We are tasked with finding the derivative of the function \( f(x) = \ln\left(\frac{1}{x}\right) \) with respect to \( x \).

### Step-by-Step Solution:

1. **Rewrite the function:**
   \[
   f(x) = \ln\left(\frac{1}{x}\right)
   \]
   Use the logarithmic identity \( \ln\left(\frac{1}{x}\right) = -\ln(x) \):
   \[
   f(x) = -\ln(x)
   \]

2. **Differentiate using the chain rule:**
   Now, find the derivative of \( f(x) = -\ln(x) \) with respect to \( x \):
   \[
   \frac{d}{dx} \left( -\ln(x) \right) = -\frac{1}{x}
   \]

Thus, the derivative is:
\[
\frac{d}{dx} \ln\left(\frac{1}{x}\right) = -\frac{1}{x}
\]

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

Here are 5 related questions to deepen your understanding:
1. What is the derivative of \( \ln(x^2) \)?
2. How would you find the derivative of \( \ln(x + 1) \)?
3. What is the second derivative of \( f(x) = \ln(1/x) \)?
4. How does the chain rule apply when differentiating logarithmic functions?
5. What is the derivative of \( \ln(a/x) \) for a constant \( a \)?

**Tip:** Always simplify logarithmic expressions when possible before differentiating, as it makes the process easier!

Understand how to differentiate ln(1/x) using logarithmic properties and the chain rule. The detailed solution simplifies the expression and guides through the process of finding the derivative of ln(1/x), resulting in -1/x. Suitable for college-level calculus or advanced high school students.

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