`sqrt(sec x) `

Explanation

To differentiate the function \( y = \sqrt{\sec x} \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in a more convenient form for differentiation:
\[ y = \sec x^{1/2} \]

### Step 2: Apply the Chain Rule
To differentiate \( y \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Here, let \( u = \sec x \), so we can express \( y \) as:
\[ y = u^{1/2} \]

### Step 3: Differentiate using the power rule
Now, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{1}{2} u^{-1/2} \]

### Step 4: Differentiate \( u = \sec x \)
Next, we need to find \( \frac{du}{dx} \). The derivative of \( \sec x \) is:
\[ \frac{du}{dx} = \sec x \tan x \]

### Step 5: Combine using the Chain Rule
Now we can combine these results using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{2} u^{-1/2} \cdot \sec x \tan x \]

### Step 6: Substitute back \( u = \sec x \)
Substituting back \( u = \sec x \) into the equation gives:
\[ \frac{dy}{dx} = \frac{1}{2} (\sec x)^{-1/2} \cdot \sec x \tan x \]

### Step 7: Simplify the expression
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{1}{2} \sec x \tan x \cdot \frac{1}{\sqrt{\sec x}} = \frac{1}{2} \tan x \sqrt{\sec x} \]

### Final Answer
Thus, the derivative of \( y = \sqrt{\sec x} \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \tan x \sqrt{\sec x} \]

---