You seem to be asking for the expression \(\frac{f(x+h) - f(x)}{h}\) where \(f(x) = \sqrt{x}\). This expression represents the difference quotient, which is used to calculate the derivative of a function.

Let's simplify the difference quotient step by step:

Given \(f(x) = \sqrt{x}\), the difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = \frac{\sqrt{x+h} - \sqrt{x}}{h}
\]

To simplify this expression, we can multiply the numerator and the denominator by the conjugate of the numerator to eliminate the square roots. The conjugate of \(\sqrt{x+h} - \sqrt{x}\) is \(\sqrt{x+h} + \sqrt{x}\). So we multiply:

\[
\frac{\sqrt{x+h} - \sqrt{x}}{h} \times \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} = \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}
\]

Using the difference of squares, the numerator simplifies as follows:

\[
(\sqrt{x+h})^2 - (\sqrt{x})^2 = (x+h) - x = h
\]

So, the expression becomes:

\[
\frac{h}{h(\sqrt{x+h} + \sqrt{x})}
\]

The \(h\) in the numerator and denominator cancels out:

\[
\frac{1}{\sqrt{x+h} + \sqrt{x}}
\]

So the simplified difference quotient is:

\[
\frac{1}{\sqrt{x+h} + \sqrt{x}}
\]

This expression approaches the derivative of \(\sqrt{x}\) as \(h\) approaches 0.

Do you have any questions or need further clarification? Here are some related questions you might explore:

1. What is the derivative of \(f(x) = \sqrt{x}\)?
2. How would you find the derivative using the definition of a derivative?
3. What happens if you set \(h = 0\) in the simplified expression?
4. How do you find the derivative of \(\sqrt{x}\) using the power rule?
5. Can you find the second derivative of \(\sqrt{x}\)?

**Tip:** Multiplying by the conjugate is a common technique used to simplify expressions involving square roots.

Explore the difference quotient \(\frac{f(x+h) - f(x)}{h}\) for the function \(f(x) = \sqrt{x}\). Learn how to simplify the expression step-by-step to find the derivative of \(\sqrt{x}\). Suitable for advanced high school students and anyone interested in calculus fundamentals.

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