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

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

Q2:

Let 𝑓 ( 𝑥 ) = 8 𝑥 6 𝑥 + 9 2 . 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 ) = 8 𝑥 , the slope of the tangent at point ( 1 , 2 ) = 𝑓 ( 1 ) = 8
  • B 𝑓 ( 𝑥 ) = 8 𝑥 6 , the slope of the tangent at point ( 1 , 2 ) = 𝑓 ( 1 ) = 2
  • C 𝑓 ( 𝑥 ) = 2 𝑥 6 , the slope of the tangent at point ( 1 , 2 ) = 𝑓 ( 1 ) = 4
  • D 𝑓 ( 𝑥 ) = 1 6 𝑥 6 , the slope of the tangent at point ( 1 , 2 ) = 𝑓 ( 1 ) = 1 0
  • E 𝑓 ( 𝑥 ) = 6 , the slope of the tangent at point ( 1 , 2 ) = 𝑓 ( 1 ) = 6

Q3:

Using the definition of a derivative, evaluate d d 𝑥 1 𝑥 + 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 ) = 6 𝑥 7 𝑥 3 2 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑓 ( 𝑥 ) = 6 𝑥 7 𝑥 2 , ( 0 , ) ,
  • B 𝑓 ( 𝑥 ) = 1 8 𝑥 1 4 𝑥 3 2 , , ( 0 , )
  • C 𝑓 ( 𝑥 ) = 1 8 𝑥 1 4 2 , ( 0 , ) , ( 0 , )
  • D 𝑓 ( 𝑥 ) = 1 8 𝑥 1 4 𝑥 2 , ,
  • E 𝑓 ( 𝑥 ) = 1 8 𝑥 1 4 𝑥 3 , ,

Q5:

Determine the derivative of the function 𝑓 ( 𝑥 ) = 2 𝑥 1 6 using the definition of the derivative.

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

Q6:

Evaluate l i m 0 𝑓 ( + 4 ) 𝑓 ( 2 ) + 𝑓 ( 2 ) 𝑓 ( 4 ) .

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

Q7:

Given a function with 𝑓 ( 3 ) = 7 and 𝑓 ( 3 ) = 3 , what is l i m 0 5 𝑓 ( 3 ) 7 ?

  • A3
  • B0
  • C 1 3
  • D 5 3
  • E15

Q8:

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

Q9:

Find the derivative of the function 𝑔 ( 𝑡 ) = 1 2 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 3 , ( 0 , ) ,
  • B 𝑔 ( 𝑡 ) = 1 4 𝑡 3 , ,
  • C 𝑔 ( 𝑡 ) = 𝑡 4 3 , ( 0 , ) , ( 0 , )
  • D 𝑔 ( 𝑡 ) = 1 4 𝑡 3 , ( 0 , ) , ( 0 , )
  • E 𝑔 ( 𝑡 ) = 1 4 𝑡 , ( 0 , ) ,

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