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
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 \( (-4, \infty) \).
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 \( (-4, \infty) \).
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