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To solve this problem using implicit differentiation, we need to differentiate both sides of the given equation with respect to \( x \). The given equation is:

\[
2x^3 + x^2 y - xy^3 = 6
\]

### Step-by-Step Implicit Differentiation:

1. **Differentiate each term with respect to \( x \), treating \( y \) as a function of \( x \).**

   - For \( 2x^3 \): The derivative is straightforward:
     \[
     \frac{d}{dx}(2x^3) = 6x^2
     \]
   
   - For \( x^2 y \): Apply the product rule (\( uv' + vu' \)), where \( u = x^2 \) and \( v = y \):
     \[
     \frac{d}{dx}(x^2 y) = 2x y + x^2 \frac{dy}{dx}
     \]
   
   - For \( -xy^3 \): Again, use the product rule, where \( u = x \) and \( v = y^3 \):
     \[
     \frac{d}{dx}(-xy^3) = -y^3 - 3xy^2 \frac{dy}{dx}
     \]
   
   - For the constant \( 6 \): The derivative is zero:
     \[
     \frac{d}{dx}(6) = 0
     \]

2. **Combine all the derivatives:**
   \[
   6x^2 + (2xy + x^2 \frac{dy}{dx}) - (y^3 + 3xy^2 \frac{dy}{dx}) = 0
   \]

3. **Collect the terms involving \( \frac{dy}{dx} \) and the terms without it:**
   \[
   6x^2 + 2xy - y^3 + (x^2 - 3xy^2)\frac{dy}{dx} = 0
   \]

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

Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\boxed{\frac{dy}{dx} = \frac{y^3 - 6x^2 - 2xy}{x^2 - 3xy^2}}
\]

Would you like more details or have any questions about this process?

### 5 Related Questions:
1. What is the product rule and how is it applied in differentiation?
2. Why do we treat \( y \) as a function of \( x \) in implicit differentiation?
3. How would this process change if there were higher powers of \( y \)?
4. Can implicit differentiation be used for equations involving trigonometric functions?
5. How can you verify your result after performing implicit differentiation?

### Tip:
Always apply the product and chain rules carefully when differentiating implicitly, especially when differentiating terms involving both \( x \) and \( y \).

Find dy/dx by implicit differentiation. The given equation is: 2x^3 + x^2*y - xy^3 = 6.

Learn how to implicitly differentiate the equation 2x^3 + x^2*y - xy^3 = 6. This detailed guide applies the product and chain rules to solve for dy/dx, suitable for students in Grades 11-12.

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