Worksheet: The Product Rule

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

Q1:

Find dd𝑦𝑥 at 𝑥=2 when 𝑦=(4𝑥1)3𝑥+7.

Q2:

Find the first derivative of 𝑓(𝑥)=2𝑥(𝑥3)(𝑥1)(𝑥+2) at (1,16).

Q3:

Find the first derivative of 𝑓(𝑥)=9𝑥𝑥77𝑥8𝑥7 at 𝑥=1.

Q4:

Find the first derivative of the function 𝑓(𝑥)=2𝑥+𝑥5𝑥+3𝑥3𝑥.

  • A 1 2 𝑥 + 2 7 𝑥 𝑥 1 5 𝑥 + 9 2 𝑥 1 0 𝑥 1 5 2 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 8 𝑥 + 2 7 𝑥 1 8 𝑥 1 5 𝑥 + 𝑥 1 5 𝑥

Q5:

Find the first derivative of 𝑓(𝑥)=𝑥+43𝑥𝑥73𝑥𝑥+7 at 𝑥=1.

Q6:

Let 𝑔(𝑥)=3𝑓(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 𝑥𝑓(𝑥)?

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

Q8:

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

  • A 𝑓 𝑔 + 𝑓 𝑔
  • B 𝑓 𝑔 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=7𝑥𝑥𝑥+7𝑥.

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

Q11:

Suppose that 𝑓(𝑥)=(2𝑥+𝑎)3𝑥𝑎 and 𝑓(1)=10. Determine 𝑎.

Q12:

Find the first derivative of the function 𝑦=3𝑥+773𝑥.

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

Q13:

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

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

Q14:

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

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

Q15:

Find the first derivative of the function 𝑦=𝑥𝑥+23𝑥+3𝑥+6.

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

Q16:

Consider the functions 𝑓(𝑥)=𝑥, 𝑔(𝑥)=𝑥.

Find 𝑓(𝑥) and 𝑔(𝑥).

  • A 𝑓 ( 𝑥 ) = 1 , 𝑔 ( 𝑥 ) = 2 𝑥
  • B 𝑓 ( 𝑥 ) = 0 , 𝑔 ( 𝑥 ) = 2 𝑥
  • C 𝑓 ( 𝑥 ) = 1 , 𝑔 ( 𝑥 ) = 2
  • D 𝑓 ( 𝑥 ) = 1 , 𝑔 ( 𝑥 ) = 2 𝑥

Find 𝑓(𝑥)𝑔(𝑥).

  • A 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) = 2
  • B 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) = 0
  • C 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) = 2 𝑥
  • D 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) = 2 𝑥

Given that 𝑓(𝑥)𝑔(𝑥)=𝑥, find its derivative.

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

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