`(1)` If `f(x) = sqrt(x)`, prove that :
`(f(x + h) - f(x))/(h)` = `(1)/(sqrt(x + h) + sqrtx)`.
`(2)` If `f(x) = x^(2)`, find :
`(f(1.1) - f(1))/(1.1 - 1)`.

Answer

Let's solve the given questions step by step.

### Question 1:
Prove that:

\[ \frac{f(x + h) - f(x)}{h} = \frac{1}{\sqrt{x + h} + \sqrt{x}} \]

where \( f(x) = \sqrt{x} \).

**Step 1:** Write down the left-hand side (LHS):

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

**Step 2:** Rationalize the numerator:

To rationalize, we multiply the numerator and denominator by the conjugate of the numerator:

\[ \text{LHS} = \frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} \]

This gives:

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

**Step 3:** Simplify the numerator:

Using the identity \( a^2 - b^2 = (a - b)(a + b) \):

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

**Step 4:** Cancel out \( h \):

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

**Step 5:** Simplify further:

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

**Conclusion:**

Thus, we have proved that:

\[ \frac{f(x + h) - f(x)}{h} = \frac{1}{\sqrt{x + h} + \sqrt{x}} \]