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

Options

A. `-x`
B. x
C. y
D. `-y`

Answer

To solve the problem where \( y = x e^y \), we need to find \( x(1 - y) y' \), where \( y' = \frac{dy}{dx} \).

### Step 1: Differentiate the equation \( y = x e^y \)

We will use implicit differentiation on both sides of the equation.

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

### Step 2: 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 3: Solve for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):

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

### Step 4: Substitute \( \frac{dy}{dx} \) into \( x(1 - y) y' \)

Now we will substitute \( y' = \frac{dy}{dx} \) into the expression \( x(1 - y) y' \):

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

### Step 5: Simplify the expression

Now we can simplify the expression:

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

### Final Result

Thus, the final result for \( x(1 - y) y' \) is:

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