Derivative of `cossqrtx` is :

Options

A. `-sinsqrtx/(2sqrtx)`
B. `sinsqrtx/(2sqrtx)`
C. `-sinsqrtx/(sqrtx)`
D. `-sinsqrtx/(2)`

Answer

To find the derivative of \( \cos(\sqrt{x}) \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:

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

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Let \( f(u) = \cos(u) \) where \( u = \sqrt{x} \).
- Thus, we have \( g(x) = \sqrt{x} \).

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

3. **Differentiate the inner function**:
- The derivative of \( g(x) = \sqrt{x} \) is:
\[ g'(x) = \frac{1}{2\sqrt{x}} \]

4. **Apply the chain rule**:
- Now, applying the chain rule:
\[ \frac{d}{dx} \cos(\sqrt{x}) = f'(g(x)) \cdot g'(x) = -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

5. **Combine the results**:
- Therefore, the derivative of \( \cos(\sqrt{x}) \) is:
\[ \frac{d}{dx} \cos(\sqrt{x}) = -\frac{\sin(\sqrt{x})}{2\sqrt{x}} \]

### Final Answer:
The derivative of \( \cos(\sqrt{x}) \) is:
\[ -\frac{\sin(\sqrt{x})}{2\sqrt{x}} \]