Differentiation of cos `(sqrt(x))` with respect to x is

Options

A. `-sinsqrt(x)`
B. `-(1)/(2sqrt(x))sinsqrt(x)`
C. `-sqrt(x)sinsqrt(x)`
D. `-(1)/(sqrt(x))sinsqrt(x)`

Answer

To differentiate \( \cos(\sqrt{x}) \) with respect to \( x \), 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 \( f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Let \( y = \cos(\sqrt{x}) \).
- Here, the outer function is \( f(\theta) = \cos(\theta) \) and the inner function is \( g(x) = \sqrt{x} \).

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

3. **Differentiate the inner function**:
- The inner function is \( g(x) = \sqrt{x} = x^{1/2} \).
- Using the power rule, the derivative \( g'(x) \) is:
\[ g'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}. \]

4. **Apply the chain rule**:
- According to the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x). \]
- Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}. \]

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

### Final Answer:
The differentiation of \( \cos(\sqrt{x}) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{\sin(\sqrt{x})}{2\sqrt{x}}. \]