2. $\left(\S 3.5,5\right.$ pts each) Find the derivative $y^{\prime}$ of each equation below using implicit differentiation. c. $x e^{y}-y=x^{2}-2$

Step 1 ：First, we need to understand the problem. We are asked to find the derivative of the given equation using implicit differentiation. Implicit differentiation is a method used to find the derivative of a relation. This method is used when it is difficult to solve the equation for y explicitly.

Step 2 ：The given equation is \(x e^{y}-y=x^{2}-2\).

Step 3 ：We will differentiate both sides of the equation with respect to x.

Step 4 ：Differentiating the left side, we get \(\frac{d}{dx}(x e^{y}-y)\). Using the product rule on \(x e^{y}\) and the chain rule on \(e^{y}\), we get \(e^{y} + x e^{y} y^{\prime} - y^{\prime}\).

Step 5 ：Differentiating the right side, we get \(\frac{d}{dx}(x^{2}-2)\), which is \(2x\).

Step 6 ：Setting these two derivatives equal to each other, we get \(e^{y} + x e^{y} y^{\prime} - y^{\prime} = 2x\).

Step 7 ：We can rearrange this equation to solve for \(y^{\prime}\). First, we move the terms involving \(y^{\prime}\) to one side and the rest to the other side. This gives us \(x e^{y} y^{\prime} - y^{\prime} = 2x - e^{y}\).

Step 8 ：We can factor out \(y^{\prime}\) from the left side, giving us \(y^{\prime}(x e^{y} - 1) = 2x - e^{y}\).

Step 9 ：Finally, we can solve for \(y^{\prime}\) by dividing both sides by \(x e^{y} - 1\). This gives us \(y^{\prime} = \frac{2x - e^{y}}{x e^{y} - 1}\).

Step 10 ：This is the derivative of the given equation. We have used implicit differentiation to find it.

Step 11 ：We can check our answer by differentiating the original equation again using our result and seeing if we get the same result. If we do, then our answer is correct.

Step 12 ：The final answer is \(\boxed{y^{\prime} = \frac{2x - e^{y}}{x e^{y} - 1}}\).