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.