Use implicit differentiation to find $y^{\prime}$, and then evaluate $y^{\prime}$ for $x^{2} y-3 x^{3}+8=0$ at the point $(2,4)$.

Step 1 ：The given equation is \(x^{2} y-3 x^{3}+8=0\). We need to find the derivative of \(y\) with respect to \(x\) using implicit differentiation.

Step 2 ：Differentiating both sides of the equation with respect to \(x\), we get \(2xy + x^{2}y' - 9x^{2} = 0\).

Step 3 ：Rearranging the terms, we get \(x^{2}y' = 9x^{2} - 2xy\).

Step 4 ：Dividing both sides by \(x^{2}\), we get \(y' = \frac{9x^{2} - 2xy}{x^{2}}\).

Step 5 ：Now, we need to evaluate \(y'\) at the point \((2,4)\).

Step 6 ：Substituting \(x = 2\) and \(y = 4\) into the equation, we get \(y' = \frac{9(2)^{2} - 2(2)(4)}{(2)^{2}}\).

Step 7 ：Simplifying the above expression, we get \(y' = \frac{36 - 16}{4} = 5\).

Step 8 ：So, the derivative of \(y\) with respect to \(x\) at the point \((2,4)\) is \(5\).

Step 9 ：Checking the result, we find that it satisfies the requirements of the problem.

Step 10 ：Therefore, the final answer is \(\boxed{5}\).