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Worksheet: An Introduction to the Definition of the Derivative

In this worksheet, we will practice calculating the derivative of a polynomial function using the formal definition of the derivative as a limiting process.

Q1:

Let . Use the definition of derivative to determine . What is the gradient of the tangent to its graph at ?

  • A , the gradient of the tangent at point
  • B , the gradient of the tangent at point
  • C , the gradient of the tangent at point
  • D , the gradient of the tangent at point
  • E , the gradient of the tangent at point

Q2:

Let . Use the definition of derivative to determine . What is the gradient of the tangent to its graph at ?

  • A , the gradient of the tangent at point
  • B , the gradient of the tangent at point
  • C , the gradient of the tangent at point
  • D , the gradient of the tangent at point
  • E , the gradient of the tangent at point

Q3:

Let . Use the definition of derivative to determine . What is the gradient of the tangent to its graph at ?

  • A , the gradient of the tangent at point
  • B , the gradient of the tangent at point
  • C , the gradient of the tangent at point
  • D , the gradient of the tangent at point

Q4:

Let . Use the definition of derivative to determine . What is the gradient of the tangent to its graph at ?

  • A , the gradient of the tangent at point
  • B , the gradient of the tangent at point
  • C , the gradient of the tangent at point
  • D , the gradient of the tangent at point

Q5:

Find the derivative of the function 𝑓 ( π‘₯ ) = 6 π‘₯ βˆ’ 7 π‘₯ 3 2 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑓 β€² ( π‘₯ ) = 6 π‘₯ βˆ’ 7 π‘₯ 2 , ( 0 , ∞ ) , ℝ
  • B 𝑓 β€² ( π‘₯ ) = 1 8 π‘₯ βˆ’ 1 4 π‘₯ 3 2 , ℝ , ( 0 , ∞ )
  • C 𝑓 β€² ( π‘₯ ) = 1 8 π‘₯ βˆ’ 1 4 2 , ( 0 , ∞ ) , ( 0 , ∞ )
  • D 𝑓 β€² ( π‘₯ ) = 1 8 π‘₯ βˆ’ 1 4 π‘₯ 2 , ℝ , ℝ
  • E 𝑓 β€² ( π‘₯ ) = 1 8 π‘₯ βˆ’ 1 4 π‘₯ 3 , ℝ , ℝ

Q6:

Find the derivative of the function 𝑓 ( π‘₯ ) = βˆ’ 8 π‘₯ + 9 π‘₯ 3 2 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ + 9 π‘₯ 2 , ( 0 , ∞ ) , ℝ
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 8 π‘₯ 3 2 , ℝ , ( 0 , ∞ )
  • C 𝑓 β€² ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 8 2 , ( 0 , ∞ ) , ( 0 , ∞ )
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 8 π‘₯ 2 , ℝ , ℝ
  • E 𝑓 β€² ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 8 π‘₯ 3 , ℝ , ℝ

Q7:

Find the derivative of the function 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 3 π‘₯ 3 2 using the definition of derivative, and then state the domain of the function and the domain of its derivative.

  • A 𝑓 β€² ( π‘₯ ) = 2 π‘₯ βˆ’ 3 π‘₯ 2 , ( 0 , ∞ ) , ℝ
  • B 𝑓 β€² ( π‘₯ ) = 6 π‘₯ βˆ’ 6 π‘₯ 3 2 , ℝ , ( 0 , ∞ )
  • C 𝑓 β€² ( π‘₯ ) = 6 π‘₯ βˆ’ 6 2 , ( 0 , ∞ ) , ( 0 , ∞ )
  • D 𝑓 β€² ( π‘₯ ) = 6 π‘₯ βˆ’ 6 π‘₯ 2 , ℝ , ℝ
  • E 𝑓 β€² ( π‘₯ ) = 6 π‘₯ βˆ’ 6 π‘₯ 3 , ℝ , ℝ

Q8:

Let . Use the definition of derivative to determine . What is the gradient of the tangent to its graph at ?

  • A , the gradient of the tangent at point
  • B , the gradient of the tangent at point
  • C , the gradient of the tangent at point
  • D , the gradient of the tangent at point

Q9:

Let 𝑓 ( π‘₯ ) = βˆ’ 6 √ π‘₯ βˆ’ 6 . Use the definition of the derivative to determine 𝑓 β€² ( π‘₯ ) .

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

Q10:

Determine the derivative of the function 𝑓 ( π‘₯ ) = √ 2 π‘₯ βˆ’ 1 6 using the definition of the derivative.

  • A 𝑓 β€² ( π‘₯ ) = 1 2 √ 2 π‘₯ βˆ’ 1 6
  • B 𝑓 β€² ( π‘₯ ) = 2 √ 2 π‘₯ βˆ’ 1 6
  • C 𝑓 β€² ( π‘₯ ) = 2 π‘₯ βˆ’ 1 6
  • D 𝑓 β€² ( π‘₯ ) = 1 √ 2 π‘₯ βˆ’ 1 6

Q11:

Evaluate l i m β„Ž β†’ 0 𝑓 ( β„Ž + 1 2 ) βˆ’ 𝑓 ( β„Ž βˆ’ 1 7 ) + 𝑓 ( βˆ’ 1 7 ) βˆ’ 𝑓 ( 1 2 ) β„Ž .

  • A 𝑓 β€² ( 1 2 )
  • B 𝑓 β€² ( 1 2 ) + 𝑓 β€² ( βˆ’ 1 7 )
  • C 𝑓 β€² ( βˆ’ 1 7 ) βˆ’ 𝑓 β€² ( 1 2 )
  • D 𝑓 β€² ( 1 2 ) βˆ’ 𝑓 β€² ( βˆ’ 1 7 )
  • E 𝑓 β€² ( βˆ’ 1 7 )

Q12:

Evaluate l i m β„Ž β†’ 0 𝑓 ( β„Ž + 4 ) βˆ’ 𝑓 ( β„Ž βˆ’ 2 ) + 𝑓 ( βˆ’ 2 ) βˆ’ 𝑓 ( 4 ) β„Ž .

  • A 𝑓 β€² ( 4 )
  • B 𝑓 β€² ( 4 ) + 𝑓 β€² ( βˆ’ 2 )
  • C 𝑓 β€² ( βˆ’ 2 ) βˆ’ 𝑓 β€² ( 4 )
  • D 𝑓 β€² ( 4 ) βˆ’ 𝑓 β€² ( βˆ’ 2 )
  • E 𝑓 β€² ( βˆ’ 2 )

Q13:

Given a function with 𝑓 ( βˆ’ 3 ) = 7 and 𝑓 β€² ( βˆ’ 3 ) = 3 , what is l i m β„Ž β†’ 0 5 β„Ž 𝑓 ( β„Ž βˆ’ 3 ) βˆ’ 7 ?

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

Q14:

Consider a function with 𝑓 ( βˆ’ 8 ) = 3 and 𝑓 β€² ( βˆ’ 8 ) = 7 . What is l i m π‘₯ β†’ βˆ’ 8 𝑓 ( π‘₯ ) ?

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