18. a) Find \( \frac{d y}{d x} \) if \( x^{3}+y^{3}=3 x y^{2} \)

To find \( \frac{dy}{dx} \), we can use implicit differentiation on the equation \( x^{3}+y^{3}=3xy^{2} \). Differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(x^{3}) + \frac{d}{dx}(y^{3}) = \frac{d}{dx}(3xy^{2}) \] This results in: \[ 3x^{2} + 3y^{2}\frac{dy}{dx} = 3\left(y^{2} + 2xy\frac{dy}{dx}\right) \] Now, distribute on the right: \[ 3x^{2} + 3y^{2}\frac{dy}{dx} = 3y^{2} + 6xy\frac{dy}{dx} \] Now, let's rearrange to isolate \( \frac{dy}{dx} \): \[ 3y^{2}\frac{dy}{dx} - 6xy\frac{dy}{dx} = 3y^{2} - 3x^{2} \] Factoring \( \frac{dy}{dx} \) out: \[ \frac{dy}{dx}(3y^{2} - 6xy) = 3y^{2} - 3x^{2} \] Finally, we can solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{3y^{2} - 3x^{2}}{3y^{2} - 6xy} \] This simplifies to: \[ \frac{dy}{dx} = \frac{y^{2} - x^{2}}{y^{2} - 2xy} \]