To find the derivative of \( f(x) = \ln(\ln(\ln(x))) \), we apply the chain rule multiple times:
1. **Outermost layer**: The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u’ \). Here, \( u = \ln(\ln(x)) \), so:
\[
f'(x) = \frac{1}{\ln(\ln(x))} \cdot \frac{d}{dx}[\ln(\ln(x))]
\]
2. **Middle layer**: The derivative of \( \ln(v) \) is \( \frac{1}{v} \cdot v’ \). Here, \( v = \ln(x) \), so:
\[
\frac{d}{dx}[\ln(\ln(x))] = \frac{1}{\ln(x)} \cdot \frac{d}{dx}[\ln(x)]
\]
3. **Innermost layer**: The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
**Combining all layers**:
\[
f'(x) = \frac{1}{\ln(\ln(x))} \cdot \frac{1}{\ln(x)} \cdot \frac{1}{x}
\]
**Final Answer**:
\[
\boxed{\frac{1}{x \ln(x) \ln(\ln(x))}}
\]
