Given the equation \( x^{8}+y^{3}=24 \), find \( \frac{d y}{d x} \) by implicit differentiation.

To find \(\frac{dy}{dx}\) using implicit differentiation, we start with the given equation: \[ x^{8} + y^{3} = 24 \] Now, we differentiate both sides of the equation with respect to \(x\): \[ \frac{d}{dx}(x^{8}) + \frac{d}{dx}(y^{3}) = \frac{d}{dx}(24) \] Differentiating the left side: \[ 8x^{7} + 3y^{2}\frac{dy}{dx} = 0 \] The right side differentiates to 0 since 24 is a constant. Now we can isolate \(\frac{dy}{dx}\): \[ 3y^{2}\frac{dy}{dx} = -8x^{7} \] Now, divide both sides by \(3y^{2}\): \[ \frac{dy}{dx} = \frac{-8x^{7}}{3y^{2}} \] Thus, the result for \(\frac{dy}{dx}\) is: \[ \frac{dy}{dx} = \frac{-8x^{7}}{3y^{2}} \]