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−6𝑥+6√𝑥
  • D−6𝑥−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 𝑥=1?

  • A𝑓′(𝑥)=16𝑥−6, the slope of the tangent at 𝑥=1 is 𝑓′(1)=10
  • B𝑓′(𝑥)=−6, the slope of the tangent at 𝑥=1 is 𝑓′(1)=−6
  • C𝑓′(𝑥)=8𝑥, the slope of the tangent at 𝑥=1 is 𝑓′(1)=8
  • D𝑓′(𝑥)=2𝑥−6, the slope of the tangent at 𝑥=1 is 𝑓′(1)=−4
  • E𝑓′(𝑥)=8𝑥−6, the slope of the tangent at 𝑥=1 is 𝑓′(1)=2

Q3:

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

  • A1(𝑥+1)
  • B−1(𝑥+1)
  • C𝑥+1
  • D−1𝑥+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.

  • A𝑓′(𝑥)=18𝑥−14𝑥, ℝ, ℝ
  • B𝑓′(𝑥)=6𝑥−7𝑥, (0,∞), ℝ
  • C𝑓′(𝑥)=18𝑥−14𝑥, ℝ, ℝ
  • D𝑓′(𝑥)=18𝑥−14𝑥, ℝ, (0,∞)
  • E𝑓′(𝑥)=18𝑥−14, (0,∞), (0,∞)

Q5:

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

  • A𝑓′(𝑥)=12√2𝑥−16
  • B𝑓′(𝑥)=2𝑥−16
  • C𝑓′(𝑥)=2√2𝑥−16
  • D𝑓′(𝑥)=1√2𝑥−16

Q6:

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

  • A𝑓′(−2)
  • B𝑓′(−2)−𝑓′(4)
  • C𝑓′(4)−𝑓′(−2)
  • D𝑓′(4)
  • E𝑓′(4)+𝑓′(−2)

Q7:

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

  • A13
  • B15
  • C53
  • 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.

  • A𝑔(𝑡)=14√𝑡, (0,∞), (0,∞)
  • B𝑔(𝑡)=√𝑡4, (0,∞), (0,∞)
  • C𝑔(𝑡)=−14√𝑡, (0,∞), ℝ
  • D𝑔(𝑡)=14√𝑡, ℝ, ℝ
  • E𝑔(𝑡)=−14√𝑡, (0,∞), ℝ

Q10:

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

  • A−3√−𝑥+9
  • B3√−𝑥+9
  • C−3(−𝑥+9)
  • D32√−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.

  • A𝑔′(𝑥)=4√−𝑥+9, ℝ, ℝ
  • B𝑔′(𝑥)=−4√−𝑥+9, (−∞,9], (−∞,9]
  • C𝑔′(𝑥)=4√−𝑥+9, (−∞,9], (−∞,9)
  • D𝑔′(𝑥)=16√−𝑥+9, (−∞,9], ℝ
  • E𝑔′(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.

Q14:

Find the derivative of the function 𝑓(𝑥)=2𝑥−11𝑥 using the definition of the derivative.

  • A−6𝑥−11
  • B−6𝑥−11
  • C−3𝑥−11𝑥
  • D−6𝑥−11𝑥
  • E−3𝑥−11

Q15:

Find the derivative of the function 𝑓(𝑡)=2𝑡+8𝑡−1 using the definition of the derivative.

  • A2𝑡+7
  • B6𝑡+8
  • C2𝑡+8
  • D6𝑡+8𝑡
  • E6𝑡+8𝑡−1

Q16:

Find the derivative of the function 𝑔(𝑥)=2𝑥 using the definition of the derivative.

  • A−4𝑥
  • B−2𝑥
  • C4𝑥
  • D−𝑥
  • E−𝑥

Find the equation of the tangent to the graph of 𝑔 at 𝑥=3.

  • A𝑦=−427𝑥−29
  • B𝑦=427𝑥−29
  • C𝑦=−481𝑥+1027
  • D𝑦=−427𝑥+23
  • E𝑦=427𝑥+23

Q17:

Find the derivative of the function 𝑓(𝑡)=12𝑡+7𝑡 using the definition of the derivative.

  • A4𝑡+7(2𝑡+7𝑡)
  • B4𝑡−7(2𝑡+7𝑡)
  • C−4𝑡−7(2𝑡+7𝑡)
  • D2𝑡+7(2𝑡+7𝑡)
  • E4𝑡+7(2𝑡+7𝑡)

Q18:

Find the derivative of the function 𝑔(𝑥)=2(−5𝑥−17) using the definition of the derivative.

  • A−2(−5𝑥−17)
  • B−10(−5𝑥−17)
  • C10(−5𝑥−17)
  • D5(−5𝑥−17)
  • E10𝑥(−5𝑥−17)

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