • D βˆ’ 6 3 5

Q19:

Find d d 𝑦 π‘₯ , given that 𝑦 = 8 𝑧 + 1 𝑧 and π‘₯ 𝑧 = 9 .

  • A 1 9 + 7 2 π‘₯ 2
  • B βˆ’ 1 8 1 π‘₯ βˆ’ 9 π‘₯ + 8 2 2
  • C 1 8 1 π‘₯ + 9 π‘₯ + 8 2 2
  • D 1 9 βˆ’ 7 2 π‘₯ 2

Q20:

Given that 𝑦 = 2 π‘₯ βˆ’ 2 2 and π‘₯ = 𝑧 βˆ’ 1 3 , determine d d d d 𝑦 𝑧 + 4 π‘₯ 𝑧 .

  • A 2 𝑧 + 2 𝑧 5 2
  • B 1 2 𝑧 1 3
  • C 2 𝑧 + 2 𝑧 2 3 4
  • D 1 2 𝑧 5

Q21:

Given that 𝑦 = √ 7 βˆ’ 4 𝑧 and 𝑧 = 2 π‘₯ t a n , determine d d 𝑦 π‘₯ at π‘₯ = πœ‹ 8 .

  • A βˆ’ 1 6 √ 3 3
  • B βˆ’ 2 √ 3 3
  • C βˆ’ 8
  • D βˆ’ 8 √ 3 3

Q22:

Evaluate d d 𝑦 π‘₯ at π‘₯ = 2 if 𝑦 = 2 √ 𝑧 + 9 √ 𝑧 and 𝑧 = 2 π‘₯ + 1 2 .

  • A 2 3
  • B 1 3
  • C4
  • D 4 3
  • E2

Q23:

Given that 𝑦 = 8 𝑧 βˆ’ 6 𝑧 βˆ’ 9 3 and 𝑧 = 3 π‘₯ βˆ’ 2 7 π‘₯ , determine d d 𝑦 π‘₯ when π‘₯ = βˆ’ 3 .

  • A6
  • B βˆ’ 6
  • C0
  • D βˆ’ 3 6
  • E βˆ’ 1 2

Q24:

Given that π‘₯ = 𝑑 + 1 2 and 𝑦 = 𝑒 βˆ’ 1 𝑑 , find d d 𝑦 π‘₯ .

  • A 2 𝑑 𝑒 𝑑
  • B 2 𝑒 𝑑
  • C 𝑒 𝑑 𝑑
  • D 𝑒 2 𝑑 𝑑
  • E 𝑒 𝑑

Q25:

Determine d d 𝑦 π‘₯ at 𝑑 = 0 , given that π‘₯ = ( 𝑑 βˆ’ 2 ) ( 4 𝑑 + 3 ) , and 𝑦 = ο€Ή 3 𝑑 βˆ’ 4  ( 𝑑 βˆ’ 3 ) 2 .

  • A20
  • B 5 4
  • C βˆ’ 9
  • D 4 5