Given the equation \( x^{8}+y^{3}=24 \), find \( \frac{d y}{d x} \) by implicit differentiation. Solve the obtained equation \( 8 x^{7}+3 y^{2} \frac{d y}{d x}=0 \) for \( \frac{d y}{d x} \)

To find \( \frac{d y}{d x} \) using implicit differentiation for the equation \( x^{8} + y^{3} = 24 \), we start by differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(x^{8}) + \frac{d}{dx}(y^{3}) = \frac{d}{dx}(24) \] This results in: \[ 8x^{7} + 3y^{2} \frac{dy}{dx} = 0 \] Now, we need to solve for \( \frac{dy}{dx} \): \[ 3y^{2} \frac{dy}{dx} = -8x^{7} \] Dividing both sides by \( 3y^{2} \), we obtain: \[ \frac{dy}{dx} = -\frac{8x^{7}}{3y^{2}} \] Thus, the derivative \( \frac{dy}{dx} \) is given by: \[ \frac{dy}{dx} = -\frac{8x^{7}}{3y^{2}} \]