To analyze the function \( y = \ln(\ln(x)) \), let's go through a few key aspects:

### 1. **Domain:**
The function \( \ln(x) \) is only defined for \( x > 0 \). Additionally, for \( y = \ln(\ln(x)) \) to be defined, the inner function \( \ln(x) \) must also be greater than 0:

\[
\ln(x) > 0 \implies x > 1
\]

Thus, the domain of \( y = \ln(\ln(x)) \) is \( x > 1 \).

### 2. **Range:**
Since the natural logarithm function \( \ln(x) \) can take on any positive value for \( x > 1 \), the function \( \ln(\ln(x)) \) can take any real value. Hence, the range of \( y = \ln(\ln(x)) \) is:

\[
(-\infty, +\infty)
\]

### 3. **Derivative:**
To find the derivative of \( y = \ln(\ln(x)) \), we use the chain rule:

\[
\frac{dy}{dx} = \frac{d}{dx} [\ln(\ln(x))] = \frac{1}{\ln(x)} \cdot \frac{d}{dx} [\ln(x)] = \frac{1}{\ln(x)} \cdot \frac{1}{x} = \frac{1}{x \ln(x)}
\]

So, the derivative of the function is:

\[
\frac{dy}{dx} = \frac{1}{x \ln(x)}
\]

### 4. **Critical Points:**
To find critical points, we set the derivative to zero:

\[
\frac{1}{x \ln(x)} = 0
\]

Since the expression \( \frac{1}{x \ln(x)} \) is never zero for \( x > 1 \), there are no critical points where the derivative is zero. However, we need to check the behavior at the boundary of the domain, \( x \to 1^+ \).

### 5. **Behavior at the Boundary:**
As \( x \to 1^+ \), we have:

\[
\ln(x) \to 0^+ \implies \ln(\ln(x)) \to -\infty
\]

Thus, the function decreases to negative infinity as \( x \) approaches 1 from the right.

### Conclusion:
- **Domain:** \( x > 1 \)
- **Range:** \( (-\infty, +\infty) \)
- **Derivative:** \( \frac{dy}{dx} = \frac{1}{x \ln(x)} \)

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. What is the second derivative of \( y = \ln(\ln(x)) \)?
2. How do you find the concavity and inflection points of \( y = \ln(\ln(x)) \)?
3. How would you plot the function \( y = \ln(\ln(x)) \)?
4. How does the function behave as \( x \to \infty \)?
5. How can you apply logarithmic differentiation to more complex functions?

**Tip:** Remember that logarithmic functions have unique properties when it comes to growth rates and asymptotic behavior!

Explore the function y = ln(ln(x)), including its domain, range, and first derivative. This detailed analysis covers key concepts in calculus, such as logarithmic functions and the chain rule, suitable for students in Grades 11-12.

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