I am trying to show that if $X:U\to\mathbb{R}^3$ is a parametrization of a coordinate patch on a refular surface $S$ and $F:U'\subset\mathbb{R}^2\to U$ such that $Y=X\circ F$ is a regular parametrization of an open subset of S, then if $q=F(q')$ and $p=X(q)$, then
$$Y_s(q')\times Y_t(q')=\det(dF_{q'})(X_u(q)\times X_v(q))$$
So far, I tried expanding the right hand side with $Y_{sx},Y_{sy},Y_{sz},Y_{tx},Y_{ty},Y_{tz}$ the respective coordinate functions for $Y_s$ and $Y_t$ and comparing it with the expansion on the left hand side. I couldn't see the connection though...
So I tried a purely algebraic approach using the fact that $$\frac{\partial Y}{\partial s}=\frac{\partial u}{\partial s}X_u+\frac{\partial v}{\partial s}X$$ and $$\frac{\partial Y}{\partial t}=\frac{\partial u}{\partial t}X_u+\frac{\partial v}{\partial t}X$$
where $$[dF]=\left[ \begin{array}{ccc} \frac{\partial u}{\partial s} & \frac{\partial u}{\partial t} \\ \frac{\partial v}{\partial s} & \frac{\partial v}{\partial t} \\ \end{array} \right].$$
Substituting this in and expanding gives:
\begin{align*} (u_sX_u+v_sX_v)\times(u_tX_u+v_tX_t)&=-((u_tX_u\times(u_sX_u+v_sX_v))+(v_tX_v\times(u_sX_u+v_sX_v)))\\ &=-((u_tX_u\times u_sX_u)+(u_tX_u\times v_sX_v)+(v_tX_v\times u_sX_u)+(v_tX_v\times v_sX_v))\\ &=-((u_tX_u\times v_sX_v)+(v_tX_v\times u_sX_u)) \end{align*}
But now I'm at a loss where to go from here.
$$ Y_s\times Y_t=(X\circ F)_s\times (X\circ F)_t = (dX\ dF \partial_s)\times (dX\ dF\partial_t)$$ $$= dX (u_s,v_s) \times dX(u_t,v_t)= (X_uu_s + X_vv_s)\times (X_uu_t+X_vv_t)= X_u\times X_v(u_sv_t-v_su_t) $$
Here ${\rm det}\ dF=u_sv_t-v_su_t$ So we complete the proof