I want to apply the Taylor for approximating function $$\begin{align}f(t,x,v)=f(t,x_{k},v_{k})+\frac{\partial f}{\partial x}(t,x_{k},v_{k}) (x-x_{k})+ \frac{\partial f}{\partial v }(t,x_{k},v_{k})(v-v_{k})\end{align}+O(h^{2})$$ where $t$ independent variable and two dependent variables $x$ and $v$.
Is that correct?
What you are really doing is this: $$ f(t,x,v) = f(t_k,x_k,v_k) + \frac{df}{dt}(t_k,x_k,v_k)(t-t_k) + \mathcal{O}(h^2) $$ i.e $$ f(t,x,v) = f(t_k,x_k,v_k) +\left [\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+ \frac{\partial f}{\partial v}\frac{\partial v}{\partial t} \right ](t_k,x_k,v_k) (t-t_k) + \mathcal{O}(h^2) $$