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?

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 $f'(x) \neq 0$.