Worksheet: Definition of the Derivative

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

  • A3𝑥
  • B6𝑥
  • C6𝑥+6𝑥
  • D6𝑥6𝑥

Q2:

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

Q3:

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

  • A1(𝑥+1)
  • B1(𝑥+1)
  • C𝑥+1
  • D1𝑥+1

Q4:

Find the derivative of the function 𝑓(𝑥)=6𝑥7𝑥 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A𝑓(𝑥)=18𝑥14𝑥, ,
  • B𝑓(𝑥)=6𝑥7𝑥, (0,),
  • C𝑓(𝑥)=18𝑥14𝑥, ,
  • D𝑓(𝑥)=18𝑥14𝑥, , (0,)
  • E𝑓(𝑥)=18𝑥14, (0,), (0,)

Q5:

Determine the derivative of the function 𝑓(𝑥)=2𝑥16 using the definition of the derivative.

  • A𝑓(𝑥)=122𝑥16
  • B𝑓(𝑥)=2𝑥16
  • C𝑓(𝑥)=22𝑥16
  • D𝑓(𝑥)=12𝑥16

Q6:

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)

Q7:

Given a function with 𝑓(3)=7 and 𝑓(3)=3, what is lim5𝑓(3)7?

  • A13
  • B15
  • C53
  • D0
  • E3

Q8:

Consider a function with 𝑓(8)=3 and 𝑓(8)=7. What is lim𝑓(𝑥)?

Q9:

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

  • A𝑔(𝑡)=14𝑡, (0,), (0,)
  • B𝑔(𝑡)=𝑡4, (0,), (0,)
  • C𝑔(𝑡)=14𝑡, (0,),
  • D𝑔(𝑡)=14𝑡, ,
  • E𝑔(𝑡)=14𝑡, (0,),

Q10:

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

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

Q11:

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)

Q12:

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.

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)=2𝑥11𝑥 using the definition of the derivative.

  • A6𝑥11
  • B6𝑥11
  • C3𝑥11𝑥
  • D6𝑥11𝑥
  • E3𝑥11

Q15:

Find the derivative of the function 𝑓(𝑡)=2𝑡+8𝑡1 using the definition of the derivative.

  • A2𝑡+7
  • B6𝑡+8
  • C2𝑡+8
  • D6𝑡+8𝑡
  • E6𝑡+8𝑡1

Q16:

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

  • A4𝑥
  • B2𝑥
  • C4𝑥
  • D𝑥
  • E𝑥

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

  • A𝑦=427𝑥29
  • B𝑦=427𝑥29
  • 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)
  • B4𝑡7(2𝑡+7𝑡)
  • C4𝑡7(2𝑡+7𝑡)
  • D2𝑡+7(2𝑡+7𝑡)
  • E4𝑡+7(2𝑡+7𝑡)

Q18:

Find the derivative of the function 𝑔(𝑥)=2(5𝑥17) using the definition of the derivative.

  • A2(5𝑥17)
  • B10(5𝑥17)
  • C10(5𝑥17)
  • D5(5𝑥17)
  • E10𝑥(5𝑥17)

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