1. The problem is to find the solution set of the inequality $\frac{d}{dx} \ln(\ln x) > 0$ without directly using the derivative. 2. First, recall the function: $y = \ln(\ln x)$. 3. The domain of $y$ requires $\ln x > 0$, which means $x > 1$. 4. Since the derivative is $\frac{d}{dx} \ln(\ln x) = \frac{1}{\ln x} \cdot \frac{1}{x}$, the inequality $\frac{d}{dx} \ln(\ln x) > 0$ simplifies to $\frac{1}{x \ln x} > 0$. 5. To solve $\frac{1}{x \ln x} > 0$, analyze the sign of the denominator $x \ln x$. 6. Since $x > 1$ (domain), $x$ is positive. 7. Therefore, the sign depends on $\ln x$. 8. For $x > 1$, $\ln x > 0$, so $x \ln x > 0$. 9. Hence, $\frac{1}{x \ln x} > 0$ for all $x > 1$. 10. The solution set of the inequality is $\boxed{(1, \infty)}$. This means the function $\ln(\ln x)$ is increasing for all $x$ greater than 1.