Question

Determine numerically the first derivative of function tabulated below,

at the point x=4

x: 1 2 4 8 10

y=f(x):0 1 5 21 27

Answer 1

Newton,s dividend difference formula is

$f(x)=f(x_o)+(x-x_o)f(x_0,x_1)+(x-x_0)(x-x_1)f(x_0,x_1,x_2)+(x-x_2)(x-x_3)f(x_2,x_3)$

    $=f(1)_(x-1)\times 1+(x-1)(x-2)\times 3 +(x-4)(x-8)\times 12$

 

   $=x-1+3x^2-9x+6+12x^2-144x+384$

    $=15x^2-152x+389$

$f'(x)=30x-152\\ f'(4)=120-152=-32$