Worksheet: The Quotient Rule

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

Q1:

Find dd𝑦𝑥 if 𝑦=𝑥+3𝑥+3.

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

Q2:

Find dd𝑦𝑥, given that 𝑦=𝑥+7𝑥+6𝑥+8.

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

Q3:

Find the first derivative of function 𝑦=4𝑥9𝑥7.

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

Q4:

Differentiate 𝑓(𝑥)=4𝑥5𝑥+83𝑥4.

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

Q5:

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

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

Q6:

Suppose that 𝑓(𝑥)=𝑥+𝑎𝑥+𝑏𝑥7𝑥+4. Given that 𝑓(0)=1 and 𝑓(0)=4, find 𝑎 and 𝑏.

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

Q7:

Find the first derivative of 𝑦=8𝑥+53𝑥+22.

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

Q8:

Find the first derivative of 𝑦=𝑥93𝑥+13.

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

Q9:

Differentiate 𝑓(𝑥)=5𝑥17𝑥+6.

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

Q10:

Find the first derivative of the function 𝑦=4𝑥+5𝑥+54𝑥2𝑥+3.

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

Q11:

Given that 𝑦=3𝑥2𝑥𝑥, determine dd𝑦𝑥.

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

Q12:

Find the first derivative of 𝑦=3𝑥2𝑥+17𝑥 with respect to 𝑥.

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

Q13:

If 𝑦=29𝑥+8, find 1𝑦𝑦𝑥dd.

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

Q14:

If 𝑦=𝑥+5𝑥5𝑥5𝑥+5, find dd𝑦𝑥.

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

Q15:

Evaluate 𝑓(3), where 𝑓(𝑥)=𝑥𝑥+2𝑥3𝑥2.

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

Q16:

Calculate 𝑥𝑦𝑥dd, given 𝑦=4𝑥58𝑥.

  • A 1 5 4
  • B 2 5 8
  • C 5 8
  • D25

Q17:

Find the first derivative of the function 𝑦=12𝑥+1.

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

Q18:

Differentiate 𝑦=(𝑥1)(𝑥+1)𝑥+1𝑥.

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

Q19:

Let 𝑔(𝑥)=𝑓(𝑥)4(𝑥)5. Given that 𝑓(2)=1, 𝑓(2)=8, (2)=2, and (2)=5, find 𝑔(2).

  • A 4 9
  • B 2 5
  • C 4 4 9
  • D 4 4 3

Q20:

If 𝑦=964𝑥+49, find dd𝑦𝑥+8𝑦3.

  • A 8 5
  • B 9 1 1 3
  • C0
  • D 1 5

Q21:

Find the values of 𝑥 for which dd𝑦𝑥=0, where 𝑦=𝑥+6𝑥+36𝑥6𝑥+36.

  • A6, 6
  • B12, 12
  • C36, 36

Q22:

Let 𝑓(𝑥)=2𝑥7𝑥1. Use the definition of derivative to determine 𝑓(𝑥).

  • A 2 ( 7 𝑥 1 )
  • B 2 8 𝑥 2 ( 7 𝑥 1 )
  • C 2 8 𝑥 + 2 ( 7 𝑥 1 )
  • D 2 ( 7 𝑥 1 )

Q23:

Find the derivative of the function 𝐺, where 𝐺(𝑡)=2𝑡2𝑡+2, using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝐺 ( 𝑡 ) = 6 ( 𝑡 + 2 ) , domain of function: , domain of derivative: (,2)(2,)
  • B 𝐺 ( 𝑡 ) = 6 𝑡 + 2 , domain of function: , domain of derivative: (,2)(2,)
  • C 𝐺 ( 𝑡 ) = 6 𝑡 + 2 , domain of function: (,2)(2,), domain of derivative: (,2)(2,)
  • D 𝐺 ( 𝑡 ) = 6 ( 𝑡 + 2 ) , domain of function: (,2)(2,), domain of derivative: (,2)(2,)
  • E 𝐺 ( 𝑡 ) = 4 𝑡 + 2 ( 𝑡 + 2 ) , domain of function: (,2)(2,), domain of derivative:

Q24:

Differentiate 𝐷(𝑡)=181𝑡(3𝑡).

  • A 𝐷 ( 𝑡 ) = 1 3 𝑡 5 2 4 3 𝑡
  • B 𝐷 ( 𝑡 ) = 1 3 𝑡 5 2 4 3 𝑡
  • C 𝐷 ( 𝑡 ) = 1 3 𝑡 5 2 4 3 𝑡
  • D 𝐷 ( 𝑡 ) = 1 3 𝑡 5 2 4 3 𝑡
  • E 𝐷 ( 𝑡 ) = 1 3 𝑡 5 2 4 3 𝑡

Q25:

Evaluate 𝑓(1), where 𝑓(𝑥)=163𝑥5.

  • A 9 2
  • B 3 2
  • C 9 2
  • D 3 2

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