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The problem you have provided asks for the derivative \( \frac{dy}{dx} \) where:

\[
y = \sqrt{\tan(2x) + \sqrt{x}}
\]

### Solution:
We'll use the chain rule to differentiate the given function.

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

2. **Differentiate** \( y \) with respect to \( x \):
   Using the chain rule, we first differentiate the outer function (which is a square root or power of \( \frac{1}{2} \)):

   \[
   \frac{dy}{dx} = \frac{1}{2} \left( \tan(2x) + \sqrt{x} \right)^{-1/2} \cdot \frac{d}{dx} \left( \tan(2x) + \sqrt{x} \right)
   \]

3. **Differentiate the inner function** \( \tan(2x) + \sqrt{x} \):

   - The derivative of \( \tan(2x) \) using the chain rule is:
     \[
     \frac{d}{dx} \left( \tan(2x) \right) = 2 \sec^2(2x)
     \]
   
   - The derivative of \( \sqrt{x} = x^{1/2} \) is:
     \[
     \frac{d}{dx} \left( \sqrt{x} \right) = \frac{1}{2}x^{-1/2}
     \]

4. **Substitute the derivatives into the equation**:
   \[
   \frac{dy}{dx} = \frac{1}{2} \left( \tan(2x) + \sqrt{x} \right)^{-1/2} \cdot \left( 2 \sec^2(2x) + \frac{1}{2}x^{-1/2} \right)
   \]

This is the derivative of the given function.

Would you like more details on any of these steps?

Here are five questions that extend this problem:

1. How does the chain rule apply when differentiating composite functions?
2. What is the derivative of the tangent function, and why does it involve the secant function?
3. Can you rewrite the final derivative in a simpler form by factoring out common terms?
4. How does the presence of a square root function affect the differentiation process?
5. How would you handle the derivative if the function involved \( \tan^{-1}(x) \) instead of \( \tan(x) \)?

### Tip:
When dealing with square roots in differentiation, always remember to rewrite them as powers (e.g., \( \sqrt{x} = x^{1/2} \)) to make use of the power rule.

Find dy/dx where y = sqrt(tan(2x) + sqrt(x))

Learn how to differentiate the function y = sqrt(tan(2x) + sqrt(x)) using the chain rule. This solution involves differentiating composite functions and applying trigonometric derivatives. Suitable for students in advanced calculus classes, Grades 11-12.

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