`y=sqrt(sinsqrt(x))`

Explanation

To find the derivative of the function \( y = \sqrt{\sin(\sqrt{x})} \), we will use the chain rule and the product rule of differentiation. Let's go through the solution step by step.

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

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

### Step 3: Differentiate \( \sin(\sqrt{x}) \)
Next, we need to differentiate \( \sin(\sqrt{x}) \). Again, we apply the chain rule:
\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]
Now, we differentiate \( \sqrt{x} \):
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
Putting it all together, 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 \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2}(\sin(\sqrt{x}))^{-1/2} \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\cos(\sqrt{x})}{4\sqrt{x} \sqrt{\sin(\sqrt{x})}} \]

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