Differentiate the following :
`sin(x^(2)+1)`

Options

A. `x cos (x^(2)+1)`
B. `2x cos (x^(2)+1)`
C. `2x cos (x^(1)+1)`
D. ` cos (x^(2)+1)`

Explanation

To differentiate the function \( y = \sin(x^2 + 1) \), 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:

\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Outer function: \( f(u) = \sin(u) \) where \( u = x^2 + 1 \)
- Inner function: \( g(x) = x^2 + 1 \)

2. **Differentiate the outer function**:
- The derivative of \( f(u) = \sin(u) \) is \( f'(u) = \cos(u) \).

3. **Differentiate the inner function**:
- The derivative of \( g(x) = x^2 + 1 \) is \( g'(x) = 2x \).

4. **Apply the chain rule**:
- Now, using the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \cos(g(x)) \cdot g'(x) \]
- Substitute \( g(x) \) back into the equation:
\[ \frac{dy}{dx} = \cos(x^2 + 1) \cdot 2x \]

5. **Final result**:
- Therefore, the derivative of \( y = \sin(x^2 + 1) \) is:
\[ \frac{dy}{dx} = 2x \cos(x^2 + 1) \]