In the left below is the original equation for the Kalman Filter and the right is the Extended Kalman Filter. The difference is we replace A by the Jacobian matrix F, and C by the Jacobian matrix H.
So instead of using f(x), we replace it with f’(xi). Here is our visualization which the orange curve (computed from f’(xi)) in the top left is our new Gaussian Distribution which approximate the blue one (computed from f(x) ).