derivative of `tan sqrt(x)`

Answer

To find the derivative of the function \( f(x) = \tan(\sqrt{x}) \), we will use the chain rule and the derivative of the tangent function. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
The function can be expressed as:
- Outer function: \( g(u) = \tan(u) \)
- Inner function: \( u = \sqrt{x} \)

### Step 2: Differentiate the outer function
The derivative of \( g(u) = \tan(u) \) with respect to \( u \) is:
\[ g'(u) = \sec^2(u) \]

### Step 3: Differentiate the inner function
The derivative of \( u = \sqrt{x} \) with respect to \( x \) is:
\[ u' = \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} = \frac{1}{2} x^{-1/2} \]

### Step 4: Apply the chain rule
Using the chain rule, the derivative of \( f(x) = g(u) \) is:
\[ f'(x) = g'(u) \cdot u' = \sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Simplify the expression
Thus, we can write:
\[ f'(x) = \frac{\sec^2(\sqrt{x})}{2\sqrt{x}} \]

### Final Answer
The derivative of \( \tan(\sqrt{x}) \) is:
\[ f'(x) = \frac{\sec^2(\sqrt{x})}{2\sqrt{x}} \]

---