If `y= cos sqrtx ,then (dy)/(dx) ` =

Options

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

Answer

To differentiate the function \( y = \cos(\sqrt{x}) \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
In the function \( y = \cos(\sqrt{x}) \), we can identify:
- The outer function is \( \cos(u) \) where \( u = \sqrt{x} \).
- The inner function is \( u = \sqrt{x} \).

### Step 2: Differentiate the outer function
The derivative of the outer function \( \cos(u) \) with respect to \( u \) is:
\[ \frac{d}{du}(\cos(u)) = -\sin(u) \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = \sqrt{x} \):
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

### Step 4: Apply the chain rule
Using the chain rule, we can now find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Simplify the expression
Thus, we can write:
\[ \frac{dy}{dx} = -\frac{\sin(\sqrt{x})}{2\sqrt{x}} \]

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

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