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We are given the function \( y = 6^{-6/x} \), and the goal is to find its derivative \( \frac{dy}{dx} \).

Let's work through the steps to solve this:

1. Start with the function: 
   \[
   y = 6^{-6/x}
   \]

2. Take the natural logarithm of both sides to simplify the exponent:
   \[
   \ln y = \ln \left( 6^{-6/x} \right)
   \]
   Using the logarithmic property \( \ln(a^b) = b \ln(a) \):
   \[
   \ln y = -\frac{6}{x} \ln 6
   \]

3. Differentiate both sides with respect to \( x \). On the left side, we apply implicit differentiation, remembering that the derivative of \( \ln y \) with respect to \( x \) is \( \frac{1}{y} \frac{dy}{dx} \):
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( -\frac{6}{x} \ln 6 \right)
   \]
   The right side becomes:
   \[
   \frac{d}{dx} \left( -\frac{6}{x} \ln 6 \right) = 6 \ln 6 \cdot \frac{1}{x^2}
   \]

4. Now, solve for \( \frac{dy}{dx} \):
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{6 \ln 6}{x^2}
   \]
   Multiply both sides by \( y \) (which is \( 6^{-6/x} \)) to isolate \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = y \cdot \frac{6 \ln 6}{x^2} = 6^{-6/x} \cdot \frac{6 \ln 6}{x^2}
   \]

Thus, the derivative is:
\[
\boxed{\frac{dy}{dx} = 6^{-6/x} \cdot \frac{6 \ln 6}{x^2}}
\]

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

Here are 5 related questions to explore further:
1. How does implicit differentiation work in more complex cases?
2. Can this derivative be simplified further for specific values of \( x \)?
3. How would the solution change if the base 6 were replaced by another number?
4. What is the significance of using the natural logarithm in the differentiation process?
5. How can we apply the chain rule to derivatives involving exponential functions?

**Tip:** When differentiating functions with exponents involving variables, logarithmic differentiation can simplify the process significantly.

Learn how to differentiate the function y = 6^(-6/x) using logarithmic differentiation. This solution involves logarithmic properties and the chain rule, suitable for advanced high school or early college calculus students.

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