Use implicit differentiation to determine \( \frac{d y}{d x} \) for the equation \( \frac{8}{x}-\frac{3}{y}=4 \)

To use implicit differentiation on the equation \( \frac{8}{x} - \frac{3}{y} = 4 \), we first differentiate both sides with respect to \( x \). Starting with the left side: 1. For \( \frac{8}{x} \), using the power rule, we get \( -\frac{8}{x^2} \). 2. For \( -\frac{3}{y} \), we apply the chain rule: differentiate \( -3y^{-1} \) to get \( 3y^{-2} \frac{dy}{dx} \). Now, differentiating both sides gives us: \[ -\frac{8}{x^2} + 3\frac{1}{y^2}\frac{dy}{dx} = 0. \] Next, we can isolate \( \frac{dy}{dx} \): \[ 3\frac{1}{y^2}\frac{dy}{dx} = \frac{8}{x^2}. \] Now, multiply by \( y^2 \): \[ 3\frac{dy}{dx} = \frac{8y^2}{x^2}. \] Finally, solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{8y^2}{3x^2}. \] Thus, the derivative is: \[ \frac{dy}{dx} = \frac{8y^2}{3x^2}. \]