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

Options

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

Answer

To find the derivative of the function \( y = \sin(\sqrt{\sin(\sqrt{x})}) \), we will apply the chain rule multiple times. Here’s the step-by-step solution:

### Step 1: Differentiate the outer function
The outer function is \( \sin(u) \) where \( u = \sqrt{\sin(\sqrt{x})} \). The derivative of \( \sin(u) \) is \( \cos(u) \cdot \frac{du}{dx} \).

\[ \frac{dy}{dx} = \cos(\sqrt{\sin(\sqrt{x})}) \cdot \frac{d}{dx}(\sqrt{\sin(\sqrt{x})}) \]

### Step 2: Differentiate the middle function
Now we need to differentiate \( \sqrt{v} \) where \( v = \sin(\sqrt{x}) \). The derivative of \( \sqrt{v} \) is \( \frac{1}{2\sqrt{v}} \cdot \frac{dv}{dx} \).

\[ \frac{d}{dx}(\sqrt{\sin(\sqrt{x})}) = \frac{1}{2\sqrt{\sin(\sqrt{x})}} \cdot \frac{d}{dx}(\sin(\sqrt{x})) \]

### Step 3: Differentiate the innermost function
Next, we differentiate \( \sin(w) \) where \( w = \sqrt{x} \). The derivative of \( \sin(w) \) is \( \cos(w) \cdot \frac{dw}{dx} \).

\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]

### Step 4: Differentiate \( \sqrt{x} \)
The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \).

Putting it all together:

\[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Combine all parts
Now we can substitute back into our previous expressions:

1. From Step 3:
\[ \frac{d}{dx}(\sqrt{\sin(\sqrt{x})}) = \frac{1}{2\sqrt{\sin(\sqrt{x})}} \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]

2. Substitute this into Step 1:
\[ \frac{dy}{dx} = \cos(\sqrt{\sin(\sqrt{x})}) \cdot \left(\frac{1}{2\sqrt{\sin(\sqrt{x})}} \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right)\right) \]

### Final Expression
Combining everything, we get:

\[ \frac{dy}{dx} = \frac{\cos(\sqrt{\sin(\sqrt{x})}) \cdot \cos(\sqrt{x})}{4\sqrt{x} \cdot \sqrt{\sin(\sqrt{x})}} \]