The forward difference formula can be expressed as :

f'(x0) = (1/h)[f(x0+h)-f(x0)]-(h/2)f''(x0)-(h^2/6)f'''(x0)+O(h^3).

Use extrapolation to dervive an O(h^3) formula for f'(x0)

To derive an O(h^3) formula for f'(x0) using extrapolation, we can use the idea of Richardson extrapolation. Richardson extrapolation is a technique to obtain higher-order approximations by combining lower-order approximations.

Here's how we can proceed:

1. Start with the given forward difference formula:
f'(x0) = (1/h)[f(x0+h)-f(x0)] - (h/2)f''(x0) - (h^2/6)f'''(x0) + O(h^3).

2. Let's denote the step size h as h1.

3. Now, consider using a smaller step size, denoted as h2, where h2 = h1/2.

4. Using the same forward difference formula, but with step size h2, we have:
f'(x0) = (1/h2)[f(x0+h2)-f(x0)] - (h2/2)f''(x0) - (h2^2/6)f'''(x0) + O(h2^3).

5. We can multiply the entire equation by (2^p) to eliminate denominators, where p represents the power of h in the desired error term (O(h^p)).
Multiplying the equation by 2^3, we get:
8f'(x0) = 4[f(x0+h2)-f(x0)] - h2[f''(x0)] - (h2^2/2)[f'''(x0)] + O(h2^3).

6. Similarly, we can use a smaller step size h3 = h2/2 and repeat the process:
8f'(x0) = 4[f(x0+h3)-f(x0)] - h3[f''(x0)] - (h3^2/2)[f'''(x0)] + O(h3^3).

7. Now, to combine these equations and eliminate the lower-order terms, we can do the following:
8f'(x0) = 4[f(x0+h2)-f(x0)] - h2[f''(x0)] - (h2^2/2)[f'''(x0)] + O(h2^3).
- f'(x0) = 0.5[f(x0+h1)-f(x0)] - (h1/2)[f''(x0)] - (h1^2/6)[f'''(x0)] + O(h1^3).
- 8f'(x0) = -4[f(x0+h3)-f(x0)] + h3[f''(x0)] + (h3^2/2)[f'''(x0)] - O(h3^3).

8. By adding these three equations, the lower-order terms cancel out, and we obtain the desired O(h^3) formula for f'(x0):
6f'(x0) = (9/2)[f(x0+h1)-f(x0)] - (h1/2)[f''(x0)] - (h1^2/3)[f'''(x0)] + O(h1^3).

Therefore, combining the forward difference formula with the process of Richardson extrapolation, we have derived an O(h^3) formula for f'(x0) as:

f'(x0) = (9/6)[f(x0+h1)-f(x0)] - (h1/6)[f''(x0)] - (h1^2/18)[f'''(x0)] + O(h1^3).

This formula provides a higher-order approximation for the derivative f'(x0) with an error term of O(h^3).