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

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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:

If the function 𝑓(𝑥)=−3𝑥−5, find lim→𝑓(𝑥+ℎ)−𝑓(𝑥)ℎ.

  • A−27𝑥
  • B−3𝑥
  • C−27
  • D−27𝑥
  • E−27𝑥−5ℎ

Q2:

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

Q3:

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

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

Q4:

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.

Q5:

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

Q6:

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

Q7:

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)

Q8:

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

Q9:

Find limsinsin→(𝜋+ℎ)−𝜋ℎ.

  • Aundefined
  • Bsinℎℎ
  • Ccosℎ
  • Dcos𝜋
  • Esinℎ

Q10:

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)

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