Given the equation \( 4 x^{2}-y^{2}=8 \) that defines \( y \) as a function of \( x \), use implicit differentiation to find the derivative \( \frac{d y}{d x} \). Simplify your answer. \( d y / d x= \)

To find the derivative \( \frac{dy}{dx} \) using implicit differentiation for the equation \( 4x^{2} - y^{2} = 8 \), follow these steps: 1. **Differentiate both sides with respect to \( x \):** \[ \frac{d}{dx}(4x^{2}) - \frac{d}{dx}(y^{2}) = \frac{d}{dx}(8) \] 2. **Compute the derivatives:** - The derivative of \( 4x^{2} \) with respect to \( x \) is \( 8x \). - The derivative of \( y^{2} \) with respect to \( x \) is \( 2y \frac{dy}{dx} \) (using the chain rule). - The derivative of the constant \( 8 \) is \( 0 \). Substituting these into the equation: \[ 8x - 2y \frac{dy}{dx} = 0 \] 3. **Solve for \( \frac{dy}{dx} \):** \[ 2y \frac{dy}{dx} = 8x \implies \frac{dy}{dx} = \frac{8x}{2y} = \frac{4x}{y} \] **Final Answer:** \[ \frac{dy}{dx} = \frac{4\,x}{\,y} \]