Worksheet: The Product Rule

In this worksheet, we will practice finding the derivative of a function using the product rule.

Q1:

Find d d 𝑦 𝑥 at 𝑥 = 2 when 𝑦 = ( 4 𝑥 1 ) 3 𝑥 + 7 .

Q2:

Find the first derivative of 𝑓 ( 𝑥 ) = 2 𝑥 ( 𝑥 3 ) ( 𝑥 1 ) ( 𝑥 + 2 ) at ( 1 , 1 6 ) .

Q3:

Find the first derivative of 𝑓 ( 𝑥 ) = 9 𝑥 𝑥 7 7 𝑥 8 𝑥 7 at 𝑥 = 1 .

Q4:

Find the first derivative of the function 𝑓 ( 𝑥 ) = 2 𝑥 + 𝑥 5 𝑥 + 3 𝑥 3 𝑥 .

  • A 8 𝑥 + 2 7 𝑥 1 8 𝑥 1 5 𝑥 + 𝑥 1 5 𝑥
  • B 8 𝑥 + 2 7 𝑥 1 8 𝑥 1 5 𝑥 + 𝑥 1 5 𝑥
  • C 8 𝑥 + 2 7 𝑥 1 8 𝑥 1 5 𝑥 + 𝑥 1 5 𝑥
  • D 1 2 𝑥 + 2 7 𝑥 𝑥 1 5 𝑥 + 9 2 𝑥 1 0 𝑥 1 5 2 𝑥 1 5 𝑥

Q5:

Find the first derivative of 𝑓 ( 𝑥 ) = 𝑥 + 4 3 𝑥 𝑥 7 3 𝑥 𝑥 + 7 at 𝑥 = 1 .

Q6:

Let 𝑔 ( 𝑥 ) = 3 𝑓 ( 𝑥 ) [ ( 𝑥 ) 1 ] . If 𝑓 ( 4 ) = 1 , ( 4 ) = 9 , ( 4 ) = 6 , and 𝑓 ( 4 ) = 1 , find 𝑔 ( 4 ) .

Q7:

Suppose that 𝑓 is differentiable. What is the derivative of 𝑥 𝑓 ( 𝑥 ) 3 ?

  • A 𝑥 𝑓 ( 𝑥 ) + 𝑥 𝑓 ( 𝑥 ) 2 3
  • B 3 𝑥 𝑓 ( 𝑥 ) 2
  • C 𝑥 𝑓 ( 𝑥 ) 2
  • D 3 𝑥 𝑓 ( 𝑥 ) + 𝑥 𝑓 ( 𝑥 ) 2 3
  • E 3 𝑥 + 𝑓 ( 𝑥 ) 2

Q8:

The product rule says that ( 𝑓 𝑔 ) = 𝑓 𝑔 + 𝑓 𝑔 . Use this to derive a formula for the derivative ( 𝑓 𝑔 ) .

  • A 𝑓 𝑔 𝑓 𝑔 𝑓 𝑔
  • B 𝑓 𝑔 + 𝑓 𝑔
  • C 𝑓 𝑔 + 𝑓 𝑔
  • D 𝑓 𝑔 + 𝑓 𝑔 + 𝑓 𝑔
  • E 𝑓 𝑔 + 𝑓 𝑔 + 𝑓 𝑔

Q9:

Suppose that 𝑓 ( 2 ) = 3 , 𝑔 ( 2 ) = 5 , 𝑓 ( 2 ) = 1 , and 𝑔 ( 2 ) = 6 . Evaluate ( 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ) 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) at 𝑥 = 2 .

Q10:

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

  • A 2 1 𝑥 + 2 5 𝑥 7
  • B 2 8 𝑥 + 1 2 5 𝑥 7 𝑥
  • C 7 𝑥 + 5 0 𝑥 𝑥 7
  • D 2 8 𝑥 + 1 2 5 𝑥 𝑥 7

Q11:

Suppose that 𝑓 ( 𝑥 ) = ( 2 𝑥 + 𝑎 ) 3 𝑥 𝑎 and 𝑓 ( 1 ) = 1 0 . Determine 𝑎 .

Q12:

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

Q13:

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

Q14:

Find the first derivative of the function 𝑦 = 𝑥 + 9 ( 8 𝑥 + 3 ) .

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

Q15:

Find the first derivative of the function 𝑦 = 𝑥 𝑥 + 2 3 𝑥 + 3 𝑥 + 6 .

  • A 2 7 𝑥 + 6 3 𝑥 + 3 6 𝑥 + 3 0 𝑥 + 4 8 𝑥
  • B 3 𝑥 + 9 𝑥 + 6 𝑥 + 6 𝑥 + 1 2 𝑥
  • C 2 7 𝑥 + 6 3 𝑥 + 3 6 𝑥 + 3 0 𝑥 + 4 8 𝑥
  • D 3 0 𝑥 + 7 2 𝑥 + 4 2 𝑥 + 3 6 𝑥 + 6 0 𝑥

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