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To find \( y'' \) by implicit differentiation for the equation:
\[
4x^2 + y^2 = 3
\]

we’ll proceed in the following steps:

### Step 1: Differentiate Both Sides with Respect to \( x \)

Differentiating implicitly:
\[
\frac{d}{dx}(4x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(3)
\]

1. The derivative of \( 4x^2 \) with respect to \( x \) is \( 8x \).
2. For \( y^2 \), use the chain rule: \( \frac{d}{dx}(y^2) = 2y \cdot y' \).
3. The derivative of the constant \( 3 \) is \( 0 \).

This gives:
\[
8x + 2y \cdot y' = 0
\]

### Step 2: Solve for \( y' \)

Rearrange to isolate \( y' \):
\[
2y \cdot y' = -8x
\]
\[
y' = \frac{-8x}{2y} = \frac{-4x}{y}
\]

### Step 3: Differentiate \( y' \) to Find \( y'' \)

Now, differentiate \( y' = \frac{-4x}{y} \) with respect to \( x \) again to find \( y'' \). Use the quotient rule, where if \( u = -4x \) and \( v = y \), then \( y' = \frac{u}{v} \) and:
\[
y'' = \frac{v \cdot u' - u \cdot v'}{v^2}
\]

1. \( u = -4x \) so \( u' = -4 \).
2. \( v = y \) so \( v' = y' = \frac{-4x}{y} \).

Substitute these values:
\[
y'' = \frac{y \cdot (-4) - (-4x) \cdot \frac{-4x}{y}}{y^2}
\]
Simplify each term:
\[
y'' = \frac{-4y - \frac{16x^2}{y}}{y^2}
\]
Combine terms:
\[
y'' = \frac{-4y^2 - 16x^2}{y^3}
\]

So, the second derivative \( y'' \) is:
\[
y'' = \frac{-4(y^2 + 4x^2)}{y^3}
\]

Since \( 4x^2 + y^2 = 3 \) from the original equation, substitute \( y^2 + 4x^2 = 3 \):
\[
y'' = \frac{-4 \cdot 3}{y^3} = \frac{-12}{y^3}
\]

### Final Answer
\[
y'' = \frac{-12}{y^3}
\]

Would you like further details or have any questions?

---

Here are some related questions for practice:

1. How does implicit differentiation differ from regular differentiation?
2. Why do we use the chain rule when differentiating \( y^2 \) with respect to \( x \)?
3. Can you apply the quotient rule to differentiate other types of equations?
4. What are the steps to confirm if \( y \) is a function of \( x \) in an implicit function?
5. How would this process change if the equation included \( xy \) terms?

**Tip:** Always remember to use implicit differentiation when \( y \) is a function of \( x \) in terms that aren't easily separated.

Find y'' by implicit differentiation for the equation 4x^2 + y^2 = 3.

Step-by-step solution to find the second derivative y'' using implicit differentiation for the equation 4x^2 + y^2 = 3. This solution covers the use of the chain rule and quotient rule. Ideal for students in Grades 11-12.

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