Find \( \frac{d y}{d x} \) by implicit differentiation. \[ \begin{array}{l} 4 x^{2}-7 y^{2}=1 \\ \frac{d y}{d x}= \end{array} \] Need Help? Road It

To find \( \frac{d y}{d x} \) using implicit differentiation, we start by differentiating both sides of the equation \( 4x^2 - 7y^2 = 1 \) with respect to \( x \). Differentiating term by term, we have: \[ \frac{d}{dx}(4x^2) - \frac{d}{dx}(7y^2) = \frac{d}{dx}(1) \] This results in: \[ 8x - 14y \frac{dy}{dx} = 0 \] Now, we can solve for \( \frac{dy}{dx} \): \[ 14y \frac{dy}{dx} = 8x \] \[ \frac{dy}{dx} = \frac{8x}{14y} = \frac{4x}{7y} \] Thus, \( \frac{d y}{d x} = \frac{4x}{7y} \).