Differentiate x sin x

Options

A. `sinx - x cos x `
B. `sinx + 2 x cosx`
C. `sinx +x cos x`
D. `sin2x-2xcosx`

Answer

To differentiate the function \( y = x \sin x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}(u \cdot v) = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-Step Solution:

1. **Identify the functions**:
- Let \( u = x \) and \( v = \sin x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = 1 \]
- The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = \cos x \]

3. **Apply the product rule**:
- According to the product rule:
\[ \frac{d}{dx}(x \sin x) = u \frac{dv}{dx} + v \frac{du}{dx} \]
- Substitute the values:
\[ = x \cdot \cos x + \sin x \cdot 1 \]

4. **Simplify the expression**:
- Combine the terms:
\[ = x \cos x + \sin x \]

5. **Write the final answer**:
- Thus, the derivative of \( y = x \sin x \) is:
\[ \frac{dy}{dx} = \sin x + x \cos x \]

### Conclusion:
The correct answer is \( \sin x + x \cos x \).