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 𝑓(𝑥)=3𝑥5, find lim𝑓(𝑥+)𝑓(𝑥).

  • A27𝑥
  • B3𝑥
  • C27
  • D27𝑥
  • E27𝑥5

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)
  • B1(𝑥+1)
  • C𝑥+1
  • D1𝑥+1

Q4:

Consider the function 𝑓(𝑥)=|𝑥|.

Find lim𝑓().

Find lim𝑓().

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)=8𝑥6𝑥+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)=16𝑥6, 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)=2𝑥6, the slope of the tangent at 𝑥=1 is 𝑓(1)=4
  • E𝑓(𝑥)=8𝑥6, the slope of the tangent at 𝑥=1 is 𝑓(1)=2

Q6:

Let 𝑓(𝑥)=6𝑥6. Use the definition of the derivative to determine 𝑓(𝑥).

  • A3𝑥
  • B6𝑥
  • C6𝑥+6𝑥
  • D6𝑥6𝑥

Q7:

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

  • A𝑔(𝑥)=4𝑥+9, ,
  • B𝑔(𝑥)=4𝑥+9, (,9], (,9]
  • C𝑔(𝑥)=4𝑥+9, (,9], (,9)
  • D𝑔(𝑥)=16𝑥+9, (,9],
  • E𝑔(𝑥)=4𝑥+9, , (,9)

Q8:

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

  • A3𝑥+9
  • B3𝑥+9
  • C3(𝑥+9)
  • D32𝑥+9

Q9:

Find limsinsin(𝜋+)𝜋.

  • Aundefined
  • Bsin
  • Ccos
  • Dcos𝜋
  • Esin

Q10:

Evaluate lim𝑓(+4)𝑓(2)+𝑓(2)𝑓(4).

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

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