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The problem asks for the derivative \( \frac{dy}{dx} \) of the equation:

\[
x^3 - 2xy + 3y^2 = 7
\]

### Step-by-step solution:

We need to differentiate implicitly with respect to \( x \), as \( y \) is a function of \( x \).

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

2. Differentiate \( -2xy \) using the product rule:
   \[
   \frac{d}{dx}(-2xy) = -2\left( \frac{d}{dx}(x)y + x\frac{dy}{dx} \right) = -2(y + x\frac{dy}{dx})
   \]

3. Differentiate \( 3y^2 \) using the chain rule:
   \[
   \frac{d}{dx}(3y^2) = 6y\frac{dy}{dx}
   \]

Now, differentiating both sides of the equation \( x^3 - 2xy + 3y^2 = 7 \):

\[
3x^2 - 2(y + x\frac{dy}{dx}) + 6y\frac{dy}{dx} = 0
\]

Expand and simplify:

\[
3x^2 - 2y - 2x\frac{dy}{dx} + 6y\frac{dy}{dx} = 0
\]

Now, collect terms involving \( \frac{dy}{dx} \):

\[
-2x\frac{dy}{dx} + 6y\frac{dy}{dx} = -3x^2 + 2y
\]

Factor out \( \frac{dy}{dx} \):

\[
\frac{dy}{dx}(-2x + 6y) = -3x^2 + 2y
\]

Finally, solve for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{3x^2 - 2y}{2x - 6y}
\]

This matches option **(B)**:

\[
\boxed{\frac{3x^2 - 2y}{2x - 6y}}
\]

Would you like further details or clarifications? Here are 5 related questions:

1. How does implicit differentiation work in general?
2. What is the product rule in differentiation?
3. Can you explain more examples of implicit differentiation?
4. What are the steps to solve non-linear implicit functions?
5. How do you interpret partial derivatives in multivariable calculus?

**Tip:** When differentiating implicitly, treat \( y \) as a function of \( x \), so always apply the chain rule when differentiating terms involving \( y \).

Solve for dy/dx in the equation x^3 - 2xy + 3y^2 = 7 using implicit differentiation. This problem explores the product rule, chain rule, and differentiation techniques. Suitable for high school students in Grades 10-12.

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