Given the equation $x{e}^y=x-y$, find $\frac{dy}{dx}$ using implicit differentiation.

• A. $\frac{1-e^y}{x{e}^y+1}$
• B. $\frac{x{e}^y+1}{1-e^y}$
• C. $\frac{e^y+1}{1-x{e}^y}$
• D. $\frac{1+e^y}{x{e}^y-1}$

Answer: (A)

Verified step by step guidance

1. Step 1: Start by recognizing that the equation x e^y = x - y involves both x and y, so implicit differentiation is required. Differentiate both sides of the equation with respect to x.
2. Step 2: Apply the product rule to the term x e^y on the left-hand side. Recall that the product rule states d(uv)/dx = u dv/dx + v du/dx. Here, u = x and v = e^y.
3. Step 3: Differentiate e^y with respect to x. Since y is a function of x, use the chain rule: d(e^y)/dx = e^y * dy/dx.
4. Step 4: Differentiate the right-hand side of the equation x - y. The derivative of x is 1, and the derivative of -y is -dy/dx.
5. Step 5: Combine all differentiated terms into a single equation, isolate dy/dx, and simplify algebraically to express dy/dx in terms of x, y, and e^y.