The problem you're working on asks you to find the derivative of the inverse function for $f(x) = \frac{x}{x + 6}$.

Steps to Solve:

1. Find the inverse function: To find the derivative of the inverse, we don't actually need to calculate the explicit form of the inverse. We can use the following formula:

   $\left( f^{-1} \right)'(x) = \frac{1}{f'(f^{-1}(x))}$

   This formula tells us that the derivative of the inverse function at a point $x$ is the reciprocal of the derivative of the original function evaluated at $f^{-1}(x)$.

2. Find the derivative of $f(x)$: To use the formula, we need to first differentiate $f(x)$:

   $f(x) = \frac{x}{x + 6}$

   We can differentiate this using the quotient rule:

   $f'(x) = \frac{(x + 6)(1) - x(1)}{(x + 6)^2} = \frac{x + 6 - x}{(x + 6)^2} = \frac{6}{(x + 6)^2}$

3. Apply the inverse derivative formula: Now, using the formula for the derivative of the inverse:

   $\left( f^{-1} \right)'(x) = \frac{1}{f'(f^{-1}(x))} = \frac{(f^{-1}(x) + 6)^2}{6}$

Conclusion:

To compute the exact value for a specific $x$, you'd need to know $f^{-1}(x)$, which is the inverse function evaluated at that point. However, the general form of the derivative of the inverse at $x$ is given by:

$\left( f^{-1} \right)'(x) = \frac{(f^{-1}(x) + 6)^2}{6}$

Would you like more details on how to proceed with this, or any specific value you'd like to evaluate?

Here are 5 related questions:

1. How do you find the explicit inverse of a rational function?
2. Can you compute the derivative of an inverse for other functions like exponentials or trigonometric functions?
3. How does the chain rule relate to the formula for the derivative of an inverse?
4. Why is the quotient rule necessary in differentiating $f(x) = \frac{x}{x+6}$?
5. What does the graph of the inverse function typically look like compared to the original function?

Tip: When differentiating complex functions, always check if rules like the chain rule, quotient rule, or product rule apply to simplify the process.