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.

  • A12𝑥+27𝑥𝑥15𝑥+92𝑥10𝑥152𝑥15𝑥
  • B8𝑥+27𝑥18𝑥15𝑥+𝑥15𝑥
  • C8𝑥+27𝑥18𝑥15𝑥+𝑥15𝑥
  • D8𝑥+27𝑥18𝑥15𝑥+𝑥15𝑥

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)?

  • A𝑥𝑓(𝑥)+𝑥𝑓(𝑥)
  • B3𝑥𝑓(𝑥)
  • C𝑥𝑓(𝑥)
  • D3𝑥+𝑓(𝑥)
  • E3𝑥𝑓(𝑥)+𝑥𝑓(𝑥)

Q8:

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

  • A𝑓𝑔+𝑓𝑔
  • B𝑓𝑔𝑓𝑔𝑓𝑔
  • C𝑓𝑔+𝑓𝑔
  • D𝑓𝑔+𝑓𝑔+𝑓𝑔
  • E𝑓𝑔+𝑓𝑔+𝑓𝑔

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.

  • A28𝑥+125𝑥7𝑥
  • B7𝑥+50𝑥𝑥7
  • C21𝑥+25𝑥7
  • D28𝑥+125𝑥𝑥7

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.

  • A90𝑥
  • B90𝑥
  • C90𝑥
  • D90𝑥
  • E18𝑥

Q13:

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

  • A225𝑥+144𝑥+40𝑥+12
  • B45𝑥+48𝑥+20𝑥
  • C225𝑥+144𝑥+40𝑥
  • D225𝑥+144𝑥+40𝑥

Q14:

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

  • A3𝑥+6𝑥+72
  • B16𝑥+3𝑥+72
  • C24𝑥+6𝑥+72
  • D8𝑥+3𝑥+72

Q15:

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

  • A27𝑥+63𝑥+36𝑥+30𝑥+48𝑥
  • B27𝑥+63𝑥+36𝑥+30𝑥+48𝑥
  • C30𝑥+72𝑥+42𝑥+36𝑥+60𝑥
  • D3𝑥+9𝑥+6𝑥+6𝑥+12𝑥

Q16:

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

Find 𝑓(𝑥) and 𝑔(𝑥).

  • A𝑓(𝑥)=1, 𝑔(𝑥)=2𝑥
  • B𝑓(𝑥)=0, 𝑔(𝑥)=2𝑥
  • C𝑓(𝑥)=1, 𝑔(𝑥)=2
  • D𝑓(𝑥)=1, 𝑔(𝑥)=2𝑥

Find 𝑓(𝑥)𝑔(𝑥).

  • A𝑓(𝑥)𝑔(𝑥)=2
  • B𝑓(𝑥)𝑔(𝑥)=0
  • C𝑓(𝑥)𝑔(𝑥)=2𝑥
  • D𝑓(𝑥)𝑔(𝑥)=2𝑥

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

  • A3𝑥
  • B3𝑥
  • C2𝑥
  • D3𝑥

Q17:

dd𝑥(𝑓(𝑥)𝑔(𝑥))=.

  • A𝑓(𝑥)𝑔(𝑥)
  • B𝑓(𝑥)𝑔(𝑥)+𝑓(𝑥)𝑔(𝑥)
  • C𝑓(𝑥)𝑔(𝑥)
  • D𝑓(𝑥)𝑔(𝑥)

Q18:

If 𝑓(𝑥)=(3𝑥+2)×(𝑥), 𝑓(1)=10, and (1)=8, then 𝑓(1)=.

  • A46
  • B42
  • C6
  • D30

Q19:

If the function 𝑓 is an odd function and differentiable on and 𝑓(5)=12, then 𝑓(5)=.

  • A112
  • B12
  • C12
  • D112

Q20:

dd𝑥𝑥𝑥+5= at 𝑥=3.

  • A8914
  • B7214
  • C8194
  • D9

Q21:

If 𝑦=𝑥+3(4𝑥), then dd𝑦𝑥= at 𝑥=3.

  • A6
  • B21
  • C3
  • D45

Q22:

If 𝑓𝑓(𝑥) is differentiable at 𝑥=2, 𝑓(2)=4, 𝑓(2)=12, 𝑔(𝑥)=𝑥𝑓(𝑥), then 𝑔(2)=.

  • A16
  • B2
  • C18
  • D14

Q23:

If 𝑓(𝑥)×𝑔(𝑥)+𝑔(𝑥)×𝑓(𝑥)=3𝑥+2𝑥, then dd𝑥[𝑓(𝑥)×𝑔(𝑥)]= at 𝑥=2.

  • A7
  • B72
  • C52
  • D17

Q24:

dd𝑥((𝑥2)(𝑥+5))=.

  • A2𝑥3
  • B2𝑥7
  • C2𝑥+3
  • D2𝑥+7

Q25:

dd𝑥((2𝑥3)(𝑥+5))=.

  • A4𝑥+7
  • B2𝑥+13
  • C4𝑥7
  • D2𝑥13

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.