1. **State the problem:** We need to find the solution set for the inequality $$\frac{d}{dx} \ln(\ln x) > 0$$. 2. **Find the derivative:** Using the chain rule, the derivative of $$\ln(\ln x)$$ is $$\frac{d}{dx} \ln(\ln x) = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x}.$$ 3. **Set the inequality:** We want $$\frac{1}{x \ln x} > 0.$$ 4. **Analyze the inequality:** The fraction is positive when the denominator is positive (since numerator 1 is always positive). 5. **Determine the domain:** For $$\ln(\ln x)$$ to be defined, we need $$\ln x > 0 \Rightarrow x > 1$$. 6. **Sign of denominator:** Since $$x > 1$$, $$x$$ is positive. So the sign depends on $$\ln x$$. 7. **Inequality for $$\ln x$$:** We want $$\ln x > 0$$, which means $$x > 1$$. 8. **Combine domain and inequality:** The derivative is positive when $$x > 1$$. 9. **Final solution set:** $$\boxed{(1, \infty)}$$. This means $$\ln(\ln x)$$ is increasing for all $$x$$ greater than 1.