Differentiate implicitly to find $\frac{d^{2} y}{d x^{2}}$ \[ 5 y^{2}-x y+x^{2}=9 \]

Step 1 ：Given the equation \(5y^{2} - xy + x^{2} = 9\), we need to find the second derivative \(\frac{d^{2} y}{d x^{2}}\).

Step 2 ：First, we differentiate both sides of the equation with respect to \(x\).

Step 3 ：The derivative of \(5y^{2}\) with respect to \(x\) is \(10y \frac{dy}{dx}\) by using the chain rule.

Step 4 ：The derivative of \(-xy\) with respect to \(x\) is \(-y - x \frac{dy}{dx}\) by using the product rule.

Step 5 ：The derivative of \(x^{2}\) with respect to \(x\) is \(2x\).

Step 6 ：The derivative of \(9\) with respect to \(x\) is \(0\).

Step 7 ：So, the first derivative of the given equation is \(10y \frac{dy}{dx} - y - x \frac{dy}{dx} + 2x = 0\).

Step 8 ：Solving for \(\frac{dy}{dx}\), we get \(\frac{dy}{dx} = \frac{y - 2x}{10y - x}\).

Step 9 ：Next, we differentiate \(\frac{dy}{dx}\) with respect to \(x\) to find the second derivative.

Step 10 ：The derivative of \(\frac{y - 2x}{10y - x}\) with respect to \(x\) is \(\frac{(10y - x)(\frac{dy}{dx} - 2) - (y - 2x)(10 \frac{dy}{dx} - 1)}{(10y - x)^{2}}\) by using the quotient rule.

Step 11 ：Substituting \(\frac{dy}{dx} = \frac{y - 2x}{10y - x}\) into the equation, we get \(\frac{d^{2} y}{d x^{2}} = \frac{(10y - x)(\frac{y - 2x}{10y - x} - 2) - (y - 2x)(10 \frac{y - 2x}{10y - x} - 1)}{(10y - x)^{2}}\).

Step 12 ：Simplifying the equation, we get \(\frac{d^{2} y}{d x^{2}} = \frac{-3x^{2} - 4xy + 20y^{2}}{(10y - x)^{3}}\).

Step 13 ：So, the second derivative of the given equation is \(\boxed{\frac{d^{2} y}{d x^{2}} = \frac{-3x^{2} - 4xy + 20y^{2}}{(10y - x)^{3}}}\).