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.