Given the equation \( 7 y^{2}-x^{3}=9 \), find \( \frac{d^{2} y}{d x^{2}} \) by implicit differentiation.

To find \(\frac{d^{2} y}{d x^{2}}\) for the equation \(7 y^{2} - x^{3} = 9\) using implicit differentiation, we start by differentiating both sides of the equation with respect to \(x\). 1. Differentiate the given equation: \[ \frac{d}{dx}(7y^2) - \frac{d}{dx}(x^3) = \frac{d}{dx}(9) \] This gives us: \[ 14y \frac{dy}{dx} - 3x^2 = 0 \] Rearranging this equation, we have: \[ 14y \frac{dy}{dx} = 3x^2 \] Therefore, we can solve for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{3x^2}{14y} \] 2. Now we differentiate \(\frac{dy}{dx}\) again to find the second derivative \(\frac{d^2y}{dx^2}\): Using the quotient rule, where \(u = 3x^2\) and \(v = 14y\): \[ \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \] Here, \(u' = 6x\) and \(v' = 14 \frac{dy}{dx}\). Thus, we have: \[ \frac{d^2y}{dx^2} = \frac{(6x)(14y) - (3x^2)(14 \frac{dy}{dx})}{(14y)^2} \] 3. Substitute \(\frac{dy}{dx} = \frac{3x^2}{14y}\) into the equation: \[ \frac{d^2y}{dx^2} = \frac{(6x)(14y) - (3x^2)(14 \cdot \frac{3x^2}{14y})}{(14y)^2} \] Simplifying further, we find: \[ = \frac{(84xy) - \frac{9x^4}{y}}{196y^2} \] This can be rewritten as: \[ \frac{d^2y}{dx^2} = \frac{84xy^2 - 9x^4}{196y^3} \] So, the final expression for the second derivative \(\frac{d^2y}{dx^2}\) is: \[ \frac{d^2y}{dx^2} = \frac{84xy^2 - 9x^4}{196y^3} \]