`sqrt(tanx)`

Answer

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

### Step 1: Rewrite the Function
We start by rewriting the function in a form that is easier to differentiate:
\[ y = \sqrt{\tan x} = (\tan x)^{1/2} \]

### Step 2: Differentiate Using the Chain Rule
Now we will differentiate \( y \) with respect to \( x \). We will use the chain rule for differentiation, which states that if \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let \( f(u) = u^{1/2} \) where \( u = \tan x \).

### Step 3: Differentiate the Outer Function
First, we differentiate the outer function \( f(u) = u^{1/2} \):
\[ f'(u) = \frac{1}{2}u^{-1/2} = \frac{1}{2\sqrt{u}} = \frac{1}{2\sqrt{\tan x}} \]

### Step 4: Differentiate the Inner Function
Next, we differentiate the inner function \( g(x) = \tan x \):
\[ g'(x) = \sec^2 x \]

### Step 5: Apply the Chain Rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \frac{1}{2\sqrt{\tan x}} \cdot \sec^2 x \]

### Step 6: Write the Final Answer
Thus, the derivative of \( y = \sqrt{\tan x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{\sec^2 x}{2\sqrt{\tan x}} \]