Given the equation \( -x^{2}+x e^{y}=-7 \) that defines \( y \) as a function of \( x \), use implicit differentiation to find the derivative \( \frac{d y}{d x} \). \( \frac{d y}{d x}= \)

To find \(\frac{dy}{dx}\) using implicit differentiation for the equation: \[ -x^{2} + x e^{y} = -7 \] **Step 1: Differentiate Both Sides with Respect to \(x\)** Differentiate each term separately: 1. **Differentiate \(-x^2\):** \[ \frac{d}{dx}(-x^2) = -2x \] 2. **Differentiate \(x e^y\):** Use the product rule \(\frac{d}{dx}(uv) = u'\,v + u\,v'\): \[ \frac{d}{dx}(x e^y) = \frac{d}{dx}(x) \cdot e^y + x \cdot \frac{d}{dx}(e^y) = e^y + x e^y \frac{dy}{dx} \] 3. **Differentiate \(-7\):** \[ \frac{d}{dx}(-7) = 0 \] **Step 2: Combine the Differentiated Terms** Putting it all together: \[ -2x + e^y + x e^y \frac{dy}{dx} = 0 \] **Step 3: Solve for \(\frac{dy}{dx}\)** Rearrange the equation to isolate \(\frac{dy}{dx}\): \[ x e^y \frac{dy}{dx} = 2x - e^y \] \[ \frac{dy}{dx} = \frac{2x - e^y}{x e^y} \] **Final Answer:** \[ \frac{dy}{dx} = \frac{2x - e^{y}}{x\, e^{y}} \] **Answer:** After simplifying, dy/dx is (2 x – eʸ) divided by (x times eʸ). Thus, dy/dx = (2x − e^y)/(x e^y)