Find y ″ by implicit differentiation. Simplify where possible.
x2 + 3y2 = 3

We are given the equation:

\[
x^2 + 3y^2 = 3
\]

We need to find \( y'' \) (the second derivative of \( y \) with respect to \( x \)) using implicit differentiation. Here are the steps:

### Step 1: Differentiate both sides of the equation implicitly with respect to \( x \).

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

Using the chain rule for the \( y^2 \) term:

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

\[
2x + 3 \cdot 2y \cdot \frac{dy}{dx} = 0
\]

Simplifying:

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

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

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

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

Find y ″ by implicit differentiation. Simplify where possible.
x^2 + 3y^2 = 3

Discover how to find the second derivative of y (y″) through implicit differentiation of the equation x^2 + 3y^2 = 3. This step-by-step guide covers key calculus concepts like the chain rule and implicit differentiation. Ideal for advanced high school or early university students.

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