# Lesson Worksheet: The Differentiability of a Function Mathematics • Higher Education

In this worksheet, we will practice determining whether a function is differentiable and identifying the relation between a function’s differentiability and its continuity.

Q1:

True or False: If a function is continuous at a point, then it must be differentiable at that point.

• ATrue
• BFalse

Q2:

Suppose What can be said of the differentiability of at ?

• AThe function is differentiable at as but is not continuous.
• BThe function is not differentiable at because is undefined.
• CThe function is not differentiable at .
• DThe function is not differentiable at because is continuous at .
• EThe function is continuous but not differentiable at because .

Q3:

The figure shows the graph of . What can be said of the differentiability of at ? • AThe function is not differentiable at because it is undefined at that point.
• BThe function is differentiable at because the function’s rate of change is the same on both sides of that point.
• CThe function is not differentiable at because the function’s rate of change is different on both sides of that point.
• DThe function is differentiable at because it is continuous at that point.

Q4:

Consider the function .

What is the domain of ?

• A
• B
• C
• D
• E

Find an expression for the derivative of .

• A
• B
• C
• D
• E

What is the domain for the derivative ?

• A
• B
• C
• D
• E

Q5:

The figure shows the graph of . At which points is the derivative of the function not defined? • A, , and
• B and
• C
• D
• E

Q6:

Suppose What can be said of the differentiability of at ?

• AThe function is continuous but not differentiable at because .
• BThe function is not differentiable at because is discontinuous at .
• CThe function is differentiable at .
• DThe function is differentiable at as but is not continuous.
• EThe function is not differentiable at because is undefined.

Q7:

Consider a function with and . What is ?

Q8:

Given that is a continuous function, find the value of . What can be said of the differentiability of at ?

• A, is differentiable at
• B, is differentiable at
• C, is not differentiable at
• D, is not differentiable at

Q9:

Find the values of and and discuss the differentiability of the function at given is continuous and

• A, , and is not differentiable at .
• B, , and is not differentiable at .
• C, , and is differentiable at .
• D, , and is differentiable at .
• E, , and is not differentiable at .

Q10:

Is the function differentiable at ?

• AYes
• BNo

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