If $$ -3+3 x^{2}=y^{2} $$ then find $$ \frac{d^{2} y}{d x^{2}} $$ at the point $$ (2,-3) $$ in simplest form. Answer Attempt 2 out of 2

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To find the second derivative $$\frac{d^{2} y}{d x^{2}}$$ at the point $$(2, -3)$$ for the equation $$ -3 + 3x^{2} = y^{2} $$, follow these steps:

1. **First Derivative ($$\frac{dy}{dx}$$)**

Differentiate both sides of the equation with respect to $$x$$:
   $$
   \frac{d}{dx}(3x^{2} - 3) = \frac{d}{dx}(y^{2})
   $$
   $$
   6x = 2y \frac{dy}{dx}
   $$
   Solve for $$\frac{dy}{dx}$$:
   $$
   \frac{dy}{dx} = \frac{3x}{y}
   $$
   
   At the point $$(2, -3)$$:
   $$
   \frac{dy}{dx} = \frac{3 \times 2}{-3} = -2
   $$

2. **Second Derivative ($$\frac{d^{2} y}{d x^{2}}$$)**

Differentiate $$\frac{dy}{dx} = \frac{3x}{y}$$ with respect to $$x$$:
   $$
   \frac{d^{2} y}{d x^{2}} = \frac{d}{dx}\left(\frac{3x}{y}\right)
   $$
   Use the quotient rule:
   $$
   \frac{d^{2} y}{d x^{2}} = \frac{3y - 3x \frac{dy}{dx}}{y^{2}}
   $$
   
   Substitute $$\frac{dy}{dx} = -2$$ and the point $$(2, -3)$$:
   $$
   \frac{d^{2} y}{d x^{2}} = \frac{3(-3) - 3(2)(-2)}{(-3)^{2}} = \frac{-9 + 12}{9} = \frac{3}{9} = \frac{1}{3}
   $$

**Final Answer:**
$$
\frac{d^{2} y}{d x^{2}} = \frac{1}{3} \text{ at the point } (2, -3)
$$
At the point (2, -3), the second derivative $$\frac{d^{2} y}{d x^{2}}$$ is $$\frac{1}{3}$$.

If then find at the point in simplest form.
Answer Attempt 2 out of 2

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If then find at the point in simplest form. Answer Attempt 2 out of 2