If i define a function
$$F(t)=f(x(t),y(t))$$
with
$$ x(t)=x_0+∆x * t + ∆^2x * t^2$$ $$y(t)=y_0+∆y * t + ∆^2y * t^2$$
$∆$ is the slope $dx/dt$ and $∆^2$ is the 2nd derivative $d^2x/dt^2$ of x (or y)
I know that for just F(t) = f(x(t)) the expansion around ($t$,$t_k$) looks like $$F(t) = f(x_0) + f_x\ {∆x}\ (t-t_k) + [f_x * ∆^2x + \frac{1}{2} f_{xx}(∆x)^2](t-t_k)^2 $$
How does the taylor expansion up to order 2 look like?
If you expand around $(x_0,y_0)$ (I.e. expand around $t=0$) then you would have $F(t)=f(x_0,y_0)+f_x(x_0,y_0)(\Delta xt+\Delta^2xt^2)+f_{xy}(x_0,y_0)(\Delta xt+\Delta^2xt^2)(\Delta yt+\Delta^2yt^2)$ (plus other terms in $y$ which I will leave to you, check out multi variable Taylor theorem on wikipedia.