Lesson Worksheet: Definition of the Derivative Mathematics • 12th Grade

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

Q1:

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

Q2:

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

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:

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.

Q5:

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

Q6:

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

Q7:

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)

Q8:

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

Q9:

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

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

Q10:

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)

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