`sqrt(sin x)`

Answer

To find the differential coefficient of the function \( f(x) = \sqrt{\sin x} \), we will differentiate it with respect to \( x \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
We start by rewriting the function in a more convenient form for differentiation:
\[ f(x) = (\sin x)^{1/2} \]

2. **Apply the Power Rule**:
We will use the chain rule for differentiation. According to the chain rule, if \( f(x) = g(h(x)) \), then:
\[ f'(x) = g'(h(x)) \cdot h'(x) \]
Here, let \( g(u) = u^{1/2} \) where \( u = \sin x \).

3. **Differentiate the Outer Function**:
Differentiate \( g(u) = u^{1/2} \):
\[ g'(u) = \frac{1}{2} u^{-1/2} = \frac{1}{2 \sqrt{u}} \]

4. **Differentiate the Inner Function**:
Now, differentiate the inner function \( h(x) = \sin x \):
\[ h'(x) = \cos x \]

5. **Combine the Results**:
Now, we apply the chain rule:
\[ f'(x) = g'(\sin x) \cdot h'(x) = \frac{1}{2 \sqrt{\sin x}} \cdot \cos x \]

6. **Simplify the Expression**:
Thus, we can write the derivative as:
\[ f'(x) = \frac{\cos x}{2 \sqrt{\sin x}} \]

### Final Answer:
The differential coefficient of the function \( \sqrt{\sin x} \) with respect to \( x \) is:
\[ \frac{\cos x}{2 \sqrt{\sin x}} \]