If `xe^(xy)-y=sin^(2)x` then `(dy)/(dx)` at x = 0 is

Options

A. 0
B. 1
C. `-1`
D. none of these

Explanation

To solve the problem, we need to find \( \frac{dy}{dx} \) for the equation \( xe^{xy} - y = \sin^2 x \) at \( x = 0 \).

### Step-by-Step Solution:

1. **Substitute \( x = 0 \) to find \( y \)**:
\[ 0 \cdot e^{0 \cdot y} - y = \sin^2(0) \]
This simplifies to:
\[ -y = 0 \quad \Rightarrow \quad y = 0 \]

2. **Differentiate both sides with respect to \( x \)**:
We will use implicit differentiation on the equation \( xe^{xy} - y = \sin^2 x \).

The left-hand side:
\[ \frac{d}{dx}(xe^{xy}) - \frac{dy}{dx} = \frac{d}{dx}(\sin^2 x) \]

For \( xe^{xy} \), we use the product rule:
\[ \frac{d}{dx}(xe^{xy}) = e^{xy} + x \cdot e^{xy} \cdot \frac{d}{dx}(xy) \]
Now, applying the product rule to \( xy \):
\[ \frac{d}{dx}(xy) = y + x \frac{dy}{dx} \]
Therefore,
\[ \frac{d}{dx}(xe^{xy}) = e^{xy} + x \cdot e^{xy} \cdot (y + x \frac{dy}{dx}) \]

The right-hand side:
\[ \frac{d}{dx}(\sin^2 x) = 2 \sin x \cos x = \sin(2x) \]

3. **Set up the equation**:
Combining both sides, we have:
\[ e^{xy} + x e^{xy} (y + x \frac{dy}{dx}) - \frac{dy}{dx} = \sin(2x) \]

4. **Substitute \( x = 0 \) and \( y = 0 \)**:
Substitute \( x = 0 \) and \( y = 0 \) into the differentiated equation:
\[ e^{0} + 0 \cdot e^{0} (0 + 0 \cdot \frac{dy}{dx}) - \frac{dy}{dx} = \sin(0) \]
This simplifies to:
\[ 1 - \frac{dy}{dx} = 0 \]

5. **Solve for \( \frac{dy}{dx} \)**:
Rearranging gives:
\[ \frac{dy}{dx} = 1 \]

### Final Answer:
Thus, the value of \( \frac{dy}{dx} \) at \( x = 0 \) is \( \boxed{1} \).