We have the Taylor expansion for the position given by $$ x_{i+1} = x_i+hx^{(1)}_i+\frac{h^2}{2}x^{(2)}_i+O(h^3). $$ The corresponding expansion for the velocity is $$ v_{i+1} = v_i+hv^{(1)}_i+\frac{h^2}{2}v^{(2)}_i+O(h^3). $$ Via Newton's second law we have normally an analytical expression for the derivative of the velocity, namely $$ v^{(1)}_i = \frac{d^2 x}{dt^2}\vert_{i}= \frac{F(x_i,t_i)}{m}, $$