`d/dx(sinx^2)=?`

Options

A. `2x cosx^2`
B. `2x sin x^2`
C. 2xcosx
D. 2xcos(2x)

Answer

To find the derivative of the function \( \sin(x^2) \) with respect to \( x \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

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

2. **Differentiate the outer function**:
- The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).
- Therefore, \( \frac{d}{du}(\sin(u)) = \cos(u) \).

3. **Differentiate the inner function**:
- The derivative of \( u = x^2 \) with respect to \( x \) is \( 2x \).
- Therefore, \( \frac{du}{dx} = 2x \).

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

5. **Combine the results**:
- Thus, the final result is:
\[ \frac{d}{dx}(\sin(x^2)) = 2x \cos(x^2) \]

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

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