Worksheet: Combining the Product, Quotient, and Chain Rules

In this worksheet, we will practice finding the first derivative of a function using combinations of the product, quotient, and chain rules.

Q1:

Find , given that .

  • A
  • B
  • C
  • D

Q2:

Find d d 𝑦 𝑥 , given that 𝑦 = 3 𝑥 + 4 𝑥 + 6 7 𝑥 8 𝑥 4 2 7 8 .

  • A 1 2 𝑥 + 8 𝑥 + 6 + 4 9 𝑥 + 6 4 𝑥 3 6 7
  • B 1 2 𝑥 + 8 𝑥 + 6 + 4 9 𝑥 + 6 4 𝑥 3 8 9
  • C 3 𝑥 + 4 𝑥 + 7 𝑥 + 8 𝑥 3 8 9
  • D 1 2 𝑥 + 8 𝑥 + 4 9 𝑥 + 6 4 𝑥 3 8 9
  • E 9 𝑥 + 4 𝑥 + 5 6 𝑥 + 7 2 𝑥 3 8 9

Q3:

Differentiate 𝑓 ( 𝑥 ) = 5 𝑎 𝑥 9 𝑏 2 , where 𝑎 and 𝑏 are two constants.

  • A 5 𝑎 𝑥
  • B 1 0 𝑎 𝑥
  • C 5 𝑎 𝑥
  • D 1 0 𝑎 𝑥

Q4:

Find d d 𝑦 𝑥 , given that 𝑦 = 𝑥 4 5 + 5 𝑥 5 2 .

  • A 5 𝑥 4 1 0 𝑥 4
  • B 𝑥 4 5 𝑥 4 3
  • C 5 𝑥 4 + 1 0 𝑥 5 2
  • D 5 𝑥 4 1 0 𝑥 4 3
  • E 3 𝑥 2 + 1 0 𝑥 6

Q5:

Find the first derivative of the function 𝑦 = 3 𝑥 + 7 7 3 𝑥 5 5 .

  • A 9 0 𝑥 1 0
  • B 9 0 𝑥 9
  • C 1 8 𝑥 1 0
  • D 9 0 𝑥 9
  • E 9 0 𝑥 1 0

Q6:

Find the first derivative of the function 𝑦 = ( 5 𝑥 + 2 ) ( 9 𝑥 + 6 𝑥 + 4 ) 2 3 .

  • A 4 5 𝑥 + 4 8 𝑥 + 2 0 𝑥 4 2
  • B 2 2 5 𝑥 + 1 4 4 𝑥 + 4 0 𝑥 5 3 2
  • C 2 2 5 𝑥 + 1 4 4 𝑥 + 4 0 𝑥 6 4 3
  • D 2 2 5 𝑥 + 1 4 4 𝑥 + 4 0 𝑥 + 1 2 4 2

Q7:

Find the first derivative of the function 𝑦 = 1 2 𝑥 + 1 .

  • A 1 ( 2 𝑥 + 1 ) 2
  • B 2 ( 2 𝑥 + 1 ) 2
  • C 1 ( 2 𝑥 + 1 ) 2
  • D 2 ( 2 𝑥 + 1 ) 2

Q8:

Find the first derivative of the function 𝑦 = 9 𝑥 + 5 𝑥 4 𝑥 + 5 𝑥 2 2 .

  • A 4 0 0 𝑥 + 4 0 0 𝑥 + 1 2 5 𝑥 + 9 4 2
  • B 8 0 𝑥 + 2 0 0 𝑥 + 1 2 5 𝑥 + 9 4 2
  • C 3 2 0 𝑥 + 2 0 0 𝑥 2 5 0 𝑥 + 9 4 2
  • D 4 0 0 𝑥 + 4 0 0 𝑥 1 2 5 𝑥 + 9 4 2

Q9:

Differentiate 𝑦 = 𝑥 ( 2 𝑥 + 1 ) 3 .

  • A 𝑦 = 8 𝑥 3 + 𝑥 3 7 3 4 3
  • B 𝑦 = 2 𝑥 + 1 𝑥 3 2 3
  • C 𝑦 = 2 𝑥 + 𝑥 7 3 4 3
  • D 𝑦 = 8 𝑥 3 + 1 3 𝑥 3 2 3
  • E 𝑦 = 2 𝑥 3 2 3 𝑥 3 2 3

Q10:

Let 𝑔 ( 𝑥 ) = 2 𝑓 ( 𝑥 ) + 5 ( 𝑥 ) . If 𝑓 ( 8 ) = 9 and ( 8 ) = 1 , find 𝑔 ( 8 ) .

Q11:

If 𝑦 = 𝑥 7 𝑥 , find d d 𝑦 𝑥 .

  • A 𝑥 7 𝑥
  • B 3 𝑥 3 5 𝑥 2
  • C 4 𝑥 + 7 𝑥 2
  • D 4 𝑥 4 9 𝑥 2

Q12:

Find the first derivative of the function 𝑦 = 𝑥 + 7 𝑥 5 5 .

