To find the derivative of the function \( f(x) = \frac{1}{\ln(x)} \), we can use the chain rule and quotient rule.

### Steps:

1. Write the function in a more convenient form:  
   \[
   f(x) = (\ln(x))^{-1}
   \]
   
2. Apply the chain rule:  
   The derivative of \( (\ln(x))^{-1} \) is:
   \[
   f'(x) = -(\ln(x))^{-2} \cdot \frac{d}{dx}[\ln(x)]
   \]
   
3. The derivative of \( \ln(x) \) is \( \frac{1}{x} \), so:
   \[
   f'(x) = -\frac{1}{x} \cdot (\ln(x))^{-2}
   \]

4. Simplify the expression:
   \[
   f'(x) = -\frac{1}{x (\ln(x))^2}
   \]

### Final Answer:
\[
f'(x) = -\frac{1}{x (\ln(x))^2}
\]

Would you like more details on this derivation, or do you have any questions?

### Related Questions:
1. What is the derivative of \( \ln(x^2 + 1) \)?
2. How do you differentiate \( x \ln(x) \)?
3. What is the second derivative of \( \frac{1}{\ln(x)} \)?
4. Can you integrate \( \frac{1}{\ln(x)} \)?
5. How would you differentiate \( \ln(\sin(x)) \)?

### Tip:
When dealing with derivatives of logarithmic functions, always check if the chain rule applies for nested functions.

Find the derivative of 1/ln(x) using the chain rule and quotient rule. This detailed solution explains the steps, including key logarithmic function concepts and differentiation techniques. Perfect for college calculus or advanced high school students.

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