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To differentiate the given function \( y = \sqrt{\frac{x^3 - x}{4 - x^2}} \), we can proceed by applying the chain rule and quotient rule. Let's go through the steps to find \( \frac{dy}{dx} \):

Given:
\[
y = \sqrt{\frac{x^3 - x}{4 - x^2}}
\]

1. **Rewrite the square root as a power:**
   \[
   y = \left( \frac{x^3 - x}{4 - x^2} \right)^{\frac{1}{2}}
   \]

2. **Apply the chain rule:**
   To differentiate \( y = \left( u \right)^{\frac{1}{2}} \), where \( u = \frac{x^3 - x}{4 - x^2} \), we get:
   \[
   \frac{dy}{dx} = \frac{1}{2} \left( \frac{x^3 - x}{4 - x^2} \right)^{-\frac{1}{2}} \cdot \frac{d}{dx}\left( \frac{x^3 - x}{4 - x^2} \right)
   \]

3. **Differentiate the inner function using the quotient rule:**
   For \( u = \frac{x^3 - x}{4 - x^2} \), we use the quotient rule: \( \frac{d}{dx}\left( \frac{f}{g} \right) = \frac{f'g - fg'}{g^2} \).
   
   - Let \( f = x^3 - x \) and \( g = 4 - x^2 \).
   - Then, \( f' = 3x^2 - 1 \) and \( g' = -2x \).

   Plugging these into the quotient rule:
   \[
   \frac{d}{dx}\left( \frac{x^3 - x}{4 - x^2} \right) = \frac{(3x^2 - 1)(4 - x^2) - (x^3 - x)(-2x)}{(4 - x^2)^2}
   \]

4. **Simplify the derivative:**
   Now we substitute this result back into the chain rule expression:
   \[
   \frac{dy}{dx} = \frac{1}{2} \left( \frac{x^3 - x}{4 - x^2} \right)^{-\frac{1}{2}} \cdot \frac{(3x^2 - 1)(4 - x^2) + 2x(x^3 - x)}{(4 - x^2)^2}
   \]

5. **Express the final result:**
   Putting it all together, we have:
   \[
   \frac{dy}{dx} = \frac{1}{2} \cdot \frac{(3x^2 - 1)(4 - x^2) + 2x(x^3 - x)}{(4 - x^2)^2} \cdot \left( \frac{x^3 - x}{4 - x^2} \right)^{-\frac{1}{2}}
   \]

This is the derivative of the function. Let me know if you would like further simplification or a step-by-step breakdown of any part.

Would you like additional details or explanations? 

Here are five follow-up questions to explore this topic further:

1. How can we simplify the expression further if needed?
2. What would the second derivative of this function look like?
3. Can this function be differentiated using a substitution to simplify the process?
4. How would we approach finding critical points for this function?
5. How does the domain of the function affect the differentiation process?

**Tip:** Always double-check if there’s a simpler form of the function before differentiating; simplification beforehand can sometimes make the differentiation process easier.

Differentiate y = sqrt((x^3 - x) / (4 - x^2))

Find the derivative of y = sqrt((x^3 - x) / (4 - x^2)) by applying the chain rule and quotient rule. This solution includes a step-by-step breakdown of the differentiation process, making it ideal for Calculus I students.

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