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The question asks to calculate the derivative of \( y = \sqrt{x + \sqrt{x}} \), and it refers to the use of the **Chain Rule** in differentiation.

Let's analyze this:

### Step 1: Rewrite the function for clarity.
We have:
\[
y = \sqrt{x + \sqrt{x}} = (x + \sqrt{x})^{1/2}
\]
This involves two layers: the outer square root (power of \(1/2\)) and the inner function \(x + \sqrt{x}\).

### Step 2: Differentiate using the Chain Rule.
The Chain Rule tells us to differentiate the outer function first and then multiply by the derivative of the inner function.

1. **Outer function**: \( (x + \sqrt{x})^{1/2} \)
   - Derivative of \( u^{1/2} \) (where \( u = x + \sqrt{x} \)) is:
   \[
   \frac{1}{2}(x + \sqrt{x})^{-1/2}
   \]

2. **Inner function**: \( x + \sqrt{x} \)
   - The derivative of \( x \) is 1.
   - The derivative of \( \sqrt{x} \) is:
   \[
   \frac{1}{2}x^{-1/2}
   \]

### Step 3: Combine the results.
Now, multiply the derivative of the outer function by the derivative of the inner function:
\[
\frac{1}{2}(x + \sqrt{x})^{-1/2} \cdot \left( 1 + \frac{1}{2}x^{-1/2} \right)
\]

This is the derivative of \( y = \sqrt{x + \sqrt{x}} \).

### Choosing the correct answer:
From the options listed (though the image doesn't show all clearly), it seems the Chain Rule is applied more than once, first on the square root and then on the inner \( \sqrt{x} \). Based on the breakdown above, the correct choice should involve this sequence of functions:
- First, differentiate the outer \( \sqrt{\cdot} \).
- Then, differentiate the inner function which contains another \( \sqrt{x} \).

It seems like **Option A** (though not fully visible) would align with this approach, as it suggests defining the functions \( g(x) = \sqrt{x} \) and then \( f(u) = \sqrt{x + u} \), applying the Chain Rule multiple times.

Would you like more details or have any questions about this? Here are a few related questions to consider:

1. How does the Chain Rule apply to other composite functions?
2. What are some examples of multi-step chain rule applications?
3. How can you simplify expressions after applying the Chain Rule?
4. What role does implicit differentiation play in problems like this?
5. Can this problem be approached using another differentiation technique?

**Tip:** When applying the Chain Rule, always start from the outermost function and work your way inward, applying the rule recursively.

Calculate the derivative of y = sqrt(x + sqrt(x)). If the Chain Rule only needs to be used once, which functions should be used? If it must be used more than once, which functions should be used the first time?

Learn how to calculate the derivative of y = sqrt(x + sqrt(x)) using the Chain Rule. This problem applies the Chain Rule multiple times, covering key differentiation concepts. Suitable for college-level calculus students.

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