Differentiate the function.

*F*(*s*) = ln(ln(5*s*))

To differentiate the function \( F(s) = \ln(\ln(5s)) \), we'll apply the chain rule twice.

1. Let \( u = \ln(5s) \), then \( F(s) = \ln(u) \).
2. Differentiate \( F(s) \) with respect to \( u \):
   \[
   \frac{dF}{du} = \frac{1}{u}
   \]
3. Now differentiate \( u = \ln(5s) \) with respect to \( s \):
   \[
   \frac{du}{ds} = \frac{d}{ds} \ln(5s) = \frac{1}{5s} \cdot 5 = \frac{1}{s}
   \]
4. Finally, apply the chain rule to differentiate \( F(s) \) with respect to \( s \):
   \[
   \frac{dF}{ds} = \frac{dF}{du} \cdot \frac{du}{ds} = \frac{1}{\ln(5s)} \cdot \frac{1}{s} = \frac{1}{s \ln(5s)}
   \]

Therefore, the derivative of \( F(s) = \ln(\ln(5s)) \) is:

\[
\frac{dF}{ds} = \frac{1}{s \ln(5s)}
\]

Would you like details or have any questions about this differentiation?

Here are 8 related questions to expand your understanding:

1. How do you differentiate \( F(s) = \ln(s) \)?
2. What is the derivative of \( F(s) = \ln(5s) \)?
3. How does the chain rule work in differentiation?
4. What is the derivative of \( F(s) = \ln(\ln(s)) \)?
5. Can you differentiate \( F(s) = \ln(\ln(s^2 + 1)) \)?
6. What is the product rule in differentiation?
7. How do you differentiate \( F(s) = \ln(5) \)?
8. What is the second derivative of \( F(s) = \ln(\ln(5s)) \)?

**Tip:** Always simplify the inner function first when using the chain rule to make differentiation easier.

Learn how to differentiate the function F(s) = ln(ln(5s)) step-by-step using the chain rule. This problem involves differentiation of nested natural logarithms and is suitable for undergraduate-level mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation