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.