Problem

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

Solution

Step 1 :Differentiate the given equation implicitly with respect to x. This involves applying the product rule to the term \(x^{2} y\) and then differentiating the remaining terms normally. The differentiated equation is \(2xy - 6x\).

Step 2 :Solve the resulting equation for \(y^{\prime}\) to find the derivative of y with respect to x. The derivative, \(y^{\prime}\), is \(2x(y - 3)\).

Step 3 :Substitute the coordinates of the given point into the expression for \(y^{\prime}\) to find the value of the derivative at that point. The value of \(y^{\prime}\) at the point \((2,4)\) is \(4\).

Step 4 :So, the final answer is: \(y^{\prime}=2x(y - 3)\) and \(\boxed{\left.y^{\prime}\right|_{(2,4)}=4}\).

From Solvely APP
Source: https://solvelyapp.com/problems/7332/

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