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Let's solve the given problem using **implicit differentiation**. 

The equation is:
\[
x^3 + y^3 = 3xy^2
\]

### Step 1: Differentiate both sides with respect to \(x\)
Since \(y\) is implicitly dependent on \(x\), when differentiating terms involving \(y\), we must apply the **chain rule**. Let’s proceed.

1. Differentiate \(x^3\):
   \[
   \frac{d}{dx}(x^3) = 3x^2
   \]

2. Differentiate \(y^3\) using the chain rule:
   \[
   \frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}
   \]

3. Differentiate \(3xy^2\). This requires the **product rule**:
   \[
   \frac{d}{dx}(3xy^2) = 3 \left( \frac{d}{dx}(x) \cdot y^2 + x \cdot \frac{d}{dx}(y^2) \right)
   \]
   - \(\frac{d}{dx}(x) = 1\), so \(3 \cdot y^2\) becomes \(3y^2\).
   - \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\), so \(3x \cdot 2y \frac{dy}{dx} = 6xy \frac{dy}{dx}\).

   Thus:
   \[
   \frac{d}{dx}(3xy^2) = 3y^2 + 6xy \frac{dy}{dx}
   \]

### Step 2: Combine all differentiated terms
The differentiated equation becomes:
\[
3x^2 + 3y^2 \frac{dy}{dx} = 3y^2 + 6xy \frac{dy}{dx}
\]

### Step 3: Rearrange to solve for \(\frac{dy}{dx}\)
Move all terms involving \(\frac{dy}{dx}\) to one side and other terms to the opposite side:
\[
3y^2 \frac{dy}{dx} - 6xy \frac{dy}{dx} = 3y^2 - 3x^2
\]

Factor out \(\frac{dy}{dx}\) on the left:
\[
(3y^2 - 6xy) \frac{dy}{dx} = 3y^2 - 3x^2
\]

### Step 4: Solve for \(\frac{dy}{dx}\)
\[
\frac{dy}{dx} = \frac{3y^2 - 3x^2}{3y^2 - 6xy}
\]

### Final Answer:
\[
\boxed{\frac{dy}{dx} = \frac{y^2 - x^2}{y^2 - 2xy}}
\]

Would you like further clarification or detailed steps on any part? 

Here are some related questions to explore further:
1. How does the chain rule apply when differentiating \(y^n\) implicitly?
2. What are other examples of implicit differentiation problems?
3. How can you find the second derivative using implicit differentiation?
4. What is the geometric meaning of \(\frac{dy}{dx}\) for this curve?
5. How does the product rule work in implicit differentiation?

**Tip:** Always rearrange the equation carefully before differentiating when both \(x\) and \(y\) are involved. This minimizes algebra errors.

Find dy/dx by differentiating implicitly if x^3 + y^3 = 3xy^2.

Find the derivative dy/dx of the implicit equation x^3 + y^3 = 3xy^2 using implicit differentiation, applying the chain rule and product rule. Detailed step-by-step solution provided, suitable for Grade 12 or college-level calculus students.

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