Use implicit differentiation to find \( \frac{d y}{d x} \) \[ x^{3}+y^{3}=12 x y \]

Find the first order derivative with respect to \( x \) for \( x^{3}+y^{3}=12xy \). Find the first derivative by following steps: - step0: Find the derivative with respect to \(x\): \(x^{3}+y^{3}=12xy\) - step1: Take the derivative: \(\frac{d}{dx}\left(x^{3}+y^{3}\right)=\frac{d}{dx}\left(12xy\right)\) - step2: Calculate the derivative: \(3x^{2}+3y^{2}\frac{dy}{dx}=\frac{d}{dx}\left(12xy\right)\) - step3: Calculate the derivative: \(3x^{2}+3y^{2}\frac{dy}{dx}=12y+12x\frac{dy}{dx}\) - step4: Move the expression to the left side: \(3y^{2}\frac{dy}{dx}-12x\frac{dy}{dx}=12y-3x^{2}\) - step5: Add and subtract: \(\left(3y^{2}-12x\right)\frac{dy}{dx}=12y-3x^{2}\) - step6: Divide both sides: \(\frac{\left(3y^{2}-12x\right)\frac{dy}{dx}}{3y^{2}-12x}=\frac{12y-3x^{2}}{3y^{2}-12x}\) - step7: Divide the numbers: \(\frac{dy}{dx}=\frac{4y-x^{2}}{y^{2}-4x}\) The derivative of \( y \) with respect to \( x \) using implicit differentiation is given by: \[ \frac{dy}{dx} = \frac{4y-x^{2}}{y^{2}-4x} \]