Find the derivative and state the domain of f(x) = In(x + 4) %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the derivative and state the domain.
![**Question:**
Find the derivative and state the domain of \( f(x) = \ln(x + 4) \).
**Solution:**
To find the derivative of \( f(x) = \ln(x + 4) \), we apply the chain rule. The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Therefore, the derivative of \( \ln(x + 4) \) with respect to \( x \) is:
\[
f'(x) = \frac{1}{x + 4} \cdot (1) = \frac{1}{x + 4}
\]
**Domain:**
The natural logarithm function, \(\ln(u)\), is defined for \( u > 0 \). Therefore, for \( \ln(x + 4) \) to be defined, we need:
\[
x + 4 > 0
\]
Solving for \( x \), we find:
\[
x > -4
\]
Thus, the domain of \( f(x) = \ln(x + 4) \) is \( x > -4 \). In interval notation, the domain is \( (-4, \infty) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4077feb7-b26e-441f-89fa-ea823c6ca6c5%2Fa9af2aba-30da-4a40-a7b3-2e99df14d9bf%2F0of4uvr_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Find the derivative and state the domain of \( f(x) = \ln(x + 4) \).
**Solution:**
To find the derivative of \( f(x) = \ln(x + 4) \), we apply the chain rule. The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Therefore, the derivative of \( \ln(x + 4) \) with respect to \( x \) is:
\[
f'(x) = \frac{1}{x + 4} \cdot (1) = \frac{1}{x + 4}
\]
**Domain:**
The natural logarithm function, \(\ln(u)\), is defined for \( u > 0 \). Therefore, for \( \ln(x + 4) \) to be defined, we need:
\[
x + 4 > 0
\]
Solving for \( x \), we find:
\[
x > -4
\]
Thus, the domain of \( f(x) = \ln(x + 4) \) is \( x > -4 \). In interval notation, the domain is \( (-4, \infty) \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
