If ` y= sin (x^(2) +5) , then (dy)/(dx) =`

Options

A. ` xcos (x^(2) +5)`
B. ` (x+ 5) cos (x^(2)+ 5) `
C. ` 2xcos (x^(2) +5) `
D. ` (2x+ 5)cos (x^(2) +5)`

Answer

To find the derivative of the function \( y = \sin(x^2 + 5) \), we will use the chain rule of differentiation. 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**:
- Here, the outer function \( f(u) = \sin(u) \) where \( u = g(x) = x^2 + 5 \).

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

3. **Differentiate the inner function**:
- The inner function is \( g(x) = x^2 + 5 \).
- The derivative of \( g(x) \) is \( g'(x) = 2x \).

4. **Apply the chain rule**:
- Now, using the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \cos(x^2 + 5) \cdot 2x \]

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

### Final Answer:
\[ \frac{dy}{dx} = 2x \cos(x^2 + 5) \]