`"If "xy=e^((x-y))," then find "(dy)/(dx).`

Explanation

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

### Step 1: Differentiate both sides with respect to \( x \)

We start with the equation:
\[ xy = e^{(x - y)} \]

Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(e^{(x - y)}) \]

### Step 2: Apply the product rule on the left side

Using the product rule on the left side:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]

### Step 3: Differentiate the right side using the chain rule

For the right side, we apply the chain rule:
\[ \frac{d}{dx}(e^{(x - y)}) = e^{(x - y)} \left( \frac{d}{dx}(x - y) \right) = e^{(x - y)} \left( 1 - \frac{dy}{dx} \right) \]

### Step 4: Set the derivatives equal to each other

Now we have:
\[ x \frac{dy}{dx} + y = e^{(x - y)} \left( 1 - \frac{dy}{dx} \right) \]

### Step 5: Expand and rearrange the equation

Expanding the right side:
\[ x \frac{dy}{dx} + y = e^{(x - y)} - e^{(x - y)} \frac{dy}{dx} \]

Now, rearranging the equation to isolate terms involving \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} + e^{(x - y)} \frac{dy}{dx} = e^{(x - y)} - y \]

### Step 6: Factor out \( \frac{dy}{dx} \)

Factoring \( \frac{dy}{dx} \) from the left side:
\[ \frac{dy}{dx} (x + e^{(x - y)}) = e^{(x - y)} - y \]

### Step 7: Solve for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{e^{(x - y)} - y}{x + e^{(x - y)}} \]

### Final Result

Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{e^{(x - y)} - y}{x + e^{(x - y)}} \]

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