Differentiate the following w.r.t.x :
`sin(x^(2)+5)`

Answer

To differentiate the function \( y = \sin(x^2 + 5) \) with respect to \( x \), 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 \( u = x^2 + 5 \) (inner function).
- Then, \( y = \sin(u) \) (outer function).

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

3. **Differentiate the inner function with respect to \( x \)**:
- The derivative of \( u = x^2 + 5 \) with respect to \( x \) is:
\[ \frac{du}{dx} = 2x \]

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

5. **Substitute back the value of \( u \)**:
- Replace \( u \) with \( x^2 + 5 \):
\[ \frac{dy}{dx} = 2x \cos(x^2 + 5) \]

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