Lesson Worksheet: Definition of the Derivative Mathematics • Higher Education

In this worksheet, we will practice calculating the derivative of a function using the formal definition of the derivative as a limit.

Q1:

Let ๐‘“(๐‘ฅ)=โˆ’6โˆš๐‘ฅโˆ’6. Use the definition of the derivative to determine ๐‘“โ€ฒ(๐‘ฅ).

  • Aโˆ’3โˆš๐‘ฅ
  • Bโˆ’6โˆš๐‘ฅ
  • Cโˆ’6๐‘ฅ+6โˆš๐‘ฅ๏Šฉ
  • Dโˆ’6๐‘ฅโˆ’6โˆš๐‘ฅ๏Šฉ

Q2:

Let ๐‘“(๐‘ฅ)=8๐‘ฅโˆ’6๐‘ฅ+9๏Šจ. Use the definition of derivative to determine ๐‘“โ€ฒ(๐‘ฅ). What is the slope of the tangent to its graph at ๐‘ฅ=1?

  • A๐‘“โ€ฒ(๐‘ฅ)=16๐‘ฅโˆ’6, the slope of the tangent at ๐‘ฅ=1 is ๐‘“โ€ฒ(1)=10
  • B๐‘“โ€ฒ(๐‘ฅ)=โˆ’6, the slope of the tangent at ๐‘ฅ=1 is ๐‘“โ€ฒ(1)=โˆ’6
  • C๐‘“โ€ฒ(๐‘ฅ)=8๐‘ฅ, the slope of the tangent at ๐‘ฅ=1 is ๐‘“โ€ฒ(1)=8
  • D๐‘“โ€ฒ(๐‘ฅ)=2๐‘ฅโˆ’6, the slope of the tangent at ๐‘ฅ=1 is ๐‘“โ€ฒ(1)=โˆ’4
  • E๐‘“โ€ฒ(๐‘ฅ)=8๐‘ฅโˆ’6, the slope of the tangent at ๐‘ฅ=1 is ๐‘“โ€ฒ(1)=2

Q3:

Using the definition of a derivative, evaluate dd๐‘ฅ๏€ผ1๐‘ฅ+1๏ˆ.

  • A1(๐‘ฅ+1)๏Šจ
  • Bโˆ’1(๐‘ฅ+1)๏Šจ
  • C๐‘ฅ+1
  • Dโˆ’1๐‘ฅ+1

Q4:

Find the derivative of the function ๐‘“(๐‘ฅ)=6๐‘ฅโˆ’7๐‘ฅ๏Šฉ๏Šจ using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A๐‘“โ€ฒ(๐‘ฅ)=18๐‘ฅโˆ’14๐‘ฅ๏Šจ, โ„, โ„
  • B๐‘“โ€ฒ(๐‘ฅ)=6๐‘ฅโˆ’7๐‘ฅ๏Šจ, (0,โˆž), โ„
  • C๐‘“โ€ฒ(๐‘ฅ)=18๐‘ฅโˆ’14๐‘ฅ๏Šฉ, โ„, โ„
  • D๐‘“โ€ฒ(๐‘ฅ)=18๐‘ฅโˆ’14๐‘ฅ๏Šฉ๏Šจ, โ„, (0,โˆž)
  • E๐‘“โ€ฒ(๐‘ฅ)=18๐‘ฅโˆ’14๏Šจ, (0,โˆž), (0,โˆž)

Q5:

Determine the derivative of the function ๐‘“(๐‘ฅ)=โˆš2๐‘ฅโˆ’16 using the definition of the derivative.

  • A๐‘“โ€ฒ(๐‘ฅ)=12โˆš2๐‘ฅโˆ’16
  • B๐‘“โ€ฒ(๐‘ฅ)=2๐‘ฅโˆ’16
  • C๐‘“โ€ฒ(๐‘ฅ)=2โˆš2๐‘ฅโˆ’16
  • D๐‘“โ€ฒ(๐‘ฅ)=1โˆš2๐‘ฅโˆ’16

Q6:

Evaluate lim๏‚โ†’๏Šฆ๐‘“(โ„Ž+4)โˆ’๐‘“(โ„Žโˆ’2)+๐‘“(โˆ’2)โˆ’๐‘“(4)โ„Ž.

  • A๐‘“โ€ฒ(โˆ’2)
  • B๐‘“โ€ฒ(โˆ’2)โˆ’๐‘“โ€ฒ(4)
  • C๐‘“โ€ฒ(4)โˆ’๐‘“โ€ฒ(โˆ’2)
  • D๐‘“โ€ฒ(4)
  • E๐‘“โ€ฒ(4)+๐‘“โ€ฒ(โˆ’2)

Q7:

Given a function with ๐‘“(โˆ’3)=7 and ๐‘“โ€ฒ(โˆ’3)=3, what is lim๏‚โ†’๏Šฆ5โ„Ž๐‘“(โ„Žโˆ’3)โˆ’7?

