We are given that $F^{-1}(2) = 1$, and we are asked to find the derivative of the inverse function at $x = 2$, i.e., $(F^{-1})'(2)$.

To solve this, we can use the following property of inverse functions and their derivatives:

$(F^{-1})'(y) = \frac{1}{F'(x)}$

where $x = F^{-1}(y)$. In this case, we are given that $F^{-1}(2) = 1$, which means $F(1) = 2$.

Thus, the derivative of the inverse at $y = 2$ is given by:

$(F^{-1})'(2) = \frac{1}{F'(1)}$

To calculate $(F^{-1})'(2)$, we need to know $F'(1)$. However, since the problem does not provide this value, we cannot determine the exact value of $(F^{-1})'(2)$ without more information about the function $F$.

Summary of steps:

1. $F^{-1}(2) = 1$ implies that $F(1) = 2$.
2. The formula for the derivative of the inverse is $(F^{-1})'(y) = \frac{1}{F'(x)}$, where $x = F^{-1}(y)$.
3. For $y = 2$, we have $x = 1$, so $(F^{-1})'(2) = \frac{1}{F'(1)}$.

Therefore, to find $(F^{-1})'(2)$, we need the value of $F'(1)$.

Would you like to explore this problem further with more information on $F(x)$, or do you have additional questions?

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Further Questions:

1. What is the definition of the derivative of an inverse function?
2. How can we apply the inverse function theorem to different functions?
3. What is the relationship between the derivatives of inverse functions at specific points?
4. How can we estimate the value of $F'(1)$ if we don't know the explicit form of $F(x)$?
5. Can you provide a specific function $F(x)$ to use in this example?

Tip:

When working with inverse functions, always remember that the derivative of the inverse function at $y$ is the reciprocal of the derivative of the original function at the point $x = F^{-1}(y)$.