https://file.aiquickdraw.com/mathbot/file/17263150651274oceedvq.jpg

The equation in the image appears to be \( f^{-1}(x) = 3 \sqrt{x} \), which represents the inverse of a function. To find the derivative of this inverse function, we can follow these steps.

### 1. Differentiate \( f^{-1}(x) = 3 \sqrt{x} \)

We begin by rewriting the square root in a more convenient form:
\[
f^{-1}(x) = 3x^{1/2}
\]

Now, differentiate with respect to \(x\):
\[
\frac{d}{dx}\left(3x^{1/2}\right) = 3 \cdot \frac{1}{2}x^{-1/2} = \frac{3}{2\sqrt{x}}
\]

Thus, the derivative of \( f^{-1}(x) \) is:
\[
\boxed{\frac{3}{2\sqrt{x}}}
\]

This represents the rate of change of the inverse function with respect to \(x\).

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

### 5 Related Questions:
1. What is the significance of differentiating an inverse function?
2. How do you find the derivative of a composite function involving square roots?
3. How does the chain rule apply to differentiating functions like this one?
4. What are the general rules for differentiating inverse functions?
5. How do you interpret the derivative of \( f^{-1}(x) \) geometrically?

### Tip:
When dealing with square roots, it often helps to rewrite them as fractional exponents before differentiating.

Learn how to differentiate the inverse function f^{-1}(x) = 3√x, rewritten as 3x^(1/2), and apply the power rule to find the derivative. This step-by-step guide is suitable for high school students learning calculus in Grades 11-12.

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