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

Options

A. `((1+xy)y)/((1-xy)x)`
B. `(1+xy)/(1-xy)`
C. `((1-xy)y)/((1+xy)x)`
D. `(1-xy)/(1+xy)`

Answer

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

### Step 1: Differentiate both sides with respect to \(x\)
Starting with the equation:
\[ y = x e^{xy} \]
We will differentiate both sides with respect to \(x\). Remember that \(y\) is a function of \(x\), so we will use the product rule and chain rule.

### Step 2: Apply the product rule
Differentiating the right-hand side using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) \cdot e^{xy} + x \cdot \frac{d}{dx}(e^{xy}) \]
The derivative of \(x\) is \(1\), so we have:
\[ \frac{dy}{dx} = e^{xy} + x \cdot \frac{d}{dx}(e^{xy}) \]

### Step 3: Differentiate \(e^{xy}\) using the chain rule
Using the chain rule for \(e^{xy}\):
\[ \frac{d}{dx}(e^{xy}) = e^{xy} \cdot \frac{d}{dx}(xy) \]
Now, we need to differentiate \(xy\) using the product rule:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]
So,
\[ \frac{d}{dx}(e^{xy}) = e^{xy} (x \frac{dy}{dx} + y) \]

### Step 4: Substitute back into the equation
Now substituting back into our equation:
\[ \frac{dy}{dx} = e^{xy} + x \cdot e^{xy} (x \frac{dy}{dx} + y) \]
This simplifies to:
\[ \frac{dy}{dx} = e^{xy} + x e^{xy} (x \frac{dy}{dx} + y) \]

### Step 5: Rearranging the equation
Now we can rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} - x^2 e^{xy} \frac{dy}{dx} = e^{xy} + x y e^{xy} \]
Factoring out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} (1 - x^2 e^{xy}) = e^{xy} + x y e^{xy} \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Finally, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{e^{xy} + x y e^{xy}}{1 - x^2 e^{xy}} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{e^{xy}(1 + xy)}{1 - x^2 e^{xy}} \]