Derivative of ` sqrt( tan sqrt(x))` with respect to x is

Options

A. `(sec^(2) sqrt(x))/(4sqrt(x)sqrttansqrt(x))`
B. `(sec^(2) sqrt(x))/(4sqrt(x)tansqrt(x))`
C. `(sec^(2) sqrt(x))/(sqrt(x)sqrttansqrt(x))`
D. `(4sec^(2) sqrt(x))/(sqrt(x)sqrttansqrt(x))`

Explanation

To find the derivative of \( y = \sqrt{\tan(\sqrt{x})} \) with respect to \( x \), we will use the chain rule and the derivative formulas for trigonometric and square root functions.

### Step-by-Step Solution:

1. **Rewrite the function**:
\[ y = (\tan(\sqrt{x}))^{1/2} \]

2. **Differentiate using the chain rule**:
Using the chain rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2} (\tan(\sqrt{x}))^{-1/2} \cdot \frac{d}{dx}[\tan(\sqrt{x})] \]

3. **Differentiate \( \tan(\sqrt{x}) \)**:
To differentiate \( \tan(\sqrt{x}) \), we again use the chain rule:
\[ \frac{d}{dx}[\tan(\sqrt{x})] = \sec^2(\sqrt{x}) \cdot \frac{d}{dx}[\sqrt{x}] \]

4. **Differentiate \( \sqrt{x} \)**:
The derivative of \( \sqrt{x} \) is:
\[ \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} \]

5. **Combine the derivatives**:
Now, substituting back, we have:
\[ \frac{d}{dx}[\tan(\sqrt{x})] = \sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

6. **Substitute back into the derivative of \( y \)**:
Now substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2} (\tan(\sqrt{x}))^{-1/2} \cdot \left(\sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]

7. **Simplify the expression**:
Thus, we can simplify:
\[ \frac{dy}{dx} = \frac{1}{4\sqrt{x}} (\tan(\sqrt{x}))^{-1/2} \sec^2(\sqrt{x}) \]

### Final Answer:
The derivative of \( \sqrt{\tan(\sqrt{x})} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{\sec^2(\sqrt{x})}{4\sqrt{x} \sqrt{\tan(\sqrt{x})}} \]