Question

If \(x\,e^{xy} = y+sin^2x\), then the value of \(\frac{dy}{dx}\) at \(x = 0\) is equal to

• A. \(-1\)
• B. \(1\)
• C. \(-2\)
• D. \(2\)

Hint:

Find y at x = 0 first, then differentiate both sides.

Answer 1

The Correct Option is B

Step 1: Expand near zero:
Near $x = 0$, $y$ is small, so $e^{xy} \approx 1$ and the equation reads $x \approx y + x^2$, giving $y \approx x$.

Step 2: Slope:
So $\frac{dy}{dx}$ at $0$ is $1$, in agreement with the exact differentiation.

Final Answer:
The value is $1$, option (B). \[ \boxed{1} \]