Worksheet: Definition of the Derivative

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

Q1:

Let 𝑓(𝑥)=−6√𝑥−6. Use the definition of the derivative to determine 𝑓′(𝑥).

  • A − 3 √ 𝑥
  • B − 6 √ 𝑥
  • C − 6 𝑥 + 6 √ 𝑥
  • D − 6 𝑥 − 6 √ 𝑥

Q2:

Let 𝑓(𝑥)=8𝑥−6𝑥+9. Use the definition of derivative to determine 𝑓′(𝑥). What is the slope of the tangent to its graph at (1,2)?

  • A 𝑓 ′ ( 𝑥 ) = 1 6 𝑥 − 6 , the slope of the tangent at point (1,2)=𝑓′(1)=10
  • B 𝑓 ′ ( 𝑥 ) = − 6 , the slope of the tangent at point (1,2)=𝑓′(1)=−6
  • C 𝑓 ′ ( 𝑥 ) = 8 𝑥 , the slope of the tangent at point (1,2)=𝑓′(1)=8
  • D 𝑓 ′ ( 𝑥 ) = 2 𝑥 − 6 , the slope of the tangent at point (1,2)=𝑓′(1)=−4
  • E 𝑓 ′ ( 𝑥 ) = 8 𝑥 − 6 , the slope of the tangent at point (1,2)=𝑓′(1)=2

Q3:

Using the definition of a derivative, evaluate dd𝑥1𝑥+1.

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

Q4:

Find the derivative of the function 𝑓(𝑥)=6𝑥−7𝑥 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑓 ′ ( 𝑥 ) = 1 8 𝑥 − 1 4 𝑥 , ℝ , ℝ
  • B 𝑓 ′ ( 𝑥 ) = 6 𝑥 − 7 𝑥 , ( 0 , ∞ ) , ℝ
  • C 𝑓 ′ ( 𝑥 ) = 1 8 𝑥 − 1 4 𝑥 , ℝ , ℝ
  • D 𝑓 ′ ( 𝑥 ) = 1 8 𝑥 − 1 4 𝑥 , ℝ , ( 0 , ∞ )
  • E 𝑓 ′ ( 𝑥 ) = 1 8 𝑥 − 1 4 , ( 0 , ∞ ) , ( 0 , ∞ )

Q5:

Determine the derivative of the function 𝑓(𝑥)=√2𝑥−16 using the definition of the derivative.

  • A 𝑓 ′ ( 𝑥 ) = 1 2 √ 2 𝑥 − 1 6
  • B 𝑓 ′ ( 𝑥 ) = 2 𝑥 − 1 6
  • C 𝑓 ′ ( 𝑥 ) = 2 √ 2 𝑥 − 1 6
  • D 𝑓 ′ ( 𝑥 ) = 1 √ 2 𝑥 − 1 6

Q6:

Evaluate lim→𝑓(ℎ+4)−𝑓(ℎ−2)+𝑓(−2)−𝑓(4)ℎ.

  • A 𝑓 ′ ( − 2 )
  • B 𝑓 ′ ( − 2 ) − 𝑓 ′ ( 4 )
  • C 𝑓 ′ ( 4 ) − 𝑓 ′ ( − 2 )
  • D 𝑓 ′ ( 4 )
  • E 𝑓 ′ ( 4 ) + 𝑓 ′ ( − 2 )

Q7:

Given a function with 𝑓(−3)=7 and 𝑓′(−3)=3, what is lim→5ℎ𝑓(ℎ−3)−7?

  • A 1 3
  • B15
  • C 5 3
  • D0
  • E3

Q8:

Consider a function with 𝑓(−8)=3 and 𝑓′(−8)=7. What is lim→𝑓(𝑥)?

Q9:

Find the derivative of the function 𝑔(𝑡)=−12√𝑡 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑔 ( 𝑡 ) = 1 4 √ 𝑡 , ( 0 , ∞ ) , ( 0 , ∞ )
  • B 𝑔 ( 𝑡 ) = √ 𝑡 4 , ( 0 , ∞ ) , ( 0 , ∞ )
  • C 𝑔 ( 𝑡 ) = − 1 4 √ 𝑡 , ( 0 , ∞ ) , ℝ
  • D 𝑔 ( 𝑡 ) = 1 4 √ 𝑡 , ℝ , ℝ
  • E 𝑔 ( 𝑡 ) = − 1 4 √ 𝑡 , ( 0 , ∞ ) , ℝ

Q10:

Let 𝑓(𝑥)=−3√−𝑥+9. Use the definition of derivative to determine 𝑓′(𝑥).

  • A − 3 √ − 𝑥 + 9
  • B 3 √ − 𝑥 + 9
  • C − 3 ( − 𝑥 + 9 )
  • D 3 2 √ − 𝑥 + 9

Q11:

Find the derivative of the function 𝑔(𝑥)=−8√−𝑥+9 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑔 ′ ( 𝑥 ) = 4 √ − 𝑥 + 9 , ℝ , ℝ
  • B 𝑔 ′ ( 𝑥 ) = − 4 √ − 𝑥 + 9 , ( − ∞ , 9 ] , ( − ∞ , 9 ]
  • C 𝑔 ′ ( 𝑥 ) = 4 √ − 𝑥 + 9 , ( − ∞ , 9 ] , ( − ∞ , 9 )
  • D 𝑔 ′ ( 𝑥 ) = 1 6 √ − 𝑥 + 9 , ( − ∞ , 9 ] , ℝ
  • E 𝑔 ′ ( 𝑥 ) = − 4 √ − 𝑥 + 9 , ℝ , ( − ∞ , 9 )

Q12:

Consider the function 𝑓(𝑥)=|𝑥|.

Find lim→𝑓(ℎ)ℎ.

Find lim→𝑓(ℎ)ℎ.

What can you conclude about the derivative of 𝑓(𝑥) at 𝑥=0?

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

Q13:

Find the derivative of 𝑓(𝑥)=𝑥 at the point 𝑥=2 from first principles.

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