The problem asks you to differentiate the function $F(s) = \ln(\ln(2s))$.

Solution:

To differentiate $F(s) = \ln(\ln(2s))$, we will apply the chain rule multiple times.

1. Start by differentiating the outer function, which is $\ln(x)$, where $x = \ln(2s)$. The derivative of $\ln(x)$ is $\frac{1}{x}$, so we get: $F'(s) = \frac{1}{\ln(2s)} \cdot \frac{d}{ds} [\ln(2s)]$

2. Now differentiate the inner function $\ln(2s)$. The derivative of $\ln(2s)$ is: [ \frac{d}{ds} [\ln(2s)] = \frac{1}{2s} \cdot \frac{d}{ds} [2s] = \frac{1}{2s} \cdot 2 = \frac{1}{s} ]

3. Combine the two results: $F'(s) = \frac{1}{\ln(2s)} \cdot \frac{1}{s}$

Thus, the derivative is: $F'(s) = \frac{1}{s \ln(2s)}$

Final answer:

$F'(s) = \frac{1}{s \ln(2s)}$

Would you like further details or have any questions on this differentiation?

Here are 5 related questions to expand your understanding:

1. What is the chain rule, and how does it apply to nested logarithmic functions?
2. How do you differentiate functions involving logarithms of products, such as $\ln(2s)$?
3. Can you differentiate a function like $\ln(\ln(\ln(s)))$ using similar methods?
4. What is the general approach to differentiating logarithmic functions with different bases?
5. How do you find the second derivative of $F(s) = \ln(\ln(2s))$?

Tip: Always remember to apply the chain rule carefully when dealing with composite functions like nested logarithms.