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Worksheet: Definition of the Derivative

Q1:

Using the definition of a derivative, evaluate d d π‘₯ ο€Ό 1 π‘₯ + 1  .

  • A βˆ’ 1 π‘₯ + 1
  • B 1 ( π‘₯ + 1 ) 2
  • C π‘₯ + 1
  • D βˆ’ 1 ( π‘₯ + 1 ) 2

Q2:

Let If 𝑓 is differentiable at π‘₯ = 4 , find π‘Ž and 𝑏 .

  • A π‘Ž = 0 , 𝑏 = 1
  • B π‘Ž = 2 , 𝑏 = 1
  • C π‘Ž = 2 , 𝑏 = 3
  • D π‘Ž = 0 , 𝑏 = βˆ’ 1
  • E π‘Ž = βˆ’ 1 , 𝑏 = βˆ’ 2

Q3:

Find d d 𝑦 π‘₯ if 𝑦 = 6 √ π‘₯ 7 .

  • A 6 7 √ π‘₯
  • B 3 √ π‘₯ 7
  • C βˆ’ 3 7 √ π‘₯
  • D 3 7 √ π‘₯

Q4:

Differentiate 𝑓 ( π‘₯ ) = 1 π‘₯ , and identify the value of π‘₯ at which the function is not differentiable.

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ 1 π‘₯ 2 , and the function is not differentiable at π‘₯ = βˆ’ 4 .
  • B 𝑓 β€² ( π‘₯ ) = 1 π‘₯ 2 , and the function is not differentiable at π‘₯ = βˆ’ 4 .
  • C 𝑓 β€² ( π‘₯ ) = 1 π‘₯ 2 , and the function is not differentiable at π‘₯ = 0 .
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ 1 π‘₯ 2 , and the function is not differentiable at π‘₯ = 0 .

Q5:

Differentiate 𝑓 ( π‘₯ ) = 7 βˆ’ 8 π‘₯ + 5 , and identify the value of π‘₯ at which the function is not differentiable.

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ 5 6 ( βˆ’ 8 π‘₯ + 5 ) 2 , and the function is not differentiable at π‘₯ = 5 8 .
  • B 𝑓 β€² ( π‘₯ ) = 5 6 ( βˆ’ 8 π‘₯ + 5 ) 2 , and the function is not differentiable at π‘₯ = βˆ’ 5 8 .
  • C 𝑓 β€² ( π‘₯ ) = βˆ’ 5 6 ( βˆ’ 8 π‘₯ + 5 ) 2 , and the function is not differentiable at π‘₯ = βˆ’ 5 8 .
  • D 𝑓 β€² ( π‘₯ ) = 5 6 ( βˆ’ 8 π‘₯ + 5 ) 2 , and the function is not differentiable at π‘₯ = 5 8 .

Q6:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = 1 ?

  • AThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 ( π‘₯ ) is discontinuous at 𝑓 ( 1 ) .
  • BThe function 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) β‰  𝑓 β€² ( 1 )   .
  • CThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 1 as l i m l i m  β†’   β†’  οŽͺ  𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) but is not continuous.
  • DThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 1 .
  • EThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 ( 1 ) is undefined.

Q7:

Find the derivative of the function 𝑔 ( 𝑑 ) = βˆ’ 1 2 √ 𝑑 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑔 β€² ( 𝑑 ) = βˆ’ 1 4 √ 𝑑 3 , ( 0 , ∞ ) , ℝ
  • B 𝑔 β€² ( 𝑑 ) = 1 4 √ 𝑑 3 , ℝ , ℝ
  • C 𝑔 β€² ( 𝑑 ) = √ 𝑑 4 3 , ( 0 , ∞ ) , ( 0 , ∞ )
  • D 𝑔 β€² ( 𝑑 ) = 1 4 √ 𝑑 3 , ( 0 , ∞ ) , ( 0 , ∞ )
  • E 𝑔 β€² ( 𝑑 ) = βˆ’ 1 4 √ 𝑑 , ( 0 , ∞ ) , ℝ

Q8:

Let 𝑓 ( π‘₯ ) = βˆ’ 3 √ βˆ’ π‘₯ + 9 . Use the definition of derivative to determine 𝑓 β€² ( π‘₯ ) .

  • A 3 √ βˆ’ π‘₯ + 9
  • B βˆ’ 3 √ βˆ’ π‘₯ + 9 2
  • C βˆ’ 3  ( βˆ’ π‘₯ + 9 ) 3
  • D 3 2 √ βˆ’ π‘₯ + 9

Q9:

Find the derivative of the function 𝑔 ( π‘₯ ) = βˆ’ 8 √ βˆ’ π‘₯ + 9 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑔 β€² ( π‘₯ ) = 1 6 √ βˆ’ π‘₯ + 9 , ( βˆ’ ∞ , 9 ] , ℝ
  • B 𝑔 β€² ( π‘₯ ) = βˆ’ 4 √ βˆ’ π‘₯ + 9 , ℝ , ( βˆ’ ∞ , 9 )
  • C 𝑔 β€² ( π‘₯ ) = 4 √ βˆ’ π‘₯ + 9 , ℝ , ℝ
  • D 𝑔 β€² ( π‘₯ ) = 4 √ βˆ’ π‘₯ + 9 , ( βˆ’ ∞ , 9 ] , ( βˆ’ ∞ , 9 )
  • E 𝑔 β€² ( π‘₯ ) = βˆ’ 4 √ βˆ’ π‘₯ + 9 , ( βˆ’ ∞ , 9 ] , ( βˆ’ ∞ , 9 ]

Q10:

Differentiate 𝑓 ( π‘₯ ) = 2 π‘₯ + 9 , and identify the value of π‘₯ at which the function is not differentiable.

  • A 𝑓 β€² ( π‘₯ ) = 2 ( π‘₯ + 9 ) 2 , and the function is not differentiable at π‘₯ = βˆ’ 9 .
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ 2 ( π‘₯ + 9 ) 2 , and the function is not differentiable at π‘₯ = 9 .
  • C 𝑓 β€² ( π‘₯ ) = 2 ( π‘₯ + 9 ) 2 , and the function is not differentiable at π‘₯ = 9 .
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ 2 ( π‘₯ + 9 ) 2 , and the function is not differentiable at π‘₯ = βˆ’ 9 .