The derivative of `sin ^(2) sqrtx ` with respect to x is :

Options

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

Answer

To find the derivative of \( \sin^2(\sqrt{x}) \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let \( y = \sin^2(\sqrt{x}) \). We can identify the outer function as \( u^2 \) where \( u = \sin(\sqrt{x}) \), and the inner function as \( \sqrt{x} \).

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = 2u = 2\sin(\sqrt{x}) \]

### Step 3: Differentiate the inner function
Next, we differentiate \( u = \sin(\sqrt{x}) \) with respect to \( x \):
\[ \frac{du}{dx} = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]
Now, we need to differentiate \( \sqrt{x} \):
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
Thus, we have:
\[ \frac{du}{dx} = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 4: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 2\sin(\sqrt{x}) \cdot \left(\cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\sin(\sqrt{x}) \cos(\sqrt{x})}{\sqrt{x}} \]

### Step 5: Use the double angle identity
Using the double angle identity for sine, we can express this as:
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]
Thus, we can write:
\[ \sin(\sqrt{x}) \cos(\sqrt{x}) = \frac{1}{2} \sin(2\sqrt{x}) \]
So, we have:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \sin(2\sqrt{x}) \]

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

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