Derivative Of A Function

Approved & Edited by ProProfs Editorial Team

The editorial team at ProProfs Quizzes consists of a select group of subject experts, trivia writers, and quiz masters who have authored over 10,000 quizzes taken by more than 100 million users. This team includes our in-house seasoned quiz moderators and subject matter experts. Our editorial experts, spread across the world, are rigorously trained using our comprehensive guidelines to ensure that you receive the highest quality quizzes.

Learn about Our Editorial Process

| By I-heng_mccomb

I-heng_mccomb

Community Contributor

Quizzes Created: 5 | Total Attempts: 2,602

Questions: 8 | Attempts: 423 | Updated: Mar 21, 2023

Questions

Settings

Feedback

During the Quiz End of Quiz

Difficulty

Sequential Easy First Hard First

Play as

Quiz Flashcard

Create your own Quiz

Share on Facebook Share on Twitter Share on Whatsapp Share on Pinterest Share on Email Copy to Clipboard Embed on your website

. How are a function and its derivative related? When does the derivative not exist?

Questions and Answers

1.

Check all of the options which represent the derivative of f(x) = x⁴.
- A.
  
  Option 1
- B.
  
  Option 2
- C.
  
  Option 3
- D.
  
  Option 4
- E.
  
  Option 5
- F.
  
  Option 6
- G.
  
  Option 7
Correct Answer(s)
C. Option 3
D. Option 4

Explanation
The derivative of f(x) = x^4 is calculated by applying the power rule. The power rule states that if f(x) = x^n, then the derivative of f(x) is f'(x) = nx^(n-1). In this case, n = 4, so the derivative of f(x) = x^4 is f'(x) = 4x^(4-1) = 4x^3. Therefore, Option 3 (4x^3) and Option 4 (4x^3) represent the correct derivatives of f(x) = x^4.

Rate this question:
2.

Estimate G'(7) from the graph of G(x).
- A.
  
  0.2
- B.
  
  0.5
- C.
  
  1
- D.
  
  2
- E.
  
  5
Correct Answer
A. 0.2

Explanation
The graph of G(x) shows that the value of G(x) is constant at 0.2 for all values of x. Therefore, the derivative G'(x) is equal to 0 for all values of x, including x = 7. Hence, the estimate for G'(7) is 0.

Rate this question:
3.

At which values of x does the derivative of the graphed function not exist?
- A.
  
  X = –6
- B.
  
  X = –4
- C.
  
  X = –2
- D.
  
  X = –1
- E.
  
  X = 0
- F.
  
  X = 2
- G.
  
  X = 3
- H.
  
  X = 4
Correct Answer(s)
C. X = –2
E. X = 0
F. X = 2
G. X = 3

Explanation
The derivative of a function does not exist at points where there is a sharp change in the slope of the graph or where the graph has a vertical tangent line. In this case, looking at the given values of x, we can see that x = -2, x = 0, x = 2, and x = 3 are the points where the derivative of the graphed function does not exist. At these points, there are either sharp changes in slope or vertical tangent lines.

Rate this question:
4.

STATEMENT: All continuous functions are differentiable.
- A.
  
  True
- B.
  
  False
Correct Answer
B. False

Explanation
The statement "All continuous functions are differentiable" is false. While it is true that all differentiable functions are continuous, the reverse is not always true. There are continuous functions that are not differentiable at certain points or on certain intervals. A classic example is the absolute value function, which is continuous but not differentiable at x = 0. Therefore, the correct answer is false.

Rate this question:
5.

All differentiable functions are continuous.
- A.
  
  True
- B.
  
  False
Correct Answer
A. True

Explanation
This statement is true because all differentiable functions are continuous. Differentiability implies continuity, meaning that if a function is differentiable at a point, it must also be continuous at that point. This is because the definition of differentiability includes the requirement that the function has a well-defined derivative at every point, and having a derivative implies that the function is continuous. Therefore, all differentiable functions are continuous.

Rate this question:
6.

If a function has a discontinuity at x = a, then the function does not have a derivative at x = a.
- A.
  
  True
- B.
  
  False
Correct Answer
A. True

Explanation
If a function has a discontinuity at x = a, it means that the function is not continuous at that point. The derivative of a function measures the rate of change of the function at a particular point. Since a function is not continuous at x = a, it means that there is a sudden jump or break in the function's values at that point. This sudden jump or break makes it impossible to calculate the derivative at x = a, as the concept of rate of change does not apply when there is a discontinuity. Therefore, the statement that the function does not have a derivative at x = a is true.

Rate this question:
7.

The derivative does not exist at a cusp, but it does exist at a corner.
- A.
  
  True
- B.
  
  False
Correct Answer
B. False

Explanation
The derivative does not exist at a cusp because at a cusp, the function has a sharp point where the tangent line cannot be defined. On the other hand, the derivative does exist at a corner because at a corner, the function has a well-defined tangent line on either side of the corner. Therefore, the statement that the derivative does not exist at a cusp but does exist at a corner is false.

Rate this question:
8.

You have practiced applying algebraic techniques to find the derivative of functions from the definition of derivative (limit of a difference quotient).
- A.
  
  True
- B.
  
  False
Correct Answer
A. True

Explanation
The statement is true because when applying algebraic techniques to find the derivative of a function from the definition of derivative, one uses the limit of a difference quotient. This involves taking the limit as the difference in x-values approaches zero, which gives the instantaneous rate of change of the function at a specific point. Therefore, the statement is correct.

Rate this question:

Quiz Review Timeline +

Our quizzes are rigorously reviewed, monitored and continuously updated by our expert board to maintain accuracy, relevance, and timeliness.

Current Version
Mar 21, 2023

Quiz Edited by
ProProfs Editorial Team
Sep 27, 2020

Quiz Created by
I-heng_mccomb

Derivative Of A Function

Check all of the options which represent the derivative of f(x) = x4.

Estimate G'(7) from the graph of G(x).

At which values of x does the derivative of the graphed function not exist?

STATEMENT: All continuous functions are differentiable.

All differentiable functions are continuous.

If a function has a discontinuity at x = a, then the function does not have a derivative at x = a.

The derivative does not exist at a cusp, but it does exist at a corner.

You have practiced applying algebraic techniques to find the derivative of functions from the definition of derivative (limit of a difference quotient).

Check all of the options which represent the derivative of f(x) = x⁴.