If `y=xe^(y)," then "(1-y)y_(1)=`

Options

A. `e^(x)`
B. `e^(y)`
C. `ylogx`
D. `xlogy`

Answer

To solve the problem where \( y = x e^y \) and we need to find \( (1 - y) y' \), we will follow these steps:

### Step 1: Differentiate the equation \( y = x e^y \)
We start by differentiating both sides of the equation with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx}(x e^y) \]

### Step 2: Apply the product rule
Using the product rule for differentiation, we have:

\[ \frac{dy}{dx} = e^y \cdot \frac{dx}{dx} + x \cdot \frac{d}{dx}(e^y) \]

Since \( \frac{dx}{dx} = 1 \) and using the chain rule for \( e^y \):

\[ \frac{d}{dx}(e^y) = e^y \cdot \frac{dy}{dx} \]

So, we can rewrite the equation as:

\[ \frac{dy}{dx} = e^y + x e^y \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} - x e^y \frac{dy}{dx} = e^y \]

Factoring out \( \frac{dy}{dx} \):

\[ \frac{dy}{dx}(1 - x e^y) = e^y \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{e^y}{1 - x e^y} \]

### Step 5: Substitute \( y' \) into \( (1 - y) y' \)
Now we need to find \( (1 - y) y' \):

\[ (1 - y) y' = (1 - y) \cdot \frac{e^y}{1 - x e^y} \]

### Step 6: Final expression
Thus, we have:

\[ (1 - y) y' = \frac{(1 - y)e^y}{1 - x e^y} \]

### Final Answer
The value of \( (1 - y) y' \) is:

\[ (1 - y) y' = \frac{(1 - y)e^y}{1 - x e^y} \]

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