The second question in the image asks to "Show that $f'(1)$ is equal to $f(2)$ using the divided difference table."

To solve this, you will typically need a divided difference table. Here's how you can approach the problem:

Step 1: Understand the Divided Difference Table

The divided difference table is a recursive division of differences, used primarily in Newton's method of interpolation.

Step 2: Set Up the Table

You start with the values $x_i$ and $f(x_i)$ you have for your function $f(x)$. The table is typically set up as follows:

[ \begin{array}{c|c|c|c|c} x & f(x) & f[x_0, x_1] & f[x_0, x_1, x_2] & \dots \ \hline x_0 & f(x_0) & & & \ x_1 & f(x_1) & f[x_0, x_1] & & \ x_2 & f(x_2) & f[x_1, x_2] & f[x_0, x_1, x_2] & \ \vdots & \vdots & \vdots & \vdots & \ddots \ \end{array} ]

Step 3: Fill Out the Table

From your table, you would compute the first-order differences $f[x_0, x_1] = \frac{f(x_1) - f(x_0)}{x_1 - x_0}$, then second-order, and so forth.

Step 4: Show the Relationship

To prove that $f'(1) = f(2)$, you would typically show that the first derivative at $x = 1$ using the divided difference approach equals the function value at $x = 2$.

If you could provide the specific values of $x_i$ and $f(x_i)$, I can help you build the table step-by-step.

Would you like to proceed with this approach?