To differentiate $f(x) = \ln(8x)$, we will apply the chain rule.

Step-by-step differentiation:

1. Recall that the derivative of $\ln(u)$ is $\frac{1}{u}$, where $u$ is a function of $x$.

2. For $f(x) = \ln(8x)$, $u = 8x$. Now, the derivative of $u$ with respect to $x$ is: $\frac{d}{dx}(8x) = 8$

3. Applying the chain rule, the derivative of $f(x)$ is: $f'(x) = \frac{1}{8x} \cdot 8$

4. Simplifying: $f'(x) = \frac{8}{8x} = \frac{1}{x}$

Thus, the derivative of $f(x) = \ln(8x)$ is: $f'(x) = \frac{1}{x}$

Would you like further details or clarification on this process?

Here are 5 related questions to explore further:

1. How do you differentiate other logarithmic functions like $\ln(ax + b)$?
2. How do you apply the chain rule to more complex functions?
3. What is the derivative of exponential functions like $e^{x}$?
4. Can we differentiate logarithms with different bases, such as $\log_b(x)$?
5. How does the product rule interact with logarithmic differentiation?

Tip: When differentiating logarithmic functions, always apply the chain rule if the argument of the logarithm is more complex than just $x$.