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Let's solve the problem step by step.

We are given the equation:
\[
xy + 2x - 2y^2 = -8
\]
and need to find \(\frac{dy}{dx}\) at the point \((2, 3)\).

### Step 1: Differentiate implicitly
To find \(\frac{dy}{dx}\), we will differentiate both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\).

- The derivative of \(xy\) using the product rule is:
  \[
  \frac{d}{dx}(xy) = \frac{d}{dx}(x) \cdot y + x \cdot \frac{dy}{dx} = y + x \frac{dy}{dx}
  \]
- The derivative of \(2x\) is:
  \[
  \frac{d}{dx}(2x) = 2
  \]
- The derivative of \(-2y^2\) using the chain rule is:
  \[
  \frac{d}{dx}(-2y^2) = -2 \cdot 2y \cdot \frac{dy}{dx} = -4y \frac{dy}{dx}
  \]
- The derivative of the constant \(-8\) is:
  \[
  \frac{d}{dx}(-8) = 0
  \]

Now, we can differentiate the entire equation:
\[
y + x \frac{dy}{dx} + 2 - 4y \frac{dy}{dx} = 0
\]

### Step 2: Solve for \(\frac{dy}{dx}\)
Combine like terms:
\[
x \frac{dy}{dx} - 4y \frac{dy}{dx} = -y - 2
\]
Factor out \(\frac{dy}{dx}\) on the left-hand side:
\[
(x - 4y) \frac{dy}{dx} = -y - 2
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-y - 2}{x - 4y}
\]

### Step 3: Substitute the point \((x, y) = (2, 3)\)
Substitute \(x = 2\) and \(y = 3\) into the equation:
\[
\frac{dy}{dx} = \frac{-(3) - 2}{(2) - 4(3)} = \frac{-3 - 2}{2 - 12} = \frac{-5}{-10} = \frac{1}{2}
\]

So, \(\frac{dy}{dx} = \frac{1}{2}\) at the point \((2, 3)\).

Would you like further details on any step?

Here are some related questions to deepen your understanding:
1. How does implicit differentiation differ from explicit differentiation?
2. What are other common uses of the product rule in calculus?
3. How do we apply the chain rule when differentiating \(y^2\)?
4. Can you apply implicit differentiation to the equation \(x^2 + y^2 = 25\)?
5. How would the answer change if the constant on the right-hand side were a variable?

**Tip:** Always ensure to apply the product rule correctly when differentiating terms involving products of variables like \(xy\).

Let xy + 2x - 2y^2 = -8. dy/dx at the point (2, 3) is ____.

Solve the implicit differentiation problem involving the equation xy + 2x - 2y^2 = -8 and find dy/dx at the point (2, 3). Learn step-by-step methods using the product rule and chain rule. Suitable for college-level calculus students.

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