F[u_,v_]=u^2*v/(1+v)/2

D[F[u,v],{u,v}]

F10[u_,v_]=D[F[u,v],u]

F11[u_,v_]=D[F10[u,v],v]

Simplify[%7]

D[F[u,v],u,v]

G[u_,v_]=u^2*v/(1+v)/2 -(u-1)^2/2

G10[u_,v_]=D[G[u,v],u]

G11[u_,v_]=D[G10[u,v],v]

Simplify[%11]

D[G[u,v],u,v]