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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.

If f(x) = √(2 + 2x), find f'(3) using the definition of derivative.

Solve for f'(3) where f(x) = √(2 + 2x) using the definition of derivative. This problem involves applying limits and simplifying expressions using the conjugate method.

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