https://file.aiquickdraw.com/mathbot/file/1727212487141xkr9eajr.jpg

The graph in the image shows the function \( f(x) \) and asks to estimate the value of \( (f^{-1})'(3) \), which is the derivative of the inverse function at \( x = 3 \). 

To solve this, we use the relationship between the derivative of a function and its inverse:
\[
(f^{-1})'(y) = \frac{1}{f'(x)} \quad \text{where} \ f(x) = y.
\]
In this case, we are asked to find \( (f^{-1})'(3) \), so we need to find \( x \) such that \( f(x) = 3 \). From the graph, it appears that \( f(3) \approx 3 \), meaning we need the slope of the tangent line to the curve at this point.

Looking at the graph, the slope of the tangent line at \( f(x) = 3 \) is given as 4. So we can use this in the formula:
\[
(f^{-1})'(3) = \frac{1}{f'(3)} = \frac{1}{4}.
\]

Thus, the estimated value of \( (f^{-1})'(3) \) is approximately \( \frac{1}{4} \) or 0.25.

Let me know if you would like further clarification or details on this!

### Related Questions:
1. How do you compute the derivative of an inverse function using implicit differentiation?
2. What does the slope of the tangent line represent in the context of a function's derivative?
3. Why is the reciprocal of the slope used when finding the derivative of an inverse function?
4. How can we find the inverse of a function graphically?
5. What is the significance of the point where \( f(x) = x \) on a function's graph?

### Tip:
When estimating values from a graph, carefully check the axes and labeled points to ensure accurate interpretation of values like slopes or intercepts.

The graph shows f(x). Use the graph to estimate (f^{-1})'(3).

Learn how to estimate the derivative of an inverse function at a given point using the graph of f(x). This problem uses the relationship between the slope of the tangent line and the inverse function derivative, following the Inverse Function Theorem.

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