Question

question 4
consider the curve in the xy - plane given by $y^{3}=9x - 6y$. find $\frac{dy}{dx}$ by implicit differentiation.
$\frac{dy}{dx}=\frac{9}{y^{2}+6}$
$\frac{dy}{dx}=\frac{3}{y^{2}+2}$
$\frac{dy}{dx}=\frac{9x}{y^{3}+2}$
$\frac{dy}{dx}=\frac{9}{y^{3}+6y}$

Explanation:

Step1: Differentiate both sides

Differentiate $y^{3}=9x - 6y$ with respect to $x$. Using the chain - rule on the left - hand side ($\frac{d}{dx}(y^{3}) = 3y^{2}\frac{dy}{dx}$) and the power rule on the right - hand side ($\frac{d}{dx}(9x)=9$ and $\frac{d}{dx}(-6y)=-6\frac{dy}{dx}$). So we get $3y^{2}\frac{dy}{dx}=9 - 6\frac{dy}{dx}$.

Step2: Isolate $\frac{dy}{dx}$

Add $6\frac{dy}{dx}$ to both sides: $3y^{2}\frac{dy}{dx}+6\frac{dy}{dx}=9$. Factor out $\frac{dy}{dx}$ on the left - hand side: $\frac{dy}{dx}(3y^{2}+6)=9$. Then solve for $\frac{dy}{dx}$ by dividing both sides by $3y^{2}+6$: $\frac{dy}{dx}=\frac{9}{3y^{2}+6}=\frac{3}{y^{2}+2}$.

Answer:

$\frac{dy}{dx}=\frac{3}{y^{2}+2}$ (the second option)