  • A13
  • B15
  • C53
  • D0
  • E3

Q8:

Consider a function with ๐‘“(โˆ’8)=3 and ๐‘“โ€ฒ(โˆ’8)=7. What is lim๏—โ†’๏Šฑ๏Šฎ๐‘“(๐‘ฅ)?

Q9:

Find the derivative of the function ๐‘”(๐‘ก)=โˆ’12โˆš๐‘ก using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A๐‘”(๐‘ก)=14โˆš๐‘ก๏Ž˜๏Šฉ, (0,โˆž), (0,โˆž)
  • B๐‘”(๐‘ก)=โˆš๐‘ก4๏Ž˜๏Šฉ, (0,โˆž), (0,โˆž)
  • C๐‘”(๐‘ก)=โˆ’14โˆš๐‘ก๏Ž˜, (0,โˆž), โ„
  • D๐‘”(๐‘ก)=14โˆš๐‘ก๏Ž˜๏Šฉ, โ„, โ„
  • E๐‘”(๐‘ก)=โˆ’14โˆš๐‘ก๏Ž˜๏Šฉ, (0,โˆž), โ„

Q10:

Let ๐‘“(๐‘ฅ)=โˆ’3โˆšโˆ’๐‘ฅ+9. Use the definition of derivative to determine ๐‘“โ€ฒ(๐‘ฅ).

  • Aโˆ’3โˆšโˆ’๐‘ฅ+9๏Šจ
  • B3โˆšโˆ’๐‘ฅ+9
  • Cโˆ’3๏„(โˆ’๐‘ฅ+9)๏Šฉ
  • D32โˆšโˆ’๐‘ฅ+9

Q11:

Find the derivative of the function ๐‘”(๐‘ฅ)=โˆ’8โˆšโˆ’๐‘ฅ+9 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A๐‘”โ€ฒ(๐‘ฅ)=4โˆšโˆ’๐‘ฅ+9, โ„, โ„
  • B๐‘”โ€ฒ(๐‘ฅ)=โˆ’4โˆšโˆ’๐‘ฅ+9, (โˆ’โˆž,9], (โˆ’โˆž,9]
  • C๐‘”โ€ฒ(๐‘ฅ)=4โˆšโˆ’๐‘ฅ+9, (โˆ’โˆž,9], (โˆ’โˆž,9)
  • D๐‘”โ€ฒ(๐‘ฅ)=16โˆšโˆ’๐‘ฅ+9, (โˆ’โˆž,9], โ„
  • E๐‘”โ€ฒ(๐‘ฅ)=โˆ’4โˆšโˆ’๐‘ฅ+9, โ„, (โˆ’โˆž,9)

Q12:

Consider the function ๐‘“(๐‘ฅ)=|๐‘ฅ|.

Find lim๏‚โ†’๏Šฆ๏Žฉ๐‘“(โ„Ž)โ„Ž.

Find lim๏‚โ†’๏Šฆ๏Žช๐‘“(โ„Ž)โ„Ž.

What can you conclude about the derivative of ๐‘“(๐‘ฅ) at ๐‘ฅ=0?

  • ASince the right and left limits are unequal, the derivative does not exist.
  • BThe derivative exists and is equal to โˆ’1.
  • CThe derivative exists and is equal to 1.
  • DThe derivative exists and is equal to 0.

Q13:

Find the derivative of ๐‘“(๐‘ฅ)=๐‘ฅ๏Šจ at the point ๐‘ฅ=2 from first principles.

Q14:

Find the derivative of the function ๐‘“(๐‘ฅ)=2๐‘ฅโˆ’11๐‘ฅ๏Šฑ๏Šฉ using the definition of the derivative.

  • Aโˆ’6๐‘ฅโˆ’11๏Šฑ๏Šจ
  • Bโˆ’6๐‘ฅโˆ’11๏Šฑ๏Šช
  • Cโˆ’3๐‘ฅโˆ’11๐‘ฅ๏Šฑ๏Šช
  • Dโˆ’6๐‘ฅโˆ’11๐‘ฅ๏Šฑ๏Šจ
  • Eโˆ’3๐‘ฅโˆ’11๏Šฑ๏Šช

Q15:

Find the derivative of the function ๐‘“(๐‘ก)=2๐‘ก+8๐‘กโˆ’1๏Šฉ using the definition of the derivative.

  • A2๐‘ก+7๏Šจ
  • B6๐‘ก+8๏Šจ
  • C2๐‘ก+8๏Šจ
  • D6๐‘ก+8๐‘ก๏Šจ
  • E6๐‘ก+8๐‘กโˆ’1๏Šจ

Q16:

Find the derivative of the function ๐‘”(๐‘ฅ)=2๐‘ฅ๏Šฑ๏Šจ using the definition of the derivative.

