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Question no 2 make the divided difference table 

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?

Learn how to use the divided difference table to prove that the first derivative at x = 1 equals the function value at x = 2. This problem involves advanced techniques in numerical analysis and interpolation methods.

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