Question

Explain what is wrong with the statement. The only function that has derivative $2 x$ is $x^{2}$.

Name: Problem 103, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-15T19:42:49-08:00
Duration: 1 min 54 s
Channel: Gregory Higby
Description: Explain what is wrong with the statement. The only function that has derivative 2 x is x^2.

   Explain what is wrong with the statement.
The only function that has derivative $2 x$ is $x^{2}$.

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 103

Instant Answer

Solved by Expert Gregory Higby

04/15/2021

Step 1

The derivative of this function is indeed $2x$. This is calculated as follows: $$ \begin{aligned} y &=x^{2} \\ y^{\prime} &=2 x \end{aligned} $$ Show more…

Show all steps

Thanks for your feedback!

Explain what is wrong with the statement. The only function that has derivative $2 x$ is $x^{2}$.

Play audio

Feedback

Gregory Higby and 53 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Constant of Integration

When computing an antiderivative, the process of integration introduces an arbitrary constant, known as the constant of integration. This constant represents the family of all functions that differ only by a constant and that all have the same derivative. The error in the statement arises from ignoring this constant, implying incorrectly that x² is uniquely defined as the antiderivative of 2x.

Antiderivative

An antiderivative of a function is any function whose derivative is the given function. In the context of the statement, saying that x² is the only function with derivative 2x ignores the fact that any function of the form x² + C, where C is an arbitrary constant, has the same derivative. This fundamental concept in calculus shows that antiderivatives are determined only up to an additive constant.

Please give Ace some feedback