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The equation provided is:

\[
3x^3 = y^3 - xy
\]

We are tasked with finding \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). This involves implicit differentiation.

### Step-by-Step Solution:

1. Differentiate both sides of the equation with respect to \(x\).

   - Left-hand side:
     \[
     \frac{d}{dx}(3x^3) = 9x^2
     \]

   - Right-hand side (using the product rule for \(-xy\)):
     \[
     \frac{d}{dx}(y^3 - xy) = 3y^2 \frac{dy}{dx} - \left( x \frac{dy}{dx} + y \right)
     \]

2. After differentiation, the equation becomes:
   \[
   9x^2 = 3y^2 \frac{dy}{dx} - \left( x \frac{dy}{dx} + y \right)
   \]

3. Now, collect all terms involving \(\frac{dy}{dx}\) on one side:
   \[
   9x^2 + y = 3y^2 \frac{dy}{dx} - x \frac{dy}{dx}
   \]
   \[
   9x^2 + y = \frac{dy}{dx} (3y^2 - x)
   \]

4. Solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{9x^2 + y}{3y^2 - x}
   \]

Thus, the derivative \(\frac{dy}{dx}\) in terms of \(x\) and \(y\) is:

\[
\frac{dy}{dx} = \frac{9x^2 + y}{3y^2 - x}
\]

Would you like further details on any step? 

Here are 5 related questions to consider:
1. How does the product rule apply when differentiating \(xy\)?
2. What are the conditions where this derivative might be undefined?
3. Can we explore how to solve this equation for specific values of \(x\) and \(y\)?
4. How do implicit differentiation and explicit differentiation differ in their applications?
5. How could this problem be approached if the equation were slightly altered (e.g., different exponents)?

**Tip:** When dealing with implicit differentiation, carefully apply the chain rule whenever differentiating a term involving both \(x\) and \(y\).

If 3x^3 = y^3 - xy then find dy/dx in terms of x and y.

This problem involves implicit differentiation of the equation 3x^3 = y^3 - xy to find dy/dx in terms of x and y. Step-by-step solution included, using key calculus concepts such as the product rule and chain rule.

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