Differentiation of `sin(x^2)` w.r.t. x is

Options

A. `cos(x^2)`
B. `2xcos(x^2)`
C. `x^2cos(x^2)`
D. `-cos(2x)`

Explanation

To differentiate the function \( \sin(x^2) \) with respect to \( x \), we will use the chain rule of differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), the derivative is given by \( f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Here, the outer function is \( f(u) = \sin(u) \) where \( u = x^2 \).
- The inner function is \( g(x) = x^2 \).

2. **Differentiate the outer function**:
- The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).

3. **Differentiate the inner function**:
- The derivative of \( g(x) = x^2 \) with respect to \( x \) is \( g'(x) = 2x \).

4. **Apply the chain rule**:
- According to the chain rule, the derivative of \( \sin(x^2) \) with respect to \( x \) is:
\[ \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

5. **Write the final answer**:
- Therefore, the derivative of \( \sin(x^2) \) with respect to \( x \) is:
\[ 2x \cos(x^2) \]

### Final Answer:
\[ \frac{d}{dx} \sin(x^2) = 2x \cos(x^2) \]