Step 1 :Given the equation $F(x, y)=3 x^{3}+2 x^{2} y-4 y^{3}=0$, we are asked to find $\frac{d y}{d x}$.
Step 2 :First, we need to find the partial derivatives $F_{x}$ and $F_{y}$.
Step 3 :Taking the partial derivative of $F$ with respect to $x$, we get $F_{x}=9 x^{2}+4 x y$.
Step 4 :Taking the partial derivative of $F$ with respect to $y$, we get $F_{y}=2 x^{2}-12 y^{2}$.
Step 5 :According to the implicit function theorem, $\frac{d y}{d x}=-\frac{F_{x}}{F_{y}}$.
Step 6 :Substituting $F_{x}$ and $F_{y}$ into the formula, we get $\frac{d y}{d x}=-\frac{9 x^{2}+4 x y}{2 x^{2}-12 y^{2}}$.
Step 7 :Thus, the derivative of $y$ with respect to $x$ is $\boxed{-\frac{9 x^{2}+4 x y}{2 x^{2}-12 y^{2}}}$.