If $xe^{xy}=y+sin^{2}x$ then $\frac{dy}{dx}$ at $x=0$ is:

Q: If $xe^{xy}=y+sin^{2}x$ then $\frac{dy}{dx}$ at $x=0$ is:

To determine $\frac{dy}{dx}$ at $x = 0$ for the equation $xe^{xy} = y + \sin^2 x$, we need to use implicit differentiation. Let's go through the steps: 1. **Differentiate both sides with respect to $x$:** The left side is $xe^{xy}$. Using the product rule and chain rule, the derivative is: \[ \frac{d}{dx}(xe^{xy}) = e^{xy} + x \cdot e^{xy} \cdot \left(y + x\frac{dy}{dx}\right) \] Simplifying, we get: \[ e^{xy} + xye^{xy} + x^2e^{xy}\frac{dy}{dx} \] The right side is $y + \sin^2 x$. The derivative is: \[ \frac{d}{dx}(y + \sin^2 x) = \frac{dy}{dx} + 2\sin x \cos x \] 2. **Set the derivatives equal:** \[ e^{xy} + xye^{xy} + x^2e^{xy}\frac{dy}{dx} = \frac{dy}{dx} + 2\sin x \cos x \] 3. **Evaluate at $x = 0$:** Substitute $x = 0$ into the equation: - The left side becomes $e^{0 \cdot y} = 1$. - The right side becomes $\frac{dy}{dx} + 0$. Therefore, the equation simplifies to: \[ 1 = \frac{dy}{dx} \] 4. **Conclusion:** At $x = 0$, $\frac{dy}{dx} = 1$. Thus, the correct answer is Option 4. **Explanation for Incorrect Options:** - **Option 1 (3):** This option is incorrect because the differentiation and evaluation at $x = 0$ clearly show that $\frac{dy}{dx} = 1$, not 3. - **Option 2 (-1):** This option is incorrect for the same reason. The differentiation process and substitution at $x = 0$ do not yield $-1$. - **Option 3 (4):** This option is incorrect because the calculations show that $\frac{dy}{dx} = 1$, not 4. In conclusion, the correct answer is indeed Option 4, as the differentiation and evaluation process confirms that $\frac{dy}{dx} = 1$ at $x = 0$.

Question

Accepted Answer

To determine $\frac{dy}{dx}$ at $x = 0$ for the equation $xe^{xy} = y + \sin^2 x$, we need to use implicit differentiation. Let's go through the steps:

1. **Differentiate both sides with respect to $x$:**

The left side is $xe^{xy}$. Using the product rule and chain rule, the derivative is:
   $$
   \frac{d}{dx}(xe^{xy}) = e^{xy} + x \cdot e^{xy} \cdot \left(y + x\frac{dy}{dx}\right)
   $$
   Simplifying, we get:
   $$
   e^{xy} + xye^{xy} + x^2e^{xy}\frac{dy}{dx}
   $$

The right side is $y + \sin^2 x$. The derivative is:
   $$
   \frac{d}{dx}(y + \sin^2 x) = \frac{dy}{dx} + 2\sin x \cos x
   $$

2. **Set the derivatives equal:**

$$
   e^{xy} + xye^{xy} + x^2e^{xy}\frac{dy}{dx} = \frac{dy}{dx} + 2\sin x \cos x
   $$

3. **Evaluate at $x = 0$:**

Substitute $x = 0$ into the equation:
   - The left side becomes $e^{0 \cdot y} = 1$.
   - The right side becomes $\frac{dy}{dx} + 0$.

Therefore, the equation simplifies to:
   $$
   1 = \frac{dy}{dx}
   $$

4. **Conclusion:**

At $x = 0$, $\frac{dy}{dx} = 1$. Thus, the correct answer is Option 4.

**Explanation for Incorrect Options:**

- **Option 1 (3):** This option is incorrect because the differentiation and evaluation at $x = 0$ clearly show that $\frac{dy}{dx} = 1$, not 3.
  
- **Option 2 (-1):** This option is incorrect for the same reason. The differentiation process and substitution at $x = 0$ do not yield $-1$.

- **Option 3 (4):** This option is incorrect because the calculations show that $\frac{dy}{dx} = 1$, not 4.

In conclusion, the correct answer is indeed Option 4, as the differentiation and evaluation process confirms that $\frac{dy}{dx} = 1$ at $x = 0$.