`tan sqrt(x)`

Answer

To find the differential coefficient of the function \( \tan(\sqrt{x}) \) with respect to \( x \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have the function \( y = \tan(\sqrt{x}) \).

### Step 2: Differentiate using the chain rule
The chain rule states that if you have a function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, \( f(g) = \tan(g) \) where \( g(x) = \sqrt{x} \).

### Step 3: Differentiate \( f(g) = \tan(g) \)
The derivative of \( \tan(g) \) with respect to \( g \) is:
\[ f'(g) = \sec^2(g) \]
Thus, substituting \( g = \sqrt{x} \):
\[ f'(\sqrt{x}) = \sec^2(\sqrt{x}) \]

### Step 4: Differentiate \( g(x) = \sqrt{x} \)
Now, we need to differentiate \( g(x) = \sqrt{x} \):
\[ g'(x) = \frac{1}{2\sqrt{x}} \]

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

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

### Summary of the solution:
The differential coefficient of \( \tan(\sqrt{x}) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{\sec^2(\sqrt{x})}{2\sqrt{x}} \]