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.

  • A𝑥9𝑥+6𝑥(𝑥+3)
  • B𝑥9𝑥+6𝑥𝑥+3
  • C𝑥+9𝑥6𝑥(𝑥+3)
  • D𝑥+9𝑥6𝑥𝑥+3

Q2:

Find dd𝑦𝑥, given that 𝑦=𝑥+7𝑥+6𝑥+8.

  • A2𝑥31𝑥112𝑥+6(𝑥+8)
  • B2𝑥+31𝑥+112𝑥6𝑥+8
  • C2𝑥31𝑥112𝑥+6𝑥+8
  • D2𝑥+31𝑥+112𝑥6(𝑥+8)

Q3:

Find the first derivative of function 𝑦=4𝑥9𝑥7.

  • A36𝑥+28(9𝑥7)
  • B7(9𝑥7)
  • C36𝑥28(9𝑥7)
  • D7(9𝑥7)

Q4:

Differentiate 𝑓(𝑥)=4𝑥5𝑥+83𝑥4.

  • A12𝑥+32𝑥+4(3𝑥4)
  • B12𝑥32𝑥4(3𝑥4)
  • C16𝑥4(3𝑥4)
  • D16𝑥+4(3𝑥4)

Q5:

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

  • A4,1
  • B4,1
  • C4,1
  • D4,1

Q6:

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

  • A𝑎=7, 𝑏=4
  • B𝑎=9, 𝑏=4
  • C𝑎=7, 𝑏=4
  • D𝑎=7, 𝑏=4

Q7:

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

  • A8(3𝑥+22)
  • B83
  • C161(3𝑥+22)
  • D176𝑥+153𝑥+22
  • E191(3𝑥+22)

Q8:

Find the first derivative of 𝑦=𝑥93𝑥+13.

  • A80(𝑥+13)
  • B106(𝑥+13)
  • C9313
  • D2𝑥106(𝑥+13)

Q9:

Differentiate 𝑓(𝑥)=5𝑥17𝑥+6.

  • A35𝑥+60𝑥+7(7𝑥+6)
  • B30𝑥7(7𝑥+6)
  • C30𝑥+7(7𝑥+6)
  • D35𝑥60𝑥7(7𝑥+6)

Q10:

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

  • A8𝑥+5(4𝑥2𝑥+3)
  • B28𝑥16𝑥+25(4𝑥2𝑥+3)
  • C8𝑥+58𝑥2
  • D(8𝑥2)(4𝑥+5𝑥+5)(4𝑥2𝑥+3)

Q11:

Given that 𝑦=3𝑥2𝑥𝑥, determine dd𝑦𝑥.

  • A32𝑥
  • B1𝑥
  • C2𝑥
  • D𝑥

Q12:

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

  • A9𝑥2𝑥172𝑥
  • B12𝑥6𝑥+172𝑥
  • C9𝑥+2𝑥+172𝑥
  • D12𝑥6𝑥+172𝑥
  • E9𝑥2𝑥172𝑥

Q13:

If 𝑦=29𝑥+8, find 1𝑦𝑦𝑥dd.

  • A92
  • B29
  • C92
  • D29

Q14:

If 𝑦=𝑥+5𝑥5𝑥5𝑥+5, find dd𝑦𝑥.

  • A20𝑥500(𝑥25)
  • B20𝑥+500(𝑥+500)
  • C20𝑥500𝑥25
  • D20𝑥500(𝑥500)

Q15:

Evaluate 𝑓(3), where 𝑓(𝑥)=𝑥𝑥+2𝑥3𝑥2.

  • A2725
  • B2725
  • C2325
  • D2325

Q16:

Calculate 𝑥𝑦𝑥dd, given 𝑦=4𝑥58𝑥.

  • A154
  • B258
  • C58
  • D25

Q17:

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

  • A1(2𝑥+1)
  • B2(2𝑥+1)
  • C1(2𝑥+1)
  • D2(2𝑥+1)

Q18:

Differentiate 𝑦=(𝑥1)(𝑥+1)𝑥+1𝑥.

  • A3𝑥𝑥
  • B2𝑥2𝑥
  • C3𝑥+𝑥
  • D𝑥+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).

  • A49
  • B25
  • C449
  • D443

Q20:

If 𝑦=964𝑥+49, find dd𝑦𝑥+8𝑦3.

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).

  • A2(7𝑥1)
  • B28𝑥2(7𝑥1)
  • C28𝑥+2(7𝑥1)
  • D2(7𝑥1)

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.

  • A𝐺(𝑡)=6(𝑡+2), domain of function: , domain of derivative: (,2)(2,)
  • B𝐺(𝑡)=6𝑡+2, domain of function: , domain of derivative: (,2)(2,)
  • C𝐺(𝑡)=6𝑡+2, domain of function: (,2)(2,), domain of derivative: (,2)(2,)
  • D𝐺(𝑡)=6(𝑡+2), domain of function: (,2)(2,), domain of derivative: (,2)(2,)
  • E𝐺(𝑡)=4𝑡+2(𝑡+2), domain of function: (,2)(2,), domain of derivative:

Q24:

Differentiate 𝐷(𝑡)=181𝑡(3𝑡).

  • A𝐷(𝑡)=13𝑡5243𝑡
  • B𝐷(𝑡)=13𝑡5243𝑡
  • C𝐷(𝑡)=13𝑡5243𝑡
  • D𝐷(𝑡)=13𝑡5243𝑡
  • E𝐷(𝑡)=13𝑡5243𝑡

Q25:

Evaluate 𝑓(1), where 𝑓(𝑥)=163𝑥5.

  • A92
  • B32
  • C92
  • D32

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