Lesson Worksheet: The Quotient Rule Mathematics • Higher Education

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

Q1:

Find dd𝑦π‘₯ if 𝑦=π‘₯+3π‘₯+3.

  • Aβˆ’π‘₯βˆ’9π‘₯+6π‘₯(π‘₯+3)οŠͺ
  • Bβˆ’π‘₯βˆ’9π‘₯+6π‘₯π‘₯+3οŠͺ
  • Cπ‘₯+9π‘₯βˆ’6π‘₯(π‘₯+3)οŠͺ
  • Dπ‘₯+9π‘₯βˆ’6π‘₯π‘₯+3οŠͺ

Q2:

Find dd𝑦π‘₯, given that 𝑦=π‘₯+7π‘₯+6π‘₯+8.

  • Aβˆ’2π‘₯βˆ’31π‘₯βˆ’112π‘₯+6(π‘₯+8)
  • B2π‘₯+31π‘₯+112π‘₯βˆ’6π‘₯+8
  • Cβˆ’2π‘₯βˆ’31π‘₯βˆ’112π‘₯+6π‘₯+8
  • D2π‘₯+31π‘₯+112π‘₯βˆ’6(π‘₯+8)

Q3:

Find the first derivative of function 𝑦=4π‘₯9π‘₯βˆ’7.

  • A36π‘₯+28(9π‘₯βˆ’7)
  • B7(9π‘₯βˆ’7)
  • Cβˆ’36π‘₯βˆ’28(9π‘₯βˆ’7)
  • Dβˆ’7(9π‘₯βˆ’7)

Q4:

Differentiate 𝑓(π‘₯)=4π‘₯βˆ’5π‘₯+83π‘₯βˆ’4.

  • Aβˆ’12π‘₯+32π‘₯+4(3π‘₯βˆ’4)
  • B12π‘₯βˆ’32π‘₯βˆ’4(3π‘₯βˆ’4)
  • Cβˆ’16π‘₯βˆ’4(3π‘₯βˆ’4)
  • D16π‘₯+4(3π‘₯βˆ’4)

Q5:

Suppose 𝑓(π‘₯)=π‘₯+π‘Žπ‘₯βˆ’π‘Ž and 𝑓′(2)=βˆ’2. Determine π‘Ž.

  • A4,βˆ’1
  • Bβˆ’4,1
  • Cβˆ’4,βˆ’1
  • D4,1

Q6:

Suppose that 𝑓(π‘₯)=π‘₯+π‘Žπ‘₯+𝑏π‘₯βˆ’7π‘₯+4. Given that 𝑓(0)=1 and 𝑓′(0)=4, find π‘Ž and 𝑏.

  • Aπ‘Ž=7, 𝑏=βˆ’4
  • Bπ‘Ž=9, 𝑏=4
  • Cπ‘Ž=7, 𝑏=4
  • Dπ‘Ž=βˆ’7, 𝑏=4

Q7:

Find the first derivative of 𝑦=8π‘₯+53π‘₯+22.

  • A8(3π‘₯+22)
  • B83
  • C161(3π‘₯+22)
  • D176π‘₯+153π‘₯+22
  • E191(3π‘₯+22)

Q8:

Find the first derivative of 𝑦=π‘₯βˆ’93π‘₯+13.

  • Aβˆ’80(π‘₯+13)
  • B106(π‘₯+13)
  • Cβˆ’9313
  • D2π‘₯βˆ’106(π‘₯+13)

Q9:

Differentiate 𝑓(π‘₯)=5π‘₯βˆ’17π‘₯+6.

  • A35π‘₯+60π‘₯+7(7π‘₯+6)
  • Bβˆ’30π‘₯βˆ’7(7π‘₯+6)
  • C30π‘₯+7(7π‘₯+6)
  • Dβˆ’35π‘₯βˆ’60π‘₯βˆ’7(7π‘₯+6)

Q10:

Find the first derivative of the function 𝑦=4π‘₯+5π‘₯+54π‘₯βˆ’2π‘₯+3.

  • A8π‘₯+5(4π‘₯βˆ’2π‘₯+3)
  • Bβˆ’28π‘₯βˆ’16π‘₯+25(4π‘₯βˆ’2π‘₯+3)
  • C8π‘₯+58π‘₯βˆ’2
  • D(8π‘₯βˆ’2)(4π‘₯+5π‘₯+5)(4π‘₯βˆ’2π‘₯+3)

Q11:

Given that 𝑦=3√π‘₯βˆ’2π‘₯√π‘₯, determine dd𝑦π‘₯.

  • A3βˆ’2√π‘₯
  • Bβˆ’1√π‘₯
  • Cβˆ’2√π‘₯
  • Dβˆ’βˆšπ‘₯

Q12:

Find the first derivative of 𝑦=βˆ’3π‘₯βˆ’2π‘₯+17√π‘₯ with respect to π‘₯.

