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To differentiate the function \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \) using the power rule, let's rewrite each term in terms of exponents:

1. **Rewrite the function**:
   - The square root of \( x \) can be written as \( x^{1/2} \).
   - The reciprocal of the square root of \( x \) can be written as \( x^{-1/2} \).
   
   Thus, the function becomes:
   \[
   y = x^{1/2} + x^{-1/2}
   \]

2. **Differentiate each term**:
   Using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \), we differentiate each term separately.

   - For \( x^{1/2} \):
     \[
     \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{1/2 - 1} = \frac{1}{2} x^{-1/2} = \frac{1}{2 \sqrt{x}}
     \]

   - For \( x^{-1/2} \):
     \[
     \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2} x^{-1/2 - 1} = -\frac{1}{2} x^{-3/2} = -\frac{1}{2x^{3/2}}
     \]

3. **Combine the results**:
   Putting it all together, the derivative of \( y \) with respect to \( x \) is:
   \[
   y' = \frac{1}{2 \sqrt{x}} - \frac{1}{2x^{3/2}}
   \]

So, the final answer is:
\[
\boxed{y' = \frac{1}{2 \sqrt{x}} - \frac{1}{2x^{3/2}}}
\]

Would you like more details or have any questions?

### Related Questions
1. How do we apply the power rule to other types of roots, like cube roots?
2. How can we simplify expressions after differentiating using the power rule?
3. What happens if we differentiate the same function again (find the second derivative)?
4. How would the solution change if there was a constant multiplied to each term?
5. How does the power rule differ when applied to negative exponents?

### Tip
When differentiating terms with roots, it's often easier to rewrite them as exponents before applying the power rule.

Differentiate y = √x + 1/√x using the power rule.

Learn how to differentiate the function y = √x + 1/√x using the power rule. This solution breaks down each step of the process, converting roots to exponents and applying the power rule to find the derivative. Suitable for high school calculus students.

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