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.