  • Aβˆ’9π‘₯βˆ’2π‘₯βˆ’172π‘₯
  • Bβˆ’12π‘₯βˆ’6π‘₯+172π‘₯
  • Cβˆ’9π‘₯+2π‘₯+172√π‘₯
  • Dβˆ’12π‘₯βˆ’6π‘₯+172√π‘₯
  • Eβˆ’9π‘₯βˆ’2π‘₯βˆ’172√π‘₯

Q13:

If 𝑦=29π‘₯+8, find 1𝑦𝑦π‘₯ο‰οŠ¨dd.

  • A92
  • Bβˆ’29
  • Cβˆ’92
  • D29

Q14:

If 𝑦=π‘₯+5π‘₯βˆ’5βˆ’π‘₯βˆ’5π‘₯+5, find dd𝑦π‘₯.

  • Aβˆ’20π‘₯βˆ’500(π‘₯βˆ’25)
  • Bβˆ’20π‘₯+500(π‘₯+500)
  • Cβˆ’20π‘₯βˆ’500π‘₯βˆ’25
  • D20π‘₯βˆ’500(π‘₯βˆ’500)

Q15:

Evaluate 𝑓′(3), where 𝑓(π‘₯)=π‘₯π‘₯+2βˆ’π‘₯βˆ’3π‘₯βˆ’2.

  • Aβˆ’2725
  • B2725
  • Cβˆ’2325
  • D2325

Q16:

Calculate π‘₯𝑦π‘₯ο‰οŠ¬dd, given 𝑦=4π‘₯βˆ’58π‘₯.

  • A154
  • B258
  • C58
  • D25

Q17:

Find the first derivative of the function 𝑦=12π‘₯+1.

  • A1(2π‘₯+1)
  • Bβˆ’2(2π‘₯+1)
  • Cβˆ’1(2π‘₯+1)
  • D2(2π‘₯+1)

Q18:

Differentiate 𝑦=(π‘₯βˆ’1)(π‘₯+1)ο€Ήπ‘₯+1π‘₯.

  • A3π‘₯βˆ’π‘₯
  • B2π‘₯βˆ’2π‘₯
  • C3π‘₯+π‘₯
  • Dπ‘₯+π‘₯

Q19:

Let 𝑔(π‘₯)=𝑓(π‘₯)βˆ’4β„Ž(π‘₯)βˆ’5. Given that 𝑓(βˆ’2)=βˆ’1, 𝑓′(βˆ’2)=βˆ’8, β„Ž(βˆ’2)=βˆ’2, and β„Žβ€²(βˆ’2)=5, find 𝑔′(βˆ’2).

  • Aβˆ’49
  • B25
  • Cβˆ’449
  • Dβˆ’443

Q20:

If 𝑦=964π‘₯+49, find dd𝑦π‘₯+ο€Ό8𝑦3.

Q21:

Find the values of π‘₯ for which dd𝑦π‘₯=0, where 𝑦=π‘₯+6π‘₯+36π‘₯βˆ’6π‘₯+36.

  • A6, βˆ’6
  • B12, βˆ’12
  • C36, βˆ’36

Q22:

Let 𝑓(π‘₯)=2π‘₯7π‘₯βˆ’1. Use the definition of derivative to determine 𝑓′(π‘₯).

  • Aβˆ’2(7π‘₯βˆ’1)
  • B28π‘₯βˆ’2(7π‘₯βˆ’1)
  • Cβˆ’28π‘₯+2(7π‘₯βˆ’1)
  • D2(7π‘₯βˆ’1)

Q23:

Find the derivative of the function 𝐺, where 𝐺(𝑑)=2π‘‘βˆ’2𝑑+2, using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A𝐺′(𝑑)=6(𝑑+2), domain of function: ℝ, domain of derivative: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞)
  • B𝐺′(𝑑)=6𝑑+2, domain of function: ℝ, domain of derivative: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞)
  • C𝐺′(𝑑)=6𝑑+2, domain of function: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞), domain of derivative: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞)
  • D𝐺′(𝑑)=6(𝑑+2), domain of function: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞), domain of derivative: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞)
  • E𝐺′(𝑑)=4𝑑+2(𝑑+2), domain of function: (βˆ’βˆž,βˆ’2)βˆͺ(βˆ’2,∞), domain of derivative: ℝ

Q24:

Differentiate 𝐷(𝑑)=1βˆ’81𝑑(3𝑑)οŠͺ.

  • A𝐷′(𝑑)=βˆ’13π‘‘βˆ’5243π‘‘οŠ±οŠ¨οŠ±οŠ¬
  • B𝐷′(𝑑)=13π‘‘βˆ’5243π‘‘οŠ±οŠ§οŠ±οŠ¬
  • C𝐷′(𝑑)=13π‘‘βˆ’5243π‘‘οŠ±οŠ¨οŠ±οŠͺ
  • D𝐷′(𝑑)=13π‘‘βˆ’5243π‘‘οŠ±οŠ¨οŠ±οŠ¬
  • E𝐷′(𝑑)=βˆ’13π‘‘βˆ’5243π‘‘οŠ±οŠ¨οŠ±οŠͺ

Q25:

Evaluate 𝑓′(1), where 𝑓(π‘₯)=1βˆ’63π‘₯βˆ’5.

  • A92
  • B32
  • Cβˆ’92
  • Dβˆ’32

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