Lesson: Definition of the Derivative

In this lesson, we will learn how to calculate the derivative of a polynomial function using the formal definition of the derivative as a limiting process.

Sample Question Videos

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Worksheet: Definition of the Derivative • 14 Questions • 3 Videos

Q1:

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

Q3:

Using the definition of a derivative, evaluate d d 𝑥 1 𝑥 + 1 .

Q4:

Let If 𝑓 is differentiable at 𝑥 = 4 , find 𝑎 and 𝑏 .

Q5:

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.

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

Find d d 𝑦 𝑥 if 𝑦 = 6 𝑥 7 .

Q11:

Differentiate 𝑓 ( 𝑥 ) = 1 𝑥 , and identify the value of 𝑥 at which the function is not differentiable.

Q12:

Differentiate 𝑓 ( 𝑥 ) = 7 8 𝑥 + 5 , and identify the value of 𝑥 at which the function is not differentiable.

Q13:

Suppose What can be said of the differentiability of 𝑓 at 𝑥 = 1 ?

Q14:

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.

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