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The question asks to find the derivative of \( xf(x) \) with respect to \( x \) at \( x = 1 \), using the given table.

### Problem Setup:

We need to compute:
\[
\frac{d}{dx} \left[ x f(x) \right] \bigg|_{x=1}
\]
This requires the product rule for differentiation:
\[
\frac{d}{dx} \left[ x f(x) \right] = f(x) + x f'(x)
\]
Now, substitute \( x = 1 \) into this expression.

### Step-by-Step Solution:
1. **From the table:**
   - \( f(1) = 2 \)
   - \( f'(1) = 3 \)

2. **Apply the product rule:**
   \[
   \frac{d}{dx} \left[ x f(x) \right] = f(x) + x f'(x)
   \]

   At \( x = 1 \), this becomes:
   \[
   f(1) + 1 \cdot f'(1) = 2 + 1 \cdot 3 = 2 + 3 = 5
   \]

Thus, the derivative of \( xf(x) \) at \( x = 1 \) is **5**.

Would you like any further details or explanations? Here are 5 related questions to explore:

1. What is the product rule for differentiation in general?
2. How would you apply the quotient rule for a similar problem?
3. Can you find the second derivative of \( x f(x) \) at \( x = 1 \)?
4. What is the interpretation of \( f'(x) \) in this context?
5. How does changing the value of \( f'(1) \) affect the final result?

**Tip:** Always check the differentiation rule being applied (product, chain, quotient) based on the structure of the function.

Find the derivative of the function x*f(x) at x=1 using the table of values for f(x) and f'(x).

Learn how to find the derivative of x*f(x) at x=1 using the product rule for differentiation. This step-by-step solution explains how to apply the product rule to the function, given a table of values for f(x) and its derivative f'(x). Suitable for high school calculus students.

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