  • Aโˆ’4๐‘ฅ๏Šฑ๏Šฉ
  • Bโˆ’2๐‘ฅ๏Šฑ๏Šฉ
  • C4๐‘ฅ๏Šฑ๏Šฉ
  • Dโˆ’๐‘ฅ๏Šฑ๏Šช
  • Eโˆ’๐‘ฅ๏Šฑ๏Šฉ

Find the equation of the tangent to the graph of ๐‘” at ๐‘ฅ=3.

  • A๐‘ฆ=โˆ’427๐‘ฅโˆ’29
  • B๐‘ฆ=427๐‘ฅโˆ’29
  • C๐‘ฆ=โˆ’481๐‘ฅ+1027
  • D๐‘ฆ=โˆ’427๐‘ฅ+23
  • E๐‘ฆ=427๐‘ฅ+23

Q17:

Find the derivative of the function ๐‘“(๐‘ก)=12๐‘ก+7๐‘ก๏Šจ using the definition of the derivative.

  • A4๐‘ก+7(2๐‘ก+7๐‘ก)๏Šจ๏Šจ
  • B4๐‘กโˆ’7(2๐‘ก+7๐‘ก)๏Šจ๏Šจ
  • Cโˆ’4๐‘กโˆ’7(2๐‘ก+7๐‘ก)๏Šจ๏Šจ
  • D2๐‘ก+7(2๐‘ก+7๐‘ก)๏Šจ๏Šจ
  • E4๐‘ก+7(2๐‘ก+7๐‘ก)๏Šจ

Q18:

Find the derivative of the function ๐‘”(๐‘ฅ)=2(โˆ’5๐‘ฅโˆ’17)๏Šฑ๏Šง using the definition of the derivative.

  • Aโˆ’2(โˆ’5๐‘ฅโˆ’17)๏Šจ
  • Bโˆ’10(โˆ’5๐‘ฅโˆ’17)๏Šจ
  • C10(โˆ’5๐‘ฅโˆ’17)๏Šจ
  • D5(โˆ’5๐‘ฅโˆ’17)๏Šจ
  • E10๐‘ฅ(โˆ’5๐‘ฅโˆ’17)๏Šจ

Q19:

Suppose ๐‘“(๐‘ฅ)=๐‘Ž๐‘ฅโˆ’๐‘๐‘ฅ+3๏Šจ. If the average rate of change as ๐‘ฅ changes from 3 to 3.5 is 4 and ๐‘“โ€ฒ(2)=โˆ’1, determine ๐‘Ž and ๐‘.

  • A๐‘Ž=2, ๐‘=7
  • B๐‘Ž=2, ๐‘=5
  • C๐‘Ž=2, ๐‘=9
  • D๐‘Ž=2, ๐‘=3

Q20:

What is the rate of change of ๐‘“(๐‘ฅ)=(๐‘ฅโˆ’11)โˆ’2๏Šจ with respect to ๐‘ฅ at ๐‘ฅ=11?

Q21:

Consider the function ๐‘“(๐‘ฅ)=|2๐‘ฅโˆ’1|.

Find lim๏‚โ†’๏Šฆ๏Žฉ๐‘“(๐‘ฅ) and lim๏‚โ†’๏Šฆ๏Žช๐‘“(๐‘ฅ).

What can you conclude about the derivative of ๐‘“(๐‘ฅ) at ๐‘ฅ=0?

  • AThe derivative exists and is equal to 2.
  • BSince the right and left limits are unequal, the derivative does not exist.
  • CThe derivative exists and is equal to โˆ’2.
  • DThe derivative exists and is equal to 12.
  • EThe derivative exists and is equal to โˆ’12.

Q22:

What is the slope of the tangent to ๐‘“(๐‘ฅ)=โˆš|2โˆ’๐‘ฅ| at ๐‘ฅ=2?

  • Aโˆž
  • Bโˆ’1
  • Cโˆ’12
  • Dโˆ’โˆž
  • ENone of the above

Q23:

Find limsinsin๏‚โ†’๏Šฆ(๐œ‹+โ„Ž)โˆ’๐œ‹โ„Ž.

  • Aundefined
  • Bsinโ„Žโ„Ž
  • Ccosโ„Ž
  • Dcos๐œ‹
  • Esinโ„Ž

Q24:

If the function ๐‘“(๐‘ฅ)=โˆ’3๐‘ฅโˆ’5๏Šฏ, find lim๏‚โ†’๏Šฆ๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ)โ„Ž.

  • Aโˆ’27๐‘ฅ๏Šฎ
  • Bโˆ’3๐‘ฅ๏Šฏ
  • Cโˆ’27
  • Dโˆ’27๐‘ฅ๏Šฏ
  • Eโˆ’27๐‘ฅโˆ’5โ„Ž๏Šฎ

Q25:

If lim๏‚โ†’๏Šฆ๐‘“(โˆ’1+โ„Ž)โˆ’๐‘“(โˆ’1โˆ’2โ„Ž)โ„Ž=6, find ๐‘“โ€ฒ(โˆ’1).

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