Worksheet: The Quotient Rule

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

Q1:

Find d d 𝑦 𝑥 if 𝑦 = 𝑥 + 3 𝑥 + 3 2 3 .

  • A 𝑥 + 9 𝑥 6 𝑥 ( 𝑥 + 3 ) 4 2 3 2
  • B 𝑥 9 𝑥 + 6 𝑥 𝑥 + 3 4 2 3
  • C 𝑥 + 9 𝑥 6 𝑥 𝑥 + 3 4 2 3
  • D 𝑥 9 𝑥 + 6 𝑥 ( 𝑥 + 3 ) 4 2 3 2

Q2:

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

  • A 2 𝑥 3 1 𝑥 1 1 2 𝑥 + 6 ( 𝑥 + 8 ) 3 2 2
  • B 2 𝑥 + 3 1 𝑥 + 1 1 2 𝑥 6 𝑥 + 8 3 2
  • C 2 𝑥 3 1 𝑥 1 1 2 𝑥 + 6 𝑥 + 8 3 2
  • D 2 𝑥 + 3 1 𝑥 + 1 1 2 𝑥 6 ( 𝑥 + 8 ) 3 2 2

Q3:

Find the first derivative of function 𝑦 = 4 𝑥 9 𝑥 7 2 .

  • A 7 ( 9 𝑥 7 ) 2 2
  • B 3 6 𝑥 + 2 8 ( 9 𝑥 7 ) 2 2 2
  • C 7 ( 9 𝑥 7 ) 2 2
  • D 3 6 𝑥 2 8 ( 9 𝑥 7 ) 2 2 2

Q4:

Differentiate 𝑓 ( 𝑥 ) = 4 𝑥 5 𝑥 + 8 3 𝑥 4 2 .

  • A 1 6 𝑥 + 4 ( 3 𝑥 4 ) 2
  • B 1 2 𝑥 + 3 2 𝑥 + 4 ( 3 𝑥 4 ) 2 2
  • C 1 6 𝑥 4 ( 3 𝑥 4 ) 2
  • D 1 2 𝑥 3 2 𝑥 4 ( 3 𝑥 4 ) 2 2

Q5:

Suppose 𝑓 ( 𝑥 ) = 𝑥 + 𝑎 𝑥 𝑎 and 𝑓 ( 2 ) = 2 . Determine 𝑎 .

  • A 4 , 1
  • B 4 , 1
  • C 4 , 1
  • D 4 , 1

Q6:

Suppose that 𝑓 ( 𝑥 ) = 𝑥 + 𝑎 𝑥 + 𝑏 𝑥 7 𝑥 + 4 2 2 . Given that 𝑓 ( 0 ) = 1 and 𝑓 ( 0 ) = 4 , find 𝑎 and 𝑏 .

  • A 𝑎 = 7 , 𝑏 = 4
  • B 𝑎 = 7 , 𝑏 = 4
  • C 𝑎 = 7 , 𝑏 = 4
  • D 𝑎 = 9 , 𝑏 = 4

Q7:

Find the first derivative of 𝑦 = 8 𝑥 + 5 3 𝑥 + 2 2 .

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

Q8:

Find the first derivative of 𝑦 = 𝑥 9 3 𝑥 + 1 3 .

  • A 8 0 ( 𝑥 + 1 3 ) 2
  • B 2 𝑥 1 0 6 ( 𝑥 + 1 3 ) 2
  • C 9 3 1 3
  • D 1 0 6 ( 𝑥 + 1 3 ) 2

Q9:

Differentiate 𝑓 ( 𝑥 ) = 5 𝑥 1 7 𝑥 + 6 2 .

  • A 3 0 𝑥 + 7 ( 7 𝑥 + 6 ) 2
  • B 3 5 𝑥 6 0 𝑥 7 ( 7 𝑥 + 6 ) 2 2
  • C 3 0 𝑥 7 ( 7 𝑥 + 6 ) 2
  • D 3 5 𝑥 + 6 0 𝑥 + 7 ( 7 𝑥 + 6 ) 2 2

Q10:

Find the first derivative of the function 𝑦 = 4 𝑥 + 5 𝑥 + 5 4 𝑥 2 𝑥 + 3 2 2 .

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

Q11:

Given that 𝑦 = 3 𝑥 2 𝑥 𝑥 , determine d d 𝑦 𝑥 .

  • A 3 2 𝑥
  • B 𝑥
  • C 2 𝑥 3
  • D 1 𝑥

Q12:

Find the first derivative of 𝑦 = 3 𝑥 2 𝑥 + 1 7 𝑥 2 with respect to 𝑥 .

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

Q13:

If 𝑦 = 2 9 𝑥 + 8 , find 1 𝑦 𝑦 𝑥 2 d d .

  • A 2 9
  • B 9 2
  • C 2 9
  • D 9 2

Q14:

If 𝑦 = 𝑥 + 5 𝑥 5 𝑥 5 𝑥 + 5 , find d d 𝑦 𝑥 .

  • A 2 0 𝑥 + 5 0 0 ( 𝑥 + 5 0 0 ) 2 2 2
  • B 2 0 𝑥 5 0 0 𝑥 2 5 2 2
  • C 2 0 𝑥 5 0 0 ( 𝑥 5 0 0 ) 2 2 2
  • D 2 0 𝑥 5 0 0 ( 𝑥 2 5 ) 2 2 2

Q15:

Evaluate 𝑓 ( 3 ) , where 𝑓 ( 𝑥 ) = 𝑥 𝑥 + 2 𝑥 3 𝑥 2 .

  • A 2 7 2 5
  • B 2 3 2 5
  • C 2 7 2 5
  • D 2 3 2 5

Q16:

Calculate 𝑥 𝑦 𝑥 6 d d , given 𝑦 = 4 𝑥 5 8 𝑥 5 5 .

  • A 1 5 4
  • B 5 8
  • C25
  • D 2 5 8

Q17:

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

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