To differentiate $y = \ln(7x)$, we'll use the chain rule. Here's the step-by-step process:

1. Recognize the outer function: The outer function is $\ln(u)$, where $u = 7x$.

2. Apply the derivative of the natural logarithm: The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$.

3. Differentiate the inner function: The derivative of $u = 7x$ with respect to $x$ is $7$.

Now applying the chain rule:

$\frac{dy}{dx} = \frac{1}{7x} \times 7$

Simplifying:

$\frac{dy}{dx} = \frac{7}{7x} = \frac{1}{x}$

So, the derivative of $y = \ln(7x)$ is:

$\frac{dy}{dx} = \frac{1}{x}$

Would you like more details or have any questions? Here are some related questions to consider:

1. What happens if the argument of the logarithmic function is a more complicated expression, such as $\ln(3x^2 + 1)$?
2. How does the chain rule apply to the differentiation of composite functions involving exponential functions?
3. Can you apply the derivative of logarithmic functions to solve optimization problems?
4. How do you differentiate $y = \ln(f(x))$, where $f(x)$ is a more general function?
5. What are the key properties of the natural logarithm that are useful in differentiation?

Tip: When applying the chain rule, always ensure to differentiate both the outer and inner functions carefully to avoid missing crucial steps.