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We are tasked with finding \( y'' \) by implicit differentiation of the equation:

\[
x^2 + 2y^2 = 2
\]

### Step 1: Differentiate both sides with respect to \( x \)

We apply implicit differentiation, remembering that \( y \) is a function of \( x \) (so we must use the chain rule when differentiating terms involving \( y \)):

\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(2y^2) = \frac{d}{dx}(2)
\]

This becomes:

\[
2x + 2 \cdot 2y \cdot \frac{dy}{dx} = 0
\]

Simplifying:

\[
2x + 4y \frac{dy}{dx} = 0
\]

### Step 2: Solve for \( \frac{dy}{dx} \)

Rearranging the above equation:

\[
4y \frac{dy}{dx} = -2x
\]

Dividing both sides by \( 4y \):

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

### Step 3: Differentiate again to find \( y'' \)

We now differentiate \( \frac{dy}{dx} = -\frac{x}{2y} \) implicitly to find \( y'' \):

\[
\frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d}{dx} \left( -\frac{x}{2y} \right)
\]

Using the quotient rule for the right-hand side:

\[
\frac{d}{dx} \left( -\frac{x}{2y} \right) = - \frac{(2y \cdot 1 - x \cdot 2 \frac{dy}{dx})}{(2y)^2}
\]

Substitute \( \frac{dy}{dx} = -\frac{x}{2y} \):

\[
y'' = - \frac{2y - x \cdot 2 \cdot \left( -\frac{x}{2y} \right)}{4y^2}
\]

Simplifying:

\[
y'' = - \frac{2y + x^2/y}{4y^2}
\]

Multiply through to clear the fraction inside:

\[
y'' = - \frac{2y^2 + x^2}{4y^3}
\]

Thus, the second derivative is:

\[
y'' = -\frac{2y^2 + x^2}{4y^3}
\]

### Final Answer:
\[
y'' = -\frac{2y^2 + x^2}{4y^3}
\]

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to expand on this:
1. How does the chain rule apply in implicit differentiation?
2. What are the key differences between explicit and implicit differentiation?
3. Can you explain the quotient rule used in finding \( y'' \)?
4. Why do we use implicit differentiation when \( y \) is a function of \( x \)?
5. How do we solve higher-order derivatives using implicit differentiation?

**Tip:** When differentiating implicitly, always apply the chain rule to every term involving the dependent variable.

Find y'' by implicit differentiation. Simplify where possible. The equation given is x^2 + 2y^2 = 2.

This page provides a step-by-step solution for finding the second derivative y'' of the equation x^2 + 2y^2 = 2 using implicit differentiation. It covers the chain rule, quotient rule, and implicit differentiation, with a focus on calculus concepts suitable for college-level students.

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