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The given equation is:
\[
\frac{a}{x} + \frac{b}{y} = a
\]
We need to find the derivative \(\frac{dy}{dx}\) of this implicit function when \(x = 2\), and we are given the values \(a = 4\) and \(b = 9\).

### Steps:
1. **Differentiate implicitly with respect to \(x\):**
   \[
   \frac{d}{dx}\left(\frac{a}{x}\right) + \frac{d}{dx}\left(\frac{b}{y}\right) = \frac{d}{dx}(a)
   \]
   Since \(a\) is constant, its derivative is zero. Now, compute the derivatives on both sides:
   \[
   -\frac{a}{x^2} + \left(-\frac{b}{y^2}\right)\frac{dy}{dx} = 0
   \]
2. **Solve for \(\frac{dy}{dx}\):**
   Rearranging the equation:
   \[
   \frac{b}{y^2} \frac{dy}{dx} = \frac{a}{x^2}
   \]
   So,
   \[
   \frac{dy}{dx} = \frac{a}{b} \cdot \frac{y^2}{x^2}
   \]

3. **Determine the value of \(y\) when \(x = 2\):**
   Using the original equation:
   \[
   \frac{4}{2} + \frac{9}{y} = 4
   \]
   This simplifies to:
   \[
   2 + \frac{9}{y} = 4 \quad \Rightarrow \quad \frac{9}{y} = 2 \quad \Rightarrow \quad y = \frac{9}{2}
   \]

4. **Substitute values into the derivative:**
   \[
   \frac{dy}{dx} = \frac{4}{9} \cdot \frac{\left(\frac{9}{2}\right)^2}{2^2} = \frac{4}{9} \cdot \frac{\left(\frac{81}{4}\right)}{4} = \frac{4}{9} \cdot \frac{81}{16} = \frac{324}{144} = \frac{9}{4}
   \]

### Final Answer:
\[
\frac{dy}{dx} = \frac{9}{4}
\]

Would you like further explanation or details on any of these steps? 

Here are some related questions to consider:
1. How do we differentiate implicitly in more complex functions?
2. What is the role of constants like \(a\) and \(b\) in derivatives?
3. How does the chain rule apply in implicit differentiation?
4. How can you verify the derivative result with a graphical method?
5. What happens if the values of \(a\) and \(b\) are different?

**Tip:** Always check if you can simplify the implicit function before starting the differentiation.

Calcula la derivada \(\frac{dy}{dx}\) de la función implícita \(\frac{a}{x} + \frac{b}{y} = a\) cuando la abscisa \(x = 2\). Considerando el valor de \(a = 4\) y \(b = 9\).

Learn how to find the derivative \(\frac{dy}{dx}\) of the implicit function \(\frac{a}{x} + \frac{b}{y} = a\) when \(a = 4\), \(b = 9\), and \(x = 2\). This step-by-step solution covers key concepts in implicit differentiation and the chain rule. Suitable for students in Grades 11-12.

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