Consider $q_m$ to be the generalised coordinates where m = 1,2,3....
in Cartesian coordinates $x(q_m, t)$, we can express the velocity as $$\dot{x_{p,i}} = \dfrac{\partial x_{p,i}}{\partial q_m} \dot{q_m} + \dfrac{\partial x_{p,i}}{\partial t} dt$$
This seems fine...
If $x$ is considered to be a function of $q_m$ and $\dot{q_m}$, then how does the statement below hold water,
$$\dfrac{\partial{\dot x_{p,i}}}{\partial \dot{q_m}} = \dfrac{\partial x_{p,i}}{\partial q_m}$$
This is from the textbook of https://books.google.co.in/books?id=1gxk4oq9trYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false, page 78.
