Find the derivative of `sin (sin x^(2))`.

Options

A. 1
B. 2
C. 0
D. None of these

Answer

To find the derivative of the function \( y = \sin(\sin(x^2)) \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by \( \frac{dy}{dg} \cdot \frac{dg}{dx} \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Let \( u = \sin(x^2) \) (inner function).
- Then, \( y = \sin(u) \) (outer function).

2. **Differentiate the outer function**:
- The derivative of \( y = \sin(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \cos(u) \]

3. **Differentiate the inner function**:
- Now, differentiate \( u = \sin(x^2) \) with respect to \( x \):
- First, we need to differentiate \( x^2 \):
\[ \frac{d}{dx}(x^2) = 2x \]
- Now, differentiate \( \sin(x^2) \):
\[ \frac{du}{dx} = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

4. **Apply the chain rule**:
- Now, we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot (2x \cos(x^2)) \]

5. **Substitute back for \( u \)**:
- Replace \( u \) with \( \sin(x^2) \):
\[ \frac{dy}{dx} = \cos(\sin(x^2)) \cdot (2x \cos(x^2)) \]

### Final Answer:
Thus, the derivative of \( \sin(\sin(x^2)) \) is:
\[ \frac{dy}{dx} = 2x \cos(x^2) \cos(\sin(x^2)) \]