Differentiate the following functions w.r.t. x:
`sin "" (x)/(2)`

Answer

To differentiate the function \( y = \sin\left(\frac{x}{2}\right) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
Let:
\[ y = \sin\left(\frac{x}{2}\right) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

In our case, let:
- \( f(u) = \sin(u) \) where \( u = \frac{x}{2} \)

Then:
- The derivative \( f'(u) = \cos(u) \)
- The inner function \( g(x) = \frac{x}{2} \) has the derivative \( g'(x) = \frac{1}{2} \)

### Step 3: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \cos\left(\frac{x}{2}\right) \cdot \frac{1}{2} \]

### Step 4: Write the final result
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{2} \cos\left(\frac{x}{2}\right) \]

So the final answer is:
\[ \frac{dy}{dx} = \frac{1}{2} \cos\left(\frac{x}{2}\right) \]

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