1. **Problem Statement:** Find the first derivative $f'(x)$ and second derivative $f''(x)$ at $x=1$ using numerical differentiation from the given table: $$\begin{array}{c|cccccc} x & 0 & 1 & 2 & 3 & 4 & 5 \\ f(x) & 3 & 4 & 19 & 84 & 259 & 628 \\\end{array}$$ 2. **Method:** Use finite difference formulas for numerical differentiation. - First derivative at $x=1$ using the central difference formula: $$f'(1) \approx \frac{f(2) - f(0)}{2}$$ - Second derivative at $x=1$ using the central difference formula: $$f''(1) \approx f(2) - 2f(1) + f(0)$$ 3. **Calculate first derivative $f'(1)$:** $$f'(1) \approx \frac{19 - 3}{2} = \frac{16}{2} = 8$$ 4. **Calculate second derivative $f''(1)$:** $$f''(1) \approx 19 - 2(4) + 3 = 19 - 8 + 3 = 14$$ 5. **Interpretation:** - The first derivative $f'(1) = 8$ represents the approximate slope of the function at $x=1$. - The second derivative $f''(1) = 14$ indicates the curvature or concavity of the function at $x=1$. **Final answers:** $$f'(1) = 8$$ $$f''(1) = 14$$