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

Options

A. ` (cos sqrt x)/( 2sqrtsin sqrtx ) `
B. ` (cos sqrt x)/(4sqrt sin sqrtx ) `
C. ` (cos sqrtx)/(2sqrtxsin sqrtx) `
D. ` (cos sqrtx)/( 4sqrt(xsin sqrtx ) )`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \sqrt{\sin(\sqrt{x})} \), we will use the chain rule and the product rule of differentiation. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start with the function:
\[ y = \sqrt{\sin(\sqrt{x})} \]
This can be rewritten as:
\[ y = (\sin(\sqrt{x}))^{1/2} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \), we apply the chain rule. The derivative of \( u^{n} \) is \( n \cdot u^{n-1} \cdot \frac{du}{dx} \). Here, \( u = \sin(\sqrt{x}) \) and \( n = \frac{1}{2} \):
\[ \frac{dy}{dx} = \frac{1}{2} (\sin(\sqrt{x}))^{-1/2} \cdot \frac{d}{dx}(\sin(\sqrt{x})) \]

### Step 3: Differentiate \( \sin(\sqrt{x}) \)
Now we need to differentiate \( \sin(\sqrt{x}) \). Again, we will use the chain rule:
\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]
The derivative of \( \sqrt{x} \) is:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
Thus, we have:
\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 4: Substitute back into the derivative
Now we substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2} (\sin(\sqrt{x}))^{-1/2} \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]

### Step 5: Simplify the expression
Putting it all together, we get:
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{\cos(\sqrt{x})}{2\sqrt{x} \sqrt{\sin(\sqrt{x})}} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\cos(\sqrt{x})}{4\sqrt{x} \sqrt{\sin(\sqrt{x})}} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{\cos(\sqrt{x})}{4\sqrt{x} \sqrt{\sin(\sqrt{x})}} \]