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

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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)
$$ \frac{dy}{dx} = \frac{2x - e^{y}}{x e^{y}} $$

Given the equation that defines as a function of , use implicit
differentiation to find the derivative .

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Given the equation that defines as a function of , use implicit differentiation to find the derivative .