an anti derivative of 15 F(X) = (x²) F(x) = (x²) Explain why or why not

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
**Problem Statement:**

Is \( F(x) = e^{(x^2)} \) an antiderivative of \( f(x) = e^{(x^2)} \)?

**Task:**

Explain why or why not.

**Explanation:**

To determine if \( F(x) = e^{x^2} \) is an antiderivative of \( f(x) = e^{x^2} \), we need to check if the derivative of \( F(x) \) results in \( f(x) \).

1. **Derivative of \( F(x) = e^{x^2} \):**

   To differentiate \( F(x) \), use the chain rule:

   \[
   \frac{d}{dx} e^{x^2} = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot 2x = 2xe^{x^2}
   \]

2. **Comparison with \( f(x) \):**

   The derivative, \( 2xe^{x^2} \), is not equal to \( f(x) = e^{x^2} \).

**Conclusion:**

Therefore, \( F(x) = e^{x^2} \) is not an antiderivative of \( f(x) = e^{x^2} \).
Transcribed Image Text:**Problem Statement:** Is \( F(x) = e^{(x^2)} \) an antiderivative of \( f(x) = e^{(x^2)} \)? **Task:** Explain why or why not. **Explanation:** To determine if \( F(x) = e^{x^2} \) is an antiderivative of \( f(x) = e^{x^2} \), we need to check if the derivative of \( F(x) \) results in \( f(x) \). 1. **Derivative of \( F(x) = e^{x^2} \):** To differentiate \( F(x) \), use the chain rule: \[ \frac{d}{dx} e^{x^2} = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot 2x = 2xe^{x^2} \] 2. **Comparison with \( f(x) \):** The derivative, \( 2xe^{x^2} \), is not equal to \( f(x) = e^{x^2} \). **Conclusion:** Therefore, \( F(x) = e^{x^2} \) is not an antiderivative of \( f(x) = e^{x^2} \).
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,