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).

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

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 (1,2)?

  • A 𝑓 ( 𝑥 ) = 1 6 𝑥 6 , the slope of the tangent at point (1,2)=𝑓(1)=10
  • B 𝑓 ( 𝑥 ) = 6 , the slope of the tangent at point (1,2)=𝑓(1)=6
  • C 𝑓 ( 𝑥 ) = 8 𝑥 , the slope of the tangent at point (1,2)=𝑓(1)=8
  • D 𝑓 ( 𝑥 ) = 2 𝑥 6 , the slope of the tangent at point (1,2)=𝑓(1)=4
  • E 𝑓 ( 𝑥 ) = 8 𝑥 6 , the slope of the tangent at point (1,2)=𝑓(1)=2

Q3:

Using the definition of a derivative, evaluate dd𝑥1𝑥+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)=6𝑥7𝑥 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

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

Q5:

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

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

Q6:

Evaluate lim𝑓(+4)𝑓(2)+𝑓(2)𝑓(4).

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

Q7:

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

  • A 1 3
  • B15
  • C 5 3
  • 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.

  • A 𝑔 ( 𝑡 ) = 1 4 𝑡 , ( 0 , ) , ( 0 , )
  • B 𝑔 ( 𝑡 ) = 𝑡 4 , ( 0 , ) , ( 0 , )
  • C 𝑔 ( 𝑡 ) = 1 4 𝑡 , ( 0 , ) ,
  • D 𝑔 ( 𝑡 ) = 1 4 𝑡 , ,
  • E 𝑔 ( 𝑡 ) = 1 4 𝑡 , ( 0 , ) ,

Q10:

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

  • A 3 𝑥 + 9
  • B 3 𝑥 + 9
  • C 3 ( 𝑥 + 9 )
  • D 3 2 𝑥 + 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.

  • A 𝑔 ( 𝑥 ) = 4 𝑥 + 9 , ,
  • B 𝑔 ( 𝑥 ) = 4 𝑥 + 9 , ( , 9 ] , ( , 9 ]
  • C 𝑔 ( 𝑥 ) = 4 𝑥 + 9 , ( , 9 ] , ( , 9 )
  • D 𝑔 ( 𝑥 ) = 1 6 𝑥 + 9 , ( , 9 ] ,
  • E 𝑔 ( 𝑥 ) = 4 𝑥 + 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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.