`sqrt(secx)`

Answer

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

### Step 1: Rewrite the function
We can rewrite the function using exponent notation:
\[ y = (\sec x)^{1/2} \]

### Step 2: Apply the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, \( f(u) = u^{1/2} \) where \( u = \sec x \).

### Step 3: Differentiate the outer function
First, we find the derivative of the outer function \( f(u) = u^{1/2} \):
\[ f'(u) = \frac{1}{2} u^{-1/2} = \frac{1}{2 \sqrt{u}} \]

### Step 4: Differentiate the inner function
Next, we need to differentiate the inner function \( g(x) = \sec x \):
\[ g'(x) = \sec x \tan x \]

### Step 5: Combine the derivatives
Now we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \frac{1}{2 \sqrt{\sec x}} \cdot (\sec x \tan x) \]

### Step 6: Simplify the expression
Now, we simplify the expression:
\[ \frac{dy}{dx} = \frac{\sec x \tan x}{2 \sqrt{\sec x}} \]

### Step 7: Rewrite in terms of \( \sqrt{\sec x} \)
We can also express this as:
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \sqrt{\sec x} \cdot \tan x \]

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