\frac{d}{dx}\left(\ln \left(x+4\right)\right)

To find the derivative of \(\ln(x+4)\) with respect to \(x\), we can use the chain rule. The chain rule states that if we have a composite function \(f(g(x))\), then the derivative \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

Here, \(f(u) = \ln(u)\) and \(u = x+4\). Let's apply the chain rule:

1. Find the derivative of the outer function \(f(u) = \ln(u)\) with respect to \(u\):
   \[
   \frac{d}{du} \ln(u) = \frac{1}{u}
   \]

2. Find the derivative of the inner function \(u = x+4\) with respect to \(x\):
   \[
   \frac{d}{dx} (x+4) = 1
   \]

3. Apply the chain rule:
   \[
   \frac{d}{dx} \left(\ln(x+4)\right) = \frac{d}{du} \ln(u) \cdot \frac{d}{dx} (x+4)
   \]
   Substituting \(u = x+4\), we get:
   \[
   \frac{d}{dx} \left(\ln(x+4)\right) = \frac{1}{x+4} \cdot 1 = \frac{1}{x+4}
   \]

Therefore, the derivative is:
\[
\frac{d}{dx}\left(\ln(x+4)\right) = \frac{1}{x+4}
\]

Would you like more details or have any questions? Here are five questions you might consider next:

1. How do you differentiate a function with a more complex inner function?
2. What is the chain rule and how is it applied in different scenarios?
3. How do you find the second derivative of \(\ln(x+4)\)?
4. Can you explain the derivative of the natural logarithm function \(\ln(x)\)?
5. How would you integrate \(\frac{1}{x+4}\)?

**Tip:** Remember that the chain rule is a powerful tool for differentiating composite functions, allowing you to break down complex derivatives into simpler parts.

Learn how to compute the derivative of ln(x+4) using the chain rule with a step-by-step explanation. This problem covers differentiation techniques, specifically the chain rule and derivative of the natural logarithm. Suitable for advanced high school students and calculus learners.

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