The problem asks to find the derivative $\frac{dy}{dx}$, where $y = \ln_e(\ln_e x)$, which can be written as $y = \ln(\ln x)$.

To differentiate $y = \ln(\ln x)$ with respect to $x$, we apply the chain rule. Here's how to proceed:

1. First, differentiate the outer function $\ln u$ with respect to its argument $u = \ln x$: $\frac{d}{du} \ln(u) = \frac{1}{u}$ So, $\frac{dy}{du} = \frac{1}{\ln x}$.

2. Then, differentiate the inner function $\ln x$ with respect to $x$: $\frac{d}{dx} \ln x = \frac{1}{x}$

3. Now, applying the chain rule: $\frac{dy}{dx} = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x}$

Thus, the derivative is: $\frac{dy}{dx} = \frac{1}{x \ln x}$

Would you like more details or have any further questions?

Here are some related questions:

1. What is the derivative of $\ln(\ln(\ln x))$?
2. How does the chain rule work for composite functions?
3. What is the derivative of $\ln x$ alone?
4. How would you differentiate $y = \ln(x^2)$?
5. What are some common applications of the natural logarithm function in calculus?

Tip: Remember to apply the chain rule when differentiating nested functions!