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To solve this problem, we need to find \((f^{-1})'(9)\), the derivative of the inverse function \(f^{-1}(x)\) at \(x = 9\).

We know:
- \(f(8) = 9\) (since \(f(x)\) at \(x = 8\) is given as \((8, 9)\)),
- The slope of the curve at \((8, 9)\) is \(\frac{2}{3}\), which means \(f'(8) = \frac{2}{3}\).

### Step-by-Step Solution:
1. **Inverse Function Derivative Formula**:
   The derivative of an inverse function \(f^{-1}\) at a point \(y = f(x)\) is given by:
   \[
   (f^{-1})'(y) = \frac{1}{f'(x)}
   \]
   where \(x = f^{-1}(y)\).

2. **Applying Values**:
   Since \(f(8) = 9\), we know that \(f^{-1}(9) = 8\). Thus, we want to find \((f^{-1})'(9)\), which is:
   \[
   (f^{-1})'(9) = \frac{1}{f'(8)}
   \]

3. **Substitute \(f'(8) = \frac{2}{3}\)**:
   \[
   (f^{-1})'(9) = \frac{1}{\frac{2}{3}} = \frac{3}{2}
   \]

### Final Answer:
\[
(f^{-1})'(9) = \frac{3}{2}
\]

Would you like more details on the process or have any further questions?

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Here are 5 related questions to deepen your understanding:
1. How do you interpret the derivative of an inverse function geometrically?
2. What is the significance of the slope of a function at a particular point?
3. How can we verify if two functions are indeed inverses of each other?
4. What happens to \((f^{-1})'(x)\) if \(f'(x)\) is zero at some point?
5. Can we find the derivative of the inverse function for a non-invertible function?

**Tip:** Remember, the derivative of the inverse function at a point \(y = f(x)\) is the reciprocal of the derivative of the original function at \(x\), as long as \(f'(x) \neq 0\).

Suppose the slope of the curve y = f(x) at (8, 9) is 2/3. Find (f^{-1})'(9).

Learn how to find the derivative of an inverse function at a specific point. This problem involves understanding the relationship between f(x) and its inverse using the Inverse Function Theorem, with a step-by-step solution for (f^{-1})'(9) given f'(8) = 2/3. Suitable for advanced high school or college calculus students.

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