The problem asks to find $f'(3)$ using the definition of the derivative, where the function is given as:

$f(x) = \sqrt{2 + 2x}$

The definition of the derivative is:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

We need to calculate $f'(3)$, so $a = 3$. Let’s break this down step by step.

Step 1: Plug $f(x)$ into the derivative formula

We know that $f(x) = \sqrt{2 + 2x}$. For $f'(3)$, we use:

$f'(3) = \lim_{h \to 0} \frac{f(3+h) - f(3)}{h}$

Step 2: Calculate $f(3)$

$f(3) = \sqrt{2 + 2(3)} = \sqrt{2 + 6} = \sqrt{8} = 2\sqrt{2}$

Step 3: Calculate $f(3+h)$

$f(3+h) = \sqrt{2 + 2(3+h)} = \sqrt{2 + 6 + 2h} = \sqrt{8 + 2h}$

Step 4: Apply to the derivative formula

Substitute into the derivative formula:

$f'(3) = \lim_{h \to 0} \frac{\sqrt{8 + 2h} - 2\sqrt{2}}{h}$

Step 5: Simplify the expression using the conjugate

To simplify, multiply the numerator and denominator by the conjugate of the numerator:

$\frac{\sqrt{8 + 2h} - 2\sqrt{2}}{h} \cdot \frac{\sqrt{8 + 2h} + 2\sqrt{2}}{\sqrt{8 + 2h} + 2\sqrt{2}} = \frac{(8 + 2h) - (8)}{h(\sqrt{8 + 2h} + 2\sqrt{2})}$

Simplifying the numerator:

$\frac{2h}{h(\sqrt{8 + 2h} + 2\sqrt{2})} = \frac{2}{\sqrt{8 + 2h} + 2\sqrt{2}}$

Step 6: Take the limit as $h \to 0$

As $h \to 0$, $\sqrt{8 + 2h} \to \sqrt{8} = 2\sqrt{2}$. Therefore:

$f'(3) = \frac{2}{2\sqrt{2} + 2\sqrt{2}} = \frac{2}{4\sqrt{2}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}$

Thus, the derivative at $x = 3$ is:

$f'(3) = \frac{\sqrt{2}}{4}$

Would you like further clarification or details on any steps?

Here are five related questions for practice:

1. How would you differentiate a function like $f(x) = \sqrt{3 + 5x}$ at $x = 2$?
2. Can you apply the quotient rule to find the derivative of a similar function?
3. How does the conjugate method simplify the calculation in the derivative definition?
4. What is the geometric interpretation of the derivative at a point like $x = 3$?
5. Can the derivative of a square root function be found using chain rule instead of the definition?

Tip: Always consider rationalizing expressions involving square roots when simplifying limits in derivatives.