# Lesson Worksheet: Definition of the Derivative Mathematics • Higher Education

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

Q1:

Let . Use the definition of the derivative to determine .

• A
• B
• C
• D

Q2:

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

• A, the slope of the tangent at is
• B, the slope of the tangent at is
• C, the slope of the tangent at is
• D, the slope of the tangent at is
• E, the slope of the tangent at is

Q3:

Using the definition of a derivative, evaluate .

• A
• B
• C
• D

Q4:

Find the derivative of the function using the definition of derivative, and then state the domain of the function and the domain of its derivative.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q5:

Determine the derivative of the function using the definition of the derivative.

• A
• B
• C
• D

Q6:

Evaluate .

• A
• B
• C
• D
• E

Q7:

Given a function with and , what is ?

• A
• B15
• C
• D0
• E3

Q8:

Consider a function with and . What is ?

Q9:

Find the derivative of the function using the definition of derivative, and then state the domain of the function and the domain of its derivative.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q10:

Let . Use the definition of derivative to determine .

• A
• B
• C
• D

Q11:

Find the derivative of the function using the definition of derivative, and then state the domain of the function and the domain of its derivative.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q12:

Consider the function .

Find .

Find .

What can you conclude about the derivative of at ?

• ASince the right and left limits are unequal, the derivative does not exist.
• BThe derivative exists and is equal to .
• CThe derivative exists and is equal to 1.
• DThe derivative exists and is equal to 0.

Q13:

Find the derivative of at the point from first principles.

Q14:

Find the derivative of the function using the definition of the derivative.

• A
• B
• C
• D
• E

Q15:

Find the derivative of the function using the definition of the derivative.

• A
• B
• C
• D
• E

Q16:

Find the derivative of the function using the definition of the derivative.

• A
• B
• C
• D
• E

Find the equation of the tangent to the graph of at .

• A
• B
• C
• D
• E

Q17:

Find the derivative of the function using the definition of the derivative.

• A
• B
• C
• D
• E

Q18:

Find the derivative of the function using the definition of the derivative.

• A
• B
• C
• D
• E

Q19:

Suppose . If the average rate of change as changes from 3 to 3.5 is 4 and , determine and .

• A,
• B,
• C,
• D,

Q20:

What is the rate of change of with respect to at ?

Q21:

Consider the function .

Find and .

What can you conclude about the derivative of at ?

• AThe derivative exists and is equal to 2.
• BSince the right and left limits are unequal, the derivative does not exist.
• CThe derivative exists and is equal to .
• DThe derivative exists and is equal to .
• EThe derivative exists and is equal to .

Q22:

What is the slope of the tangent to at ?

• A
• B
• C
• D
• ENone of the above

Q23:

Find .

• Aundefined
• B
• C
• D
• E

Q24:

If the function , find .

• A
• B
• C
• D
• E

Q25:

If , find .