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To determine if \( f(x) \) is differentiable at \( x = 3 \), we need to check both the continuity and differentiability conditions.

### Step 1: Continuity at \( x = 3 \)
For \( f(x) \) to be differentiable at \( x = 3 \), it must first be continuous there. This requires:
\[
\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)
\]
From the piecewise function:
- For \( x < 3 \), \( f(x) = x^2 + x \)
- For \( x \geq 3 \), \( f(x) = 2x^2 - 5x + 9 \)

1. **Left-hand limit** (\( x \to 3^- \)):
   \[
   \lim_{x \to 3^-} f(x) = 3^2 + 3 = 9 + 3 = 12
   \]

2. **Right-hand limit** (\( x \to 3^+ \)):
   \[
   \lim_{x \to 3^+} f(x) = 2(3^2) - 5(3) + 9 = 2(9) - 15 + 9 = 18 - 15 + 9 = 12
   \]

3. **Value of the function at \( x = 3 \)**:
   \[
   f(3) = 2(3^2) - 5(3) + 9 = 12
   \]

Since \( \lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 12 \), the function is continuous at \( x = 3 \).

### Step 2: Differentiability at \( x = 3 \)
Next, we need to check the left-hand and right-hand derivatives at \( x = 3 \).

1. **Left-hand derivative** (\( x < 3 \)): 
   The derivative of \( f(x) = x^2 + x \) is:
   \[
   f'(x) = 2x + 1
   \]
   So, the left-hand derivative at \( x = 3 \) is:
   \[
   f'(3^-) = 2(3) + 1 = 6 + 1 = 7
   \]

2. **Right-hand derivative** (\( x \geq 3 \)): 
   The derivative of \( f(x) = 2x^2 - 5x + 9 \) is:
   \[
   f'(x) = 4x - 5
   \]
   So, the right-hand derivative at \( x = 3 \) is:
   \[
   f'(3^+) = 4(3) - 5 = 12 - 5 = 7
   \]

Since the left-hand and right-hand derivatives are equal, \( f'(3^-) = f'(3^+) = 7 \), the function is differentiable at \( x = 3 \), and the derivative is \( f'(3) = 7 \).

### Final Answer:
\[
f'(3) = 7
\]

Would you like more details or have any further questions? Here are five related questions for deeper understanding:

1. What does it mean for a function to be continuous at a point?
2. How do you compute the derivative of a piecewise function?
3. What is the geometric interpretation of differentiability at a point?
4. Can a function be continuous but not differentiable? If so, can you provide an example?
5. How would the process change if the function was not continuous at \( x = 3 \)?

**Tip:** When checking differentiability of piecewise functions, always verify continuity first, as differentiability implies continuity.

Determine if f(x) is differentiable at x = 3, and if it is, find the value of its derivative.

This problem explores the differentiability of a piecewise function at x = 3, ensuring both continuity and differentiability. Step-by-step solutions include limits, left-hand and right-hand derivatives, and verification of conditions.

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