If ` y=1 +xe^(y) ,then (dy)/(dx) =`

Options

A. ` (e^(y))/( y-2)`
B. ` (e^(y))/( 2-y) `
C. ` (xe^(y))/( y-2)`
D. ` (xe^(y))/( 2-y)`

Answer

To solve the equation \( y = 1 + x e^y \) and find \( \frac{dy}{dx} \), we will use implicit differentiation. Here are the steps:

### Step-by-Step Solution:

1. **Differentiate both sides with respect to \( x \)**:
\[ \frac{d}{dx}(y) = \frac{d}{dx}(1 + x e^y) \]
This gives:
\[ \frac{dy}{dx} = 0 + \frac{d}{dx}(x e^y) \]

2. **Apply the product rule to \( x e^y \)**:
The product rule states that if you have two functions \( u \) and \( v \), then \( \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \).
Here, let \( u = x \) and \( v = e^y \). Thus:
\[ \frac{d}{dx}(x e^y) = x \frac{d}{dx}(e^y) + e^y \frac{d}{dx}(x) \]

3. **Differentiate \( e^y \)** using the chain rule:
\[ \frac{d}{dx}(e^y) = e^y \frac{dy}{dx} \]
Therefore, substituting back:
\[ \frac{d}{dx}(x e^y) = x e^y \frac{dy}{dx} + e^y \]

4. **Substituting back into the equation**:
Now we have:
\[ \frac{dy}{dx} = x e^y \frac{dy}{dx} + e^y \]

5. **Rearranging the equation**:
To isolate \( \frac{dy}{dx} \), we rearrange:
\[ \frac{dy}{dx} - x e^y \frac{dy}{dx} = e^y \]
Factor out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(1 - x e^y) = e^y \]

6. **Solving for \( \frac{dy}{dx} \)**:
Finally, divide both sides by \( (1 - x e^y) \):
\[ \frac{dy}{dx} = \frac{e^y}{1 - x e^y} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{e^y}{1 - x e^y} \]