Use implicit differentiation to find $y^{\prime}$, and then evaluate $y^{\prime}$ for $x^{2} y-3 x^{2}-4=0$ at the point $(2,4)$. \[ \begin{array}{l} y^{\prime}=\square \\ \left.y^{\prime}\right|_{(2,4)}=\square \text { (Simplify your answer.) } \end{array} \] (Simplify your answer.)

Step 1 ：Given the equation \(x^{2} y-3 x^{2}-4=0\), we can differentiate both sides with respect to \(x\) using the product rule and the chain rule.

Step 2 ：Differentiating \(x^{2} y\) with respect to \(x\) gives \(2xy + x^{2}y^{\prime}\).

Step 3 ：Differentiating \(-3 x^{2}\) with respect to \(x\) gives \(-6x\).

Step 4 ：Differentiating \(-4\) with respect to \(x\) gives \(0\).

Step 5 ：So, the derivative of the entire equation is \(2xy + x^{2}y^{\prime} - 6x = 0\).

Step 6 ：We can solve this equation for \(y^{\prime}\) to find the derivative of \(y\) with respect to \(x\).

Step 7 ：Rearranging the equation gives \(x^{2}y^{\prime} = 6x - 2xy\).

Step 8 ：Dividing both sides by \(x^{2}\) gives \(y^{\prime} = \frac{6x - 2xy}{x^{2}}\).

Step 9 ：This is the derivative of \(y\) with respect to \(x\).

Step 10 ：To evaluate \(y^{\prime}\) at the point \((2,4)\), we substitute \(x = 2\) and \(y = 4\) into the derivative.

Step 11 ：This gives \(y^{\prime} = \frac{6(2) - 2(2)(4)}{(2)^{2}} = -2\).

Step 12 ：So, the derivative of \(y\) with respect to \(x\) at the point \((2,4)\) is \(-2\).

Step 13 ：Therefore, the final answer is \(\boxed{-2}\).