Determine the partial derivative of the function with respect to x: $f(x, y)=x y e^{-y}$

Step 1 ：Determine the partial derivative of the function with respect to x: \(f(x, y)=x y e^{-y}\)

Step 2 ：The partial derivative of a function with respect to a variable is the derivative of the function with respect to that variable, treating all other variables as constants.

Step 3 ：In this case, we need to find the partial derivative of the function \(f(x, y)=x y e^{-y}\) with respect to \(x\). This involves applying the product rule of differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 4 ：Here, the two functions are \(x\) and \(y e^{-y}\).

Step 5 ：The partial derivative of the function \(f(x, y)=x y e^{-y}\) with respect to \(x\) is \(y e^{-y}\). This is the result of applying the product rule of differentiation, treating \(y e^{-y}\) as a constant when differentiating with respect to \(x\).

Step 6 ：Final Answer: The partial derivative of the function \(f(x, y)=x y e^{-y}\) with respect to \(x\) is \(\boxed{y e^{-y}}\).