  • A 5 2 𝑥 + 3 5 2 𝑥 3 2 7 2
  • B 𝑥 7 𝑥 3 2 7 2
  • C 3 2 𝑥 2 1 2 𝑥 3 2 7 2
  • D 5 2 𝑥 3 5 2 𝑥 3 2 7 2

Q13:

Evaluate d d 𝑥 5 𝑥 3 .

  • A 5 2 𝑥 3 4
  • B 5 𝑥 3 3 2
  • C 5 𝑥 3 4
  • D 5 3 𝑥 3 4

Q14:

Differentiate 𝑓 ( 𝑥 ) = 4 𝑥 + 8 , and identify the value of 𝑥 at which the function is NOT differentiable.

  • A 𝑓 ( 𝑥 ) = 2 𝑥 + 8 , the function is not differentiable at 𝑥 8 .
  • B 𝑓 ( 𝑥 ) = 4 𝑥 + 8 , the function is not differentiable at 𝑥 8 .
  • C 𝑓 ( 𝑥 ) = 4 𝑥 + 8 , the function is not differentiable at 𝑥 8 .
  • D 𝑓 ( 𝑥 ) = 2 𝑥 + 8 , the function is not differentiable at 𝑥 8 .

Q15:

Find , given that .

  • A
  • B
  • C
  • D

Q16:

Find the first derivative of 𝑦 = 9 𝑥 7 𝑥 6 with respect to 𝑥 .

  • A 5 4 𝑥 7 5
  • B 5 4 𝑥 7 𝑥 5
  • C 5 4 𝑥 7 2 𝑥 5
  • D 5 4 𝑥 7 2 𝑥 5
  • E 5 4 𝑥 7 𝑥 5

Q17:

Evaluate d d 𝑥 5 𝑥 1 9 .

  • A 5 8 𝑥 9 8
  • B 5 𝑥 9 9 1 0
  • C 5 𝑥 9 8
  • D 5 9 𝑥 9 8

Q18:

Differentiate 𝑓 ( 𝑥 ) = 9 𝑥 + 3 𝑥 5 𝑥 6 4 9 2 .

  • A 3 6 𝑥 + 3 2 𝑥 5 𝑥 4 9 2
  • B 9 𝑥 + 3 𝑥 5 𝑥 3 7 2
  • C 2 7 𝑥 3 𝑥 4 5 2 𝑥 3 9 2
  • D 3 6 𝑥 + 3 2 𝑥 4 5 2 𝑥 3 7 2

Q19:

If 𝑦 = 2 3 𝑥 , which of the following is the same as d d 𝑦 𝑥 ?

  • A 3 2 𝑦
  • B 2 𝑦 3
  • C 2 𝑦 3
  • D 3 2 𝑦

Q20:

Find d d 𝑥 2 𝑥 7 𝑥 .

  • A 7 + 1 𝑥
  • B 7 2 𝑥
  • C 𝑥 7
  • D 7 1 𝑥

Q21:

Find the first derivative of the function 𝑦 = 9 𝑥 + 2 𝑥 + 4 𝑥 𝑥 2 .

  • A 9 2 𝑥 1 2
  • B 9 6 𝑥 3 2
  • C 9 4 𝑥 3 2
  • D 9 2 𝑥 3 2

Q22:

Differentiate 𝑦 = 𝑥 5 𝑥 𝑥 4 𝑥 𝑥 1 .

  • A 1 0 𝑥 4 𝑥 1 2 𝑥 3 2 3
  • B 5 𝑥 4 1 𝑥
  • C 5 𝑥 4 𝑥 1 𝑥 2
  • D 1 0 𝑥 4 1 2 𝑥

Q23:

Find d d 𝑦 𝑥 , given that 𝑦 = 5 𝑥 + 3 𝑥 𝑥 + 2 1 𝑥 + 1 7 .

  • A 5 𝑥 + 3 𝑥 + 2 1 2
  • B 5 + 9 2 𝑥 + 2 1 𝑥 5 2
  • C 5 𝑥 + 9 2 𝑥 + 2 1 𝑥 3
  • D 5 𝑥 + 9 2 𝑥 + 2 1 2

Q24:

Differentiate 𝐺 ( 𝑡 ) = 5 𝑡 + 2 2 𝑡 .

  • A 𝐺 ( 𝑡 ) = 2 2 𝑡 + 5 5 𝑡 2
  • B 𝐺 ( 𝑡 ) = 2 2 𝑡 + 5 1 0 𝑡 2
  • C 𝐺 ( 𝑡 ) = 2 2 𝑡 + 5 2 𝑡 2
  • D 𝐺 ( 𝑡 ) = 2 2 𝑡 + 5 2 𝑡 2
  • E 𝐺 ( 𝑡 ) = 2 2 𝑡 + 5 1 0 𝑡 2

Q25:

If 𝑦 = 3 8 𝑥 + 𝑥 , find 6 𝑥 𝑦 𝑦 𝑥 2 d d .

  • A 3 𝑥 2 7 8 2
  • B 𝑥 2 7 6 4 2
  • C 3 𝑥 9 8 2
  • D 3 𝑥 2 7 6 4 2

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