Differentiate the following w.r.t.x :
`sin(x^(2))`

Answer

To differentiate the function \( \sin(x^2) \) with respect to \( x \), we will apply the chain rule. Here are the steps:

### Step-by-Step Solution

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

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

3. **Differentiate the inner function**:
- Now, we need to find \( \frac{dv}{dx} \).
- Since \( v = x^2 \), we differentiate it:
\[ \frac{dv}{dx} = 2x. \]

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

5. **Final result**:
- Therefore, the derivative of \( \sin(x^2) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2x \cos(x^2). \]

### Summary of the Solution
The derivative of \( \sin(x^2) \) with respect to \( x \) is \( 2x \cos(x^2) \).