Question

Derive formula for f'''(a) by differentiating $f(x) = p_k(k) + f[x_0, ... ,x_k]\phi _k(x)$ three times. Choose k = 3 and set .

derive error term.

Thank you!

Answer 1

"Given function f(x) = p_k(k) + f(x_0 ...x_k] phi_k(x) let f(x) = p_k(x) + f(x_0 ... x_k, x] phi_k (x) [k varies from x] differentiate on both sides f^'(x) = p_k^'(x) + f^'(x_0, x_1, x_2, x_3, x) phi_k(x) + f(x_0, x_1, x_2, x_3 x)phi_k^'(x) Again differential on both sides f^""(x) = f_k^""(x) + f^""(x_0, x_1, x_2, x_3, x) phi_k(x) + f^'(x_0, x_1, x_2, x_3, x) phi^'(x) + F^'(x_0, x_1, x_2, x_3, x) phi_k^'(x) + f(x_0, x_1, x_2, x_3, x) phi_k^'(x) Differentiate 3rd time on both sides. f^""'(x) = f^""'(x_0, x_1, x_2, x_3, x) phi_k(x) + f^""(x_0, x_1, x_2, x_3, x_k) phi_k^'(x) + f^""(x_0, x_1, x_2, x_3, x) phi_k^'(x) + f^'(x_0, x_1, x_2, x_3, x) phi_k^""(x) + f^""(x_0, x_1, x_2, x_3, x) phi_k^'(x) + f^'(x_0, x_1, x_2, x_3, x) phi_k^'(x) + f^'(x_0, x_1, x_2, x_3, x) phi_k^""(x) + f(x_0, x_1, x_2, x_3, x) phi^'""(x) [u v formula for d^3/dx^3] \r\n simplifying f^""'(x) = f_k^""(x) + f^""'(x_0, x